patrick suppes philosophical essays - volume 3

PATRICK SUPPES: SCIENTIFIC PHILOSOPHER

Volume 3

SYNTHESE LIBRARY

STUDIES IN EPISTEMOLOGY,

LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE

Managing Editor:

JAAKKO HINTIKKA, Boston University

Editors:

DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley

THE0 A.F. KUIPERS, University of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California

JAN WOLENSKI, Jagiellonian University, Krakdw, Poland

VOLUME 235

PATRICK SUPPES: SCIENTIFIC PHILOSOPHER

Volume 3. Language, Logic, and Psychology

Edited by

PAUL HUMPHREYS Corcoran Department of Philosophy,

University of Virginia, Charlottesville, VA, U.S.A.

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TABLE OF CONTENTS

Volume 3: Language, Logic, and Psychology

PART VI: PHILOSOPHY OF LANGUAGE AND LOGIC

DAGFINN F@LLESDAL / Patrick Suppes' Contribution to the Philosophy of Language / Comments by Patrick Suppes 3

MICHAEL BOTTNER / Open Problems in Relational Grammar / Comments by Patrick Suppes

WILLIAM C. PURDY / A Variable-Free Logic for Anaphora / Comments by Patrick Suppes 41

J. MORAVCSIK / Is Snow White? / Comments by Patrick Suppes 71

PAUL WEINGARTNER I Can here Be Reasons for Putting Limitations on Classical Logic? 1 Comments by Patrick Suppes 89

JAAKKO HINTIKKA and ILPO HALONEN / Quantum Logic as a Logic on Identification / Comments by Patrick Suppes 125

MARIA LUISA DALLA CHIARA and ROBERTO GIUNTINI / Logic and Probability in Quantum Mechanics / Comments by Patrick Suppes 147

PART VII: LEARNING THEORY, ACTION THEORY, AND ROBOTICS

W. K. ESTES / From Stimulus-Sampling to Array-Similarity Theory / Comments by Patrick Suppes 171

RAIMO TUOMELA and GABRIEL SANDU / Action as Seeing to It that Something Is the Case / Comments by Patrick Suppes

COLLEEN CRANGLE / Command Satisfaction and the Acquisition of Habits / Comments by Patrick Suppes 223

. . . V l l l TABLE OF CONTENTS

PART VIII: GENERAL PHILOSOPHY OF SCIENCE

MARIA CARLA GALAVOTTI / Some Observations on Patrick Suppes' Philosophy of Science 1 Comments by Patrick Suppes 245

EPILOGUE

PATRICK SUPPES / Postscript

Bibliography of Patrick Suppes

Name Index

Subject Index

Table of Contents to Volumes 1 and 2

PART VI

PHILOSOPHY OF LANGUAGE AND LOGIC

DAGFINN F0LLESDAL

PATRICK SUPPES' CONTRIBUTION TO THE PHILOSOPHY OF

LANGUAGE

ABSTRACT. Patrick Suppes has been productive in a large number of areas, the study of language being only one of them. His work on language falls into four areas: Psychology of Language (particularly children's acquisition and use of language), Formal Linguistics, Language and Robots, and the Philosophy of Language. After a brief survey of Suppes' contributions to the first three areas, attention is focused on the fourth area, notably on Suppes' idea of congruence of meaning, which is a core idea in his Philosophy of Language. It is argued that the numerous congruence relations reflect the various regularities that enable us to establish, learn and use a language.

Since I first met Pat, in 1964,I have greatly appreciated him as a friend and colleague, and we have taught several courses together. However, I have to make a confession: although Pat has been such a good friend and colleague over so many years, I have never come around to reading more than a fraction of his work. This does not mean that I have read very little. I must make a new confession: not even Pat's bibliography have I read in full. I have, however, done some counting, and I have found that so far he has published more than 300 articles, written 20-30 books and edited a similar number of volumes. I am here not counting his numerous mathematics text books and popular works.

These articles and books fall within a variety of fields, and within each of them Pat has made important contributions. When Pat was appointed to Stanford in 1950, as a very young man, he came to the Philosophy Department. But he was very quickly also made a Professor of Statistics and of Education. However, when we look through his bibliography and curriculum vitae, a curious oddity about chronology emerges: in 1972 Pat won the Distinguished Scientific Contribution Award of the American Psychological Association. This award is given for the best research in psychology that year. However, at that time Pat was not a member of the Psychology Department, so the award went to the Philosophy Department. This may explain why the next year the

P. Hurnphreys (ed.), Patrick Suppes: Scient$c Philosopher, Vol. 3, 3-1 8. @ 1994 Kluwer Acudemic Publishers. Printed in the Netherlands.

4 DAGFINN F0LLESDAL

Psychology Department finally came around to making him a member of their department.

In the present three volumes one can read a large number of papers about Pat's work in various areas. However, even this is not enough to cover all the areas in which he has worked. I will touch very briefly in my contribution on one in which he is working very actively now, namely robots. When we look at Pat's work within each of these many fields in which he has worked, and also when we survey his work as a whole, there are two things that strike one about Pat. One is his openness, he is always very eager to hear about new developments, to learn something new, and he is in fact particularly interested in perspectives different from his own. This is, of course, a good thing, but it can be overdone. One can become superficial and uncritical. Pat fortunately compensates for his openness with a second feature which is equally characteristic of him. As soon as he recognizes what is going on from that very different perspective, he starts asking: "What is the evidence?" This combination of enthusiastic openness and critical questions about evidence is typical of Pat.

I am going to discuss just one tiny part of what Pat has done, namely his work in the philosophy of language. Here we are in the fortunate situation that his book Language for Humans and Robots (Suppes, 1992) is just out. It collects some of Pat's main papers on language. However, even that volume is, of course, not complete. As I write, it was published two months ago, but there already have been other papers appearing that came too late to be included in the volume. I will not be able to go into all areas covered in this volume, but I will begin by giving a brief survey of the main contributions that Pat has made to the study of language, and then focus on one of these areas where his contributions have been particularly pertinent to the philosophy of language. Of course there are other areas that are pertinent to philosophy as well, but this one is very close to the main tradition in the philosophy of language in our century.

Trying to survey Pat's work on language, Pat himself has suggested in this book that his work may be divided into four areas, viz.: first the psychology of language, particularly children's acquisition of language and children's use of language. Secondly, there is work of a formal linguistic kind. Thirdly, there is work on language and robots. And fourthly, there is the area which I want to concentrate on, namely Pat's ideas on congruence of meaning. The latter has been an important focus

SUPPES' CONTRIBUTION TO PHILOSOPHY OF LANGUAGE 5

of his work in the philosophy of language, on which he has published a large number of papers.

Before I discuss congruence of meaning I will survey briefly what Pat has been doing in these other fields. First his approach to children's language. Here one typical trait of Pat's approach is illustrated, his regard for empirical data. He has carried out some experiments with children, but most of Pat's work on children's languages consists in close analysis of actual verbal communication between children in non-experimental settings. Pat has found through these studies certain features of children's use of language from which both linguists and philosophers can learn a good deal. Particularly interesting from a philosophical point of view is that semantics seem to be much more important for acquisition of language than the syntactical features of language. This is well documented through these empirical studies.

We then have the work in the second group, work of a more formal kind relating to logic and linguistics. Here one of Pat's main contributions has been to build up an alternative to the use of quantifiers and variables. In fact, Pat has shown, and many would agree with him, that quantificational structure, which has been a cornerstone of modern logic since Frege, is not very suitable for capturing some of the natural constructions in ordinary language. Pat has indicated how one can build variable-free semantics. This relates to work that has been done by Tars- ki and other logicians earlier, but Pat has applied these ideas to natural languages and shown how his particular variable-free approach gives a very natural way of handling certain patterns in natural language.

Then we get to Pat's work on language and robots. Here Pat's contributions fall mainly in two areas within that major field. The first has to do with developing robots that are able to take verbal instructions. That is not so easy, because the language of action - we want the robots to perform certain actions - turns out to be relatively complicated. It requires some thought and ingenuity to find a good way of programming a computer so that it can respond to verbal commands. The second area of Pat's work on language for computers is even more challenging and interesting. Pat has written several papers on the problem of constructing robots that actually pick up language. This area of language acquisition by robots is an area in which Pat is now working actively.

Finally, there is the area that I mentioned at the beginning and which I wanted to come back to: Pat's work on sameness of meaning. Pat here approaches the old philosophical problem of sameness of meaning

6 DAGFINN F0LLESDAL

in a new way. This is a basic problem in philosophy. Frege, in his unpublished writings, struggled for years trying to find identity criteria for meaning, that is, he wanted to determine under what conditions two expressions express the same meaning. This is still a challenge for his followers. Alonzo Church, in particular, has proposed several criteria of sameness of meaning, some of which have been found to lead to contradictions (Church, 1946, 195 1, 1973, 1974). Before Frege, this problem was discussed by Bolzano, and we find glimpses of it even earlier. It is connected with the problem of how communication via language can be possible. Both Bolzano and Frege argued that we cannot communicate unless there are intersubjective meanings that can be brought across from one person to another. They both held that there is a realm of objective meanings that in a platonistic way are waiting to be grasped and expressed. We are supposed to be capable of grasping these meanings. In order to communicate what we grasp, we have to learn a language, so that we can express these meanings.

This is a view that many of us find unsatisfactory. It seems too mysterious. Pat, coming to the problem from an empirical point of view, would certainly want to look much more closely into how it happens that we acquire a language at all. The process does not seem to be much like the one envisaged by Frege. On this point Pat is close to my teacher Quine, who also has been very critical towards that view; in fact, there are many similarities between Pat and Quine when it comes to dealing with this problem. I concur with both of them on this point, and in what follows I will be siding with them rather than coming up with criticism against their view. Yet their view is certainly still a minority view. If you look at the work that is being done in the philosophy of language, you will find that there is a majority who, for some reason or other, take a Fregean kind of view fairly much for granted. They start out from a semantic system where all these items are given, propositions and concepts and so on, and no serious question is raised concerning what they are and how we get at them.

Pat, in line with his pluralism that I mentioned in my introduction, argues that the whole traditional way of looking at sameness of meaning is wrong. Maybe there is not one notion of meaning that we are out to capture. Maybe we should start with an open mind and ask whether there might not be, for different purposes, quite different notions that we would be interested in capturing. Pat proposes that we should compare what we are after in this area with what one is after in geometry where


one talks about congruence between geometrical figures. In geometry we have an early idea in Euclid, who held that figures are congruent if they have the same shape and the same size. This idea was followed up by Hilbert and we now have a rather sharp definition of what is meant by 'congruence'. We can strengthen or weaken the notion of 'congruence' in various directions. Euclid does not do this, but Pat points out that we could add, for example, orientation as a factor, and say that in order to be congruent two figures have to have the same orientation. We do not find such a notion in Euclid because his geometry has no preferred direction - the notion of a horizontal line is, for example, foreign to Euclid's geometry.

We could also weaken the criterion for congruence. We could say that shape is all that matters, but not size. And in affine geometry all triangles are congruent. They can be mapped onto one another. It does not matter what size the angles are or how big the triangles are, etc. As the reader probably knows, Felix Klein in 1872, at the age of 23, proposed the Erlanger program where he suggested that in geometry we should concentrate on two related notions, that of transformation, and that of the features that remain invariant under a given transformation. These are, then, the congruent ones. This opens up a new and interesting field of research. We study all kinds of transformations and the congruence relations they give rise to. We can have very weak such relations, even weaker than the ones we find in geometry; we could have just a one-to- one mapping which preserves identity and nothing else. Just as we have a spectrum of possibilities in geometry, Pat has proposed that we could explore a similar spectrum of mappings and invariances when we study language. For some purposes, some of these mappings might be good candidates for meaning. For other purposes other mappings might be better candidates.

Now I will say a little bit about the multitude of features that Pat investigates. One kind of congruence that he discusses is congruence between equally probable utterances. We now look not at sentences but at utterances. That is the individual use of a particular sentence in a certain context. In a given context two utterances might be equally probable. A person might say, to use Pat's example, 'It will rain tomorrow', and also, 'There will be gusty winds in the afternoon'. From an intuitive point of view we do not think that they mean the same but there might be purposes for which we might want to treat them in the

8 DAGFINN FQILLESDAL

same way. And they would be similar in that they might be regarded as equally probable by the person who utters them.

Next we have the notion of extensional congruence, which is simply the familiar notion of extensional equivalence: two utterances are extensionally congruent if they are both true, or if they are both false. So we have two main groups of utterances under that congruence relation, the true ones and the false ones. Then we have intensional congruence, which can be defined in various ways. If we assume standard classical logic, we could say that two utterances are intensionally congruent if they are logically equivalent according to standard classical logic. Here we could obviously have variants. We could use intuitionistic logic or some other nonclassical logics, and say that utterances are intensionally congruent if they can be proved equivalent through those means.

Pat also discusses a notion that lies between the latter two and which he calls M-congruence: two utterances are M-congruent if they are congruent with respect to a fixed class M of models closed under isomorphism. M may, for example, be the set of models in which the laws of Euclidean geometry, or arithmetic, or of classical physics, hold. In this way we can develop a large number of different equivalence notions, different notions of congruence. As we all known, especially some variant of intensional congruence has been regarded by many as a good candidate for sameness of meaning. Frege, in his unpublished writings, discussed various criteria for sameness of meaning, and this is one that he explored. There is a problem with it that Camap, in particular, was worried about, namely that all logically true sentences come out equivalent, and likewise all the logically false ones. This led Carnap to introduce his notion of intensional isomorphism. This may be in classical writings the one notion that comes closest to what Pat suggests, but of course it comes close only to one of Pat's notions. There is a whole spectrum of different notions that Pat spreads out for us. One of Pat's key points is that it is useful to be open to all these different kinds of notions, and not get stuck with just one of them.

However, Pat also goes into some other notions, particularly connected with propositional attitudes. There are puzzles connected with belief that we can throw some light on by these notions. Pat introduces a notion of belief congruence, saying that two utterances will be belief congruent for a person if we can substitute the one for the other in all belief statements of that person, salva veritate.


This is reminiscent of an idea of Benson Mates (Mates, 1950). Mates proposed as a condition of adequacy for definitions of 'synonymy' that two expressions are synonymous in a language L if and only if they may be interchanged in each sentence of E without altering the truth value of that sentences. This might seem a good proposal. However, the criterion does not enable us to tell which expressions are synonymous, and Mates does not claim that it does. The problem is that if the criterion is going to work, then, as Mates points out, the sentences in one's language would have to include sentences involving necessity, belief and so on, and one would wonder under what conditions those sentences are true, and under what conditions they are false. This is the observation that Quine makes in 'Two Dogmas' (Quine, 195 1); that attempts to define or elucidate synonymy tend to move in circles: in order to define synonymy one appeals to analyticity, necessity or belief, and in order to define these notions one makes use of synonymy again.

We might wonder - and this is the first point of mine that might seem critical - we might wonder whether Pat falls into this trap, whether he is defining synonymy, according to that particular alternative - belief congruence, in a circular way. I think that Pat is actually avoiding the trap because Pat, unlike so many others who have worked in this field, does not think that any of hls congruence relations enables us to sort the meaning aspect from the factual aspect. The various notions of congruence group together utterances that for some purpose are to be treated in the same way. Pat seems to agree with Quine that the whole attempt to draw a line between meaning on the one hand, and theory on the other, is arbitrary. There does not seem to be any basis for thinking that there is such a line to be drawn. For Pat, as for Quine, each speaker of a language will speak his or her idiolect. This is what one should expect if there is no line to be drawn between meaning and theory, but this ubiquity of idiolects has been rejected by many, e.g., by Kripke, who in 'A Puzzle about Belief' (Kripke, 1976) repeatedly contrasts dejining with factual beliefs (for example on p. 245).

Attempts to define synonymy and analyticity have as a main aim the drawing of such a line without question-begging assumptions. I find that for the purposes Pat has in mind, there is no question-begging in his procedure. He just proposes a number of ways of trying to connect notions of sameness of meaning with some other notions, and depending on how one connects it with other notions, one will get quite different notions of sameness of meaning.

10 DAGFINN FOLLESDAL

Pat also suggests that we could develop a theory of coherent beliefs. In his book he begins to develop such a theory. To get started, he assumes that a person's beliefs are coherent. And this raises problems: what does it mean for a person's beliefs to be coherent? It means that there should be no obvious inconsistencies in that person's set of beliefs: no straight contradictions, and no inconsistencies that can very easily be spotted. Pat does not go into details about how we are going to delimit that notion of coherence. He suggests that we can make simple inferences, but not complicated ones (p. 38). There is here a tie between Pat's work and Jaakko Hintikka's work in Knowledge and Belief (Hintlkka, 1964), where Hintikka, discussing similar problems, develops a notion of surface tautology. Hintikka has followed this work up later, trying to delimit, in a certain way, what kind of inference we can make use of and what inferences we should not make use of.

As Pat and Hintikka both point out, we certainly do not want to require closure under logical consequence. That would be a way of requiring perfect rationality in the theoretical realm. There is no person ever who has known all the logical consequences of his or her views. I think that Pat's and Hintikka's approaches actually supplement one another quite nicely.

Pat goes on and discusses some further notions of congruence. One is orthographical congruence, where we require a mapping of expressions onto one another, letter by letter. This is of course a very strong congruence relation. It is in a way pretty trivial, but as Pat points out, what is interesting about it, is that we can consider different ways in which we establish the congruence. This then leads Pat on to the idea of processing as being an important key to what congruence relations we would be particularly interested in.

There is a further source of observation that we could bring in, Pat notes, namely the responses of people have to utterances. We could, for example, look at response time. We could say that a person, say John, responds more quickly in judging the truth of one sentence than in judging the truth of another, or that he responds in exactly the same time, when we ask him, for example, "Do you believe that Newton was a bachelor?" and "Do you believe that Newton was an unmarried man?" Pat proposes that one congruence relation, that he calls latency congruence, would be one where we group together those utterances which have the same response time plus/minus a certain narrow margin,

SUPPES' CONTRIBUTION TO PHILOSOPHY OF LANGUAGE 1 1

for example, he suggests, ten milliseconds. They would all be said to have the same response time.

This, of course, would group together sentences that do not have much to do with one another. They just have the same response time. But if we combine this and belief congruence, then we could get something that might be a little more interesting, because the similarity in response time would indicate that maybe the processing might be similar in complexity, and the belief congruence tells us that the person believes both utterances. And not only that, but the expressions can be substituted for one another, in all belief contexts of that person. Pat then suggests that we could go on further along these lines and look at other features of the responses as well, and thereby get a refinement of the responses, and thereby, perhaps, get a better clue to the actual processing that is taking place through empirical studies of these various aspects of response. For example, we could study the strength of belief and generally what Pat calls a whole profile of assent. That is simply a program to look at all the features of these responses that could give us certain clues to the way in which things are being processed.

There are still further features of language use that Pat wants to go into. Again this illustrates his pluralism. He thinks that philosophers and linguists have focused too much on just some particular feature, and this has often led them to think that this is the one notion that we are after. Pat argues that prosodic features of many different kinds have to get into the picture, for example the emphasis "I love you" as opposed to "I love you". There are very many such prosodic features, which might give rise to various congruence relations. Pat lists several of them, and of course one could keep on listing if one just looks upon nuances in language use that make a difference, such as the duration of a sound, rhythm, or intensity. One could add here: tone. Norwegian, for example, is a tonal language, where a difference in tone can make a radical semantic difference. The same is the case in Chinese and some African languages. There would also be other verbal and nonverbal cues to the purposes of the person who is talking, and the opinions that that person might have. All of these features come in as a kind of background context that might be important for other responses. So what Pat really wants to do, is to construct a large number of congruence relations that take into account this wide variety of factors that can be observed, and see whether some of them might be useful for some purposes and others for others. There is no natural stopping place, and not one concept of

12 DAGFINN F0LLESDAL

meaning, but many. Looking upon all of these congruence relations together, we might get a much better clue to many of the problems that have been bothering philosophers of language.

The congruence relations I have talked about so far are relations having to do with utterance of expressions as wholes. Pat also develops in other articles congruence relations for structures, where we look not just at the final results, but on the tree structure through which these are being generated. One can try to map these trees on to one another, and put certain restrictions on the mapping, and thereby again get different congruence classes depending upon what restrictions we have. There are many more points that Pat goes into in connection with translation. There is one point, however, which I should definitely bring up, since it is important for many issues in the philosophy of language. It has to do with the discussion concerning the primacy of literal meaning.

In order to communicate, it has been argued, there has to be some expressions that have a certain meaning regardless of what the speaker might want to communicate. On the other hand, there are philosophers like Grice, with whom Pat sides, who say that the speaker's intention is very important for understanding the communicative situation. Many critics have argued that Grice's view is like the Humpty Dumpty view on meaning. We all agree with Alice that we cannot make words mean what we want them to mean. On the other hand, as Pat observes, if we think about how a language gets started, it is very hard to uphold the literal meaning view. Probably, when language got started, people did make some tentative attempts to get other people to do things they wanted them to do, and maybe to gradually communicate this to them. And through those intentions that they had, and the actions they carried out in order to communicate them, they might gradually succeed in getting this across in certain ways, and those ways of achieving this might then gradually become entrenched. There might be a kind of sedimentation that started to take place, and in our language, as it is now, clearly the sedimentation that we have in literal meaning does play a very important role. Our ability to communicate with the aid of a well-established language clearly depends a lot on literal meaning.

However, and this is something that Pat emphasizes, there is a cre- ative aspect of language even now, when we are using an established language. Even now we have to assume an interplay between speakers' intention and literal meaning in order to account for what is going on. However, there is one argument of Pat's for this that I would like to


take issue with. I have to try to be critical at some point, and this is one of the few points where I find something to criticize. This is Pat's argument against Biro, who has challenged Grice. What Pat objects to is Biro's claim that he will not make any hypothesis about the speaker's intention. According to Biro, when we hear something, listen to what is being said, we can find out what is meant, and thereby understand what was to be communicated without bringing in the speaker's intention.

Pat objects to this that if we just hear a stream of sounds, pressure in the ear, distributed in certain ways, we have no basis for noticing the relevant features of what goes on. We know that are no observations without theory, and clearly, therefore, to understand what another person is trying to communicate, we need some hypothesis about that person's intentions. Otherwise we would not get started. Pat says that Biro's claim to manage without hypotheses is an appealing positivistic line, viz. that the data relevant to a theory or hypothesis must be known independently of the hypothesis. This is a major flaw that he finds in Biro, that Biro seems to be unfamiliar with the philosophy of science, that the data are known only through use of hypothesis.

However, if one looks more closely at what Biro says, then Biro does not say that we manage without hypotheses. What Biro says is hat some of these hypotheses are hypo.theses about literal meaning, but that none of them are hypotheses about the speaker's intention. So I think the difference here comes down to the fact that Biro thinks that in order to grasp what is happening in this stream of sounds, we would need hypotheses about the language being spoken and the phonetic laws of the language, the literal meaning of that language, etc., but we do not need in addition hypotheses about the speaker's intention. So I think that Pat here is unjust to Biro in saying that Biro thinks that we can do completely without hypotheses. I think the issue is really: can we manage with just those hypotheses that Biro accepts, or do we need some about speaker's meaning as well?

I agree with Pat that there is no way of getting the whole enterprise started by just talking about literal meaning. However, I think we have to bring in both, not just hypotheses about the kind of activity that is going on when people are uttering words, but also hypotheses about literal meaning. I think we should look at literal meaning as the sedimentation that has been brought about by the members of the community, through past use of language. In the beginning I assume that the sedimentations were very meager. People just tried to get others to understand what

14 DAGFINN FGLLESDAL

they wanted. However, gradually, through generations, our languages have become very highly developed, and when we learn a language nowadays, we learn a lot about literal meaning, about established features of the language, both phonetic, syntactic and semantic. However, we also need to make assumptions about what the speaker intends with his or her activity - but not which intention, or meaning, he has in his head and is seeking to express - and we have to make a lot of assumptions about the whole context in which this happens, about the speaker's background, etc.

To conclude then, I think that the notion of literal meaning that we need, would be not one, but several variants of the kinds of congruence that Pat has been discussing. Growing up in a linguistic community, we start noticing certain regularities, certain congruence relations, that we use as a basis for extrapolations, for hypotheses about what goes on, what to expect and what to do.

This is what I regard as the main import of Pat's theory of congruence of meaning: there is a large number of congruence relations that reflect the regularities that enable us to establish, learn and use language. A challenge and at the same time a guideline for further research is to study these processes in order to determine which congruence relations are important for the various aspects of these linguistic activities and what role they play.

Department of Philosophy, Stanford University, Stanford, CA 94305, U.S.A. and

Department of Philosophy, University of Oslo, l? b. I024 Blindern, 0315 Oslo, Norway

REFERENCES

Church, Alonzo: 1946, Abstract of 'A Formulation of the Logic of Sense and Denota- tion', Journal of Symbolic Logic, 11, 3 1.

Church, Alonzo: 195 1, 'A Formulation of the Logic of Sense and Denotation', in: Paul Henle, H. M. Kallen, and S. K. Langer (Eds.), Structure, Method and Meaning: Essays in Honor of H. M. ShefSer, New York: Liberal Arts Press, pp. 3-24.


Church, Alonzo: 1973, 'Outline of a Revised Formulation of the Logic of Sense and Denotation', Part I, Nocis, 7,24-33.

Church, Alonzo: 1974, 'Outline of a Revised Formulation of the Logic of Sense and Denotation', Part 11, Nods, 8, 135- 156.

Hintikka, Jaakko: 1964, Knowledge and Beliefi An Introduction to the Logic of the Two Notions, Ithaca: Cornell University Press.

Kripke, Saul A.: 1976, 'A Puzzle about Belief', in: A. Margalit (Ed.), Meaning and Use, Dordrecht: Reidel, pp. 239-283.

Mates, Benson: 1952, 'Synonymity', University of California Publications in Philoso- phy, 25 (1950), 201-226. Reprinted in: Leonard Linsky (Ed.), Semantics and the Philosophy of language, Urbana: The University of Illinois Press (1952).

Quine, W. V. 0 . : 1951, 'Two Dogmas of Empiricism', Philosophical Review, 60,2043. Reprinted in numerous collections.

Suppes, Patrick: 1992, Language for Humans and Robots, Oxford: Blackwell.

COMMENTS BY PATRICK SUPPES

Dagfinn has been my part-time colleague at Stanford for almost 30 years, and during that time we have taught a number of seminars together, especially on perception. In the seminars on perception we typically divide up the work, with Dagfinn talking about Husserl and phenomenology, and with me taking responsibility for talking about relevant experimental work by psychologists. I have finally begun to have a serious understanding of Husserl through Dagfinn's diligent instruction. We often disagree, but I know of no one with whom I have discussed philosophy at great length, with whom it is easier to disagree in a completely congenial way.

Dagfinn has given a very sympathetic account of my theory of congruence of meaning. What I want to do is comment on certain points that he covers by amplifying either what he says or what I have said in the past. As he notes at the beginning, a view like mine is like Quine's, very much in the minority among philosophers of language. Much further work will be needed to bring the ideas into the mainstream.

Propositions and Synonymy. The continued dedication of philosophers to the idea that a sentence or utterance expresses a proposition reminds me of the attitude towards geometry until the end of the eigh- teenth century. There was one unique Euclidean geometry, which according to Kant was not even a matter of experience but a primary basis of experience. In any case, the theorems of geometry were

16 DAGFINN F0LLESDAL

those of only one geometry, the geometry of Euclid. In a similar vein, just as there was then only one way to think about two triangles being congruent - namely the Euclidean way - so there was just one way of thinking about two words being congruent in meaning.

It took a major effort to dislodge Euclidean geometry from its favored status. In spite of the skeptical views expressed repeatedly by Quine and often by others like Tarski, it is clear that a similar major effort will be required to dislodge the favored status of propositions and synonymy. On the other hand, there are reasons for thinking that the battle may not be so difficult. It is simply that the achievements that flow from thinking about propositions and the synonymy of words are not at all comparable to the glorious history of Euclidean geometry.

It is convenient to talk about the proposition expressed by a sentence, but it is really no more than that. Moreover, asked to explain what is that proposition, there is a notable lack of clarity and confusion. There is, of course, also this mistaken tradition in philosophy of wanting to talk about the logical form of sentences, as if that were a concept that had any deep significance, Certainly if there is a logical form, it is not that of first-order logic, nor of any other logic artificially created, but is the kind of form given by our mental processing. But even here, I would conjecture that that form will vary with person and culture. This likely variation takes me to my next comment.

Primacy of Utterer's Meaning. I certainly agree with Dagfinn's remark about the importance of sedimentation in language and the creation of a body of literal meaning. This literal meaning is to be found prominent- ly in scientific and philosophical texts, careful classroom discussions, proper courtrooms well run by a learned judge, some corporate board- rooms, etc. Where literal meaning is not dominant is in the casual give-and-take of ordinary talk - sentences are left incomplete, words are muttered, but looks and gestures carry the day of communicating intention.

The emphasis on the word primacy simply means that (i) the claim is being made that it is these informal communications that came first and (ii) they probably dominate most of the spoken language, most of the time, anywhere in the world. This is a large claim and maybe it is too radical, but my own attempts even to transcribe what is thought to be the learned discussion of colleagues at conferences is testimony enough to the incoherence of the literal meaning taken word-by-word as


transcribed from spoken speech. The nuances, the gestures, the looks, the pointing, all play an essential part in understanding the meaning of even systematic speech not to mention our casual grunts and groans to one another.

I sometimes feel that philosophers take as the model of spoken language a philosopher reading a paper before an audience - note that I said reading. In other words, the written language is really in their hearts the primary language, and comes first in terms of importance and meaning, but clearly this is nonsense, and maybe I am exaggerating in attributing this view to many philosophers. In any case, the squawks and shouts of workers together, the meager verbal instruction given apprentices in most disciplines, richly supplemented by showing how things are done, the chaotic and foreshortened nature of what speech there is - all for these features are continual and natural. I claim that the observing of context and inference about intention are essential for the proper interpretation of meaning in the majority of actual speech use.

Differences from Geometry. Taking up the point I began with earlier, about how difficult it is to dislodge a received view, I want to point out another reason why the classical theory of meaning is less thoroughly entrenched than was Euclidean geometry. It is clear that the passage from the perceptual geometry of images to our working version of Euclidean geometry for the objects of experience, is itself a very complicated binocular construction. There is nothing the least bit simple about the way in which we move from retinal images, continually subject to eye and head movements, to the construction of physical objects. But this is a construction we learn to do marvelously well, and with almost uniform agreement we can describe what we see in essentially Euclidean terms.

In the case of language it is not at all the same kind of construction. Euclidean geometry is driven by the physical nature of our world to a very fine approximation. Our construction of language can be much more arbitrary and idiosyncratic. It is only philosophers who search for some uniformity like that of geometry. No doubt this is one of the reasons for the great appeal of a concept like that of logical form. But there is no reason that language need be the same across subject matters or across cultures. Even translating the tense systems of English into German or vice versa, two languages that are linguistically close, show how easy it is for languages to have different expressive features,

18 DAGFINN FCZILLESDAL

particularly among those most frequently used. Moreover, we tolerate very nicely that exactly the same word in a given language can be used in radically different ways even in systematic contexts. A good example is the use of the wordfield in mathematics and in agriculture, or the use of the word energy in classical mechanics and in art criticism.

I certainly do not deny that experience in general keeps us focused in the development of meaning, but we do not have anything like the concrete and sharp experience of geometry to force us to have uniform boundaries. Something like the concept of natural kinds is a very poor relative indeed compared to the wonderfully precise and developed concepts of Euclidean geometry.

We have, of course, come in geometry to find a wide range of uses for other than Euclidean concepts. It should be all that much easier in the case of language for there is no tradition represented by anything like the use for so long of a single geometry. Undoubtedly one reason for the search for the concept of synonymy or the concept of proposition is that ineluctable desire to have a rock-hard foundation for our intellectual discourse. But this is sadly mistaken. The only two disciplines that have ever really seriously sought such a foundation are mathematics and philosophy. It is now generally agreed in the case of mathematics that the main body of mathematics is much more secure than the foundations, paradoxical though that may seem. And the efforts of philosophers, contentious as they are, has never yielded anything like a firm agreement on foundations, and in my view never will. We are not, I hope, moving towards such agreement even at a slow pace, but are realizing how diverse and wonderful the many uses of language are, and with these many uses go a diverse collection of concepts of meaning.

MICHAEL BOTTNER

OPEN PROBLEMS IN RELATIONAL GRAMMAR

ABSTRACT. The notion of a relational grammar was proposed by Suppes for the semantics of natural language. A relational grammar is a context-free grammar with a relation algebraic semantics. While this concept provides elegant solutions for the semantic analysis of a variety of constructions, certain problems arise with its application to certain other constructions. In this paper, the focus is on proper nouns, pronouns, number, and verb phrases.

The notion of relational grammar (henceforth: RG) was proposed in Suppes (1976) for the semantics of natural language and has been expanded on in subsequent articles. RG was introduced for the purpose of giving a syntactic and semantic analysis of natural language.

The most attractive feature of RG is its closeness to the structure of natural language. The meaning of an expression can be read off from its linguistic structure: the sentence

All men like some women

gets the semantic structure

where [a] stands for the denotation of a and f is a semantic function. No use is made of variables for individuals as in quantifier logic. RG is first order, i.e. [a] is either a subset of some domain D or a binary relation on D. It thus avoids the procedural complexities involved in the computation of sets higher up in the set-theoretical hierarchy. The operations employed in RG semantics are the familiar set-theoretical operations with well-known properties that enhance the transparency and expedience of logical inferences. RG does not require any level of semantic representation, but denotations are mounted directly to natural language structures.

For all these reasons I believe that G is an interesting and attractive alternative to current theories of linguistic semantics. The purpose of

P. Humphreys (ed.), Patrick Suppes: Scientific Philosophel; Vol. 3, 19-39. @ 1994 Kluwer Academic Publishers. Printed in the Netherlunds.

20 MICHAEL BOTTNER

this contribution is to point out some problems and difficulties arising from the application of RG to natural language. The organization of this paper is as follows: in Section 1 I shall give a short introduction to RG. In Section 2 the problem of the missing category NP is addressed. In Section 3 the problem of anaphoric pronouns is taken up. In Section 4 the problem of verbal semantics is addressed.

1. RELATIONAL GRAMMAR

An RG is a context free grammar G with semantic functions f attached to each production rule

A + A1 +. . . +A, [A] = f ( [Ai l ] , . . - 7 [Aik])

where { i 1, . . . , i k } & (1, . . . , n} and denotations are restricted to elements of an extended relation algebra over the domain D as some model structure for G. The purpose of the functions f is to provide a compositional semantics for G. For details see Suppes (1973a).

An extended relation algebra over D is any collection of subsets of D and binary relations on D that is closed with respect to the operations of

(1) union: X u Y complementation: - X

conversion of a binary relation: R composition of binary relations R and S: R; S (upper) image of set A under relation R: R"A

where X, Y stand for either subsets of D or binary relations on D. An extended relation algebra is thus a subset of the following set-theoretical hierarchy

where F ( D ) denotes the power set of D. An equational characterization of extended relation algebras can be found in the notion of a Boolean module (Brink, 1981) which is a Boolean algebra together with a set of operators forming a relation algebra in the sense of Tarski (1941).

The operations (1) give rise to numerous other operations that can be defined in terms of these. It is well known that we can define

OPEN PROBLEMS IN RELATIONAL GRAMMAR 21

Boolean product, Boolean difference, and symmetric difference by Boolean union and complementation. Less known may be the following operations:

f was introduced in Riguet (1 948) under the term Coupe de deuxi2me esp2ce de R suivant A; f 2 was known already to de Morgan and Peirce under the term of Ordinary Involution, as pointed out in Brink (1 978); f 3 was also known to de Morgan and Peirce under the term Backward Involution, cf. Brink (1 978); f 4 was introduced by Suppes and Zanotti (1977) as the Lower Image RccA of A under R; f s was introduced in Suppes and Macken (1978) to compute the initial segment IS (A, R, c) of a set A with respect to an ordering relation R on A and an element c of A.

Notice that there are set-theoretical operations that cannot be defined in terms of (1) like, for example, domain restriction, Cartesian product, or cylindrification. One might suppose that they do not play an important role at all in the semantics of natural language. For the operation of Cartesian product, this was indeed pointed out in Keenan (1983, p. 71). On the other hand, there may be some use for cylindrification, cf. Woods (1991), and for restriction in a procedural semantics, cf. Bottner (1 992b).

RG was meant to serve as a tool for the analysis of everyday natural language. Since RG provides only two kinds of denotations, namely sets and binary relations, any well-formed natural language expression denotes either a set or a binary relation, provided it has a denotation at all. Sets are denoted by common nouns, proper nouns, classifying adjectives, intransitive verbs, verb phrases, and prepositional phrases. Binary relations are denoted by transitive nouns, intensifying adjectives, and prepositions. Certain words, like determiners and conjunctions, do not denote at all. They are treated syncategorematically.

22 MICHAEL BOTI'NER

For the purpose of illustration let me give an example that to my knowledge has not been used before. In Woods (1991) a certain operation called MR is introduced to derive a new concept

(3) (MR [child] [doctor] )

from a relational term like, e.g. child, and an absolute term like, e.g. doctor. The meaning of the derived concept is given in natural English by

(4) all of whose children are doctors.

In terms of conventional grammar (4) is a relative clause that can modify a noun. One of the characteristic features of this relative clause is that the relative pronoun is in the possessive case. It turns out that MR corresponds exactly to the lower image operation of Suppes and Zanotti (1977). Although no precise grammar for natural language is given by Woods, I think that such a grammar can and indeed should be given and that RG is most appropriate for this purpose. An RG that derives (4) would be:

(5 ) RC -, all+of+whose+RN+VP

[RC] = [RN]G~[VP] VP + a r e + N [ V P ] = [ N ]

where RC = relative clause, RN = relational noun and children is a relational noun. To derive the denotation of the phrase student all of whose children are doctors is completely standard using intersection.

2. PROPER NOUNS

One of the consequences of the radical restriction of the hierarchy of denotations is that RG cannot have combinations of a determiner and a common noun phrase as a constituent. A rigorous proof for that has been given in Suppes (1976). This is in striking contrast to conventional linguistic analysis where combinations of determiner and common noun are grouped as one constituent and classified under the category NP (noun phrase). In contradistinction, proper nouns do not combine with a determiner but are also classified under category NP. It is considered to be one of the merits of Montague grammar to provide a


semantic category corresponding to the syntactician's category NP and allowing the treatment of proper nouns on a par with determiner noun combinations.

Whereas the coordination of predicates or verb phrases by and or or is straightforward and translates directly into the corresponding Boolean operations, i.e. and into intersection and or into union, the coordination of nominal phrases (proper nouns, common nouns) appears to pose some problems. A rather elegant theory of constituent coordination covering all kinds of linguistic categories and noun phrases in particular was proposed by Keenan and Faltz (1 985). But this theory makes use of higher order denotations. The question is whether one can come up with an adequate theory for noun phrase combinations under the assumption of the very restricted hierarchy of RG.

One of the problems of traditional logic was the analysis of individual sentences: since sentences were classified into the four categoricals, the question was whether individual sentences are universal or particular. In RG, a proper noun denotes a singleton set. For singleton sets, both universal and particular return the same result, so it does not matter which operation is adopted.

Let us now look at the simple sentence

(6) John and Mary are ill.

In transformational grammar, it has been proposed that this sentence be derived from the coordination of two sentences by so-called conjunction reduction:

(7) John is ill and Mary is ill.

Although (6) and (7) are semantically equivalent, they are syntactically different. A semantic theory should be able to give some explanation for this fact rather than hide this fact under the term of stylistic variance. Moreover, if a semantics is claimed to fit 'hand in glove' with the structure of English sentences, as is claimed in Suppes (1982), it should be able to provide a semantic structure that is homomorphic to the structure of (6).

Assume that John and Mary form a constituent. What would be a semantic function deriving a denotation for it from the constituents denotations? For and intersection comes to mind first. However, intersection applied to two singleton sets will return the empty set, unless

24 MICHAEL BOPNER

the combined proper nouns denote the same individual. Another option is union:

To account for natural language and by union rather than intersection is indeed in line with the original proposal by Boole (1854). If I is the denotation for ill, (12) can be called true iff the following condition holds:

Notice that we now cannot replace universal quantification by existential quantification, since both return different results for non-singleton sets.

Proper nouns cannot only be combined by and but also by or: the problem is which Boolean operation could serve the purpose to return the set denoted by John or Mary. Recall that intersection cannot be used and union has already been used in the case of and. Since and-coordinations and or-coordinations have a different meaning, one would expect that the corresponding operations should differ. It may be interesting in this context to note that the case has been made by Faltz (1989) for or being of higher procedural complexity than and.

It may be instructive here to take a closer look at Suppes' solution to computing the truth-value of categorical sentences: categorical sentences are alike in that they are made up of two absolute terms but are different with respect to the computation of their truth-value: the semantic function computing the truth value for a sentence with all is different from the semantic function computing the truth value for a sentence with some. Since all is related to and and some is related to or, one might suggest the exploitation of the different mode of combination along the following lines: let a coordination of proper nouns denote the union of their constituent denotations. Make a syntactic distinction between and-coordinations and or-coordinations of proper nouns. The grammar could then look like this:

(10) CPNP -+ PN + and + PN [CPNP] = [PN] U [PN] APNP -+ PN + or + PN [APNP] = [PN] U [PN]

where CPNP = conjunctive proper noun phrase, and APNP = alternative proper noun phrase. This allows us to have different operations to compute the truth value from PN-combinations and their predicates:


universal quantification for and-combinations and existential quantification for or-combinations:

where sg and pl refer to the features of singular and plural respectively, and the Frege function IF] is defined as follows:

D if q5 is true in the model, otherwise.

The solution works for elementary cases since John and Mary are ill is true iff { j , m} I and John or Mary is ill is true iff { j , m) n I # 0. The extension of this analysis to coordinated proper nouns in object position is completely analogous. But one does not feel satisfied with the solution since a semantic distinction has somehow been solved by some syntactic trick. The shortcomings become obvious if one tries to extend this solution to the following cases:

John or Bill and Mary sing

John and all boys sing John and some boys sing John or all boys sing

John or some boys sing

Any conventional semantic analysis that has a category NP and sub- sumes the category of proper nouns under this category would be able to compute the denotation of the sentences from the respective constituents NP and VP rendering, e.g., John and some boys into an NP-constituent. No such analysis is available in RG, though. It is likely that we can come up with some local solution for each case, but at the same time fail to grasp the relevant generalizations that follow from having a category NP.

Now consider the sentence

(1 2) John likes Mary and Bill Susan.

Structures like these have become known under the term 'gapping', since they can be conceived of as being derived from a conjunction of two sentences

(1 3 ) John likes Mary and Bill likes Susan

26 MICHAEL BOTTNER

by some syntactic transformation that leaves a gap where the verb should have occurred for the second time. If (6) is generated from its constituents why not do the same for (l2)? Notice that (12) need not be considered to be derived from a conjunction of sentences but can equally well be considered as a coordination of a sentence and a pair of proper nouns. The question is how to provide a semantic structure for (12) where L enters only once in the structure. A possible candidate is:

Two points may be raised against this proposal: first, (14) is not an expression of an extended relation algebra for the reason that the Carte- sian product cannot be defined in terms of the fundamental operations of an extended relation algebra (1). Second, (14) is not in line with the analysis of other sentences: the same phrase likes Mary occurs as a constituent of John likes Mary but does not occur as a constituent of the semantic structure of (12), if we accept (14) as the semantic structure of (12). It is not plausible that one and the same phrase occurs as a constituent in one structure but not as a constituent in a similar structure.

A similar problem arises with the structure

(1 5) John likes Mary and Bill too

that has at least the two readings

John Likes Mary and Bill likes Mary

John likes Mary and John likes Bill

depending on the prosodic features with which the above sentence is uttered. The problem is which structures can be provided for (15) that are homomorphic to its syntactic structure.

3. NUMBER

According to the standard set-theoretical definition, a cardinal number n denotes the set of all sets having n elements. This denotation is not within the limits of the set-theoretical hierarchy (2). As one can almost predict, RG will have trouble analyzing natural language constructions that somehow involve the notion of number. The notion of number occurs in a variety of contexts in natural language, like expressions with


cardinal adjectives, e.g., ten books, or the morphological category of number exhibited by the distinction between singular book and plural books.

Cardinality

Suppes (1974) proposed for the phrase two redflowers the following structure:

(16) 2 n P ( R n F )

where R denotes the set of red objects and F the set of flowers. This solution is not feasible in RG, since it falls outside (2).

Assume we think of numbers as a set of basic entities that fulfil Peano's axioms. But then the following problems arise: (i) what would the denotation be for two redflowers?; and (ii) how is this denotation derived from constituent denotations 2, R, and F?

Collectivity

The conjunction and can be construed by intersection in verbal contexts and by union in nominal contexts as we have just seen. There are however cases where and cannot be construed as a Boolean operation:

(17) Mary and Susan are sisters.

Predicates such as those occurring in (17) are conventionally referred to under the term 'collective'. Collectives therefore pose a problem to relational grammar. A way out could be along the lines of Link (1983) where collectives occur as elements of the domain D. However, each model structure would then have to be enriched by a relation between collective individuals and the non-collective individuals occurring as their parts, since (17) implies

(18) Mary has a sister.

If the denotations of both Mary and Mary and Susan occur as elements of the domain then there is no way to make the inference from (17) to (1 8) explicit.

Singular and Plural

The natural language fragments presented by Suppes so far are in the singular if dealing with proper nouns and in the plural if dealing with

28 MICHAEL BOTTNER

common nouns as occurring in categorical sentences. One would like to know whether singular and plural can be given a unified analysis in relation algebraic semantics. What is desired is a grammar that derives semantic trees for All (some, no) dogs bark and Every (some, no) dog barks at the same time.

The set-theoretically straightforward procedure would be to have the singular denote the set of all one-elemented sets

Singular = {X 5 D I X is an atom of P ( D ) ]

and have the plural denote the set of all sets with more than one element

Plural = P ( D ) - (Singular u {0) ) .

One could then define:

[girl] = P ( G ) n Singular [girls] = P(G) n Plural

where G is the set of girls of D. However, this is not feasible due to the restriction to the set-theoretical hierarchy in (2). It is unclear how this difference can be accounted for in RG.

Notice that there are semantic differences between singular and plural.

(1 9) A limousine was provided for all guests

(20) A limousine was provided for every guest

(1 9) has at least a collective reading whereas (20) has at least a distributive reading.

Bare Plurals

Bare plurals pose a problem to linguistic analysis, since they have to be construed by universal quantification on some occasions and by existential quantification on other occasions. In Suppes (1979) bare plurals in there are-structures are analyzed by existential quantification: the sentence There are trees is given the same denotation as the sentence There are some trees. In Suppes (1 976) bare plurals in object position are also analyzed by existential quantification: the verb phrase date juniors


is given the same denotation as the verb phrase date some juniors. On the other hand, sentences such as Birds are smaller than foxes, as occurring in Schubert's steamroller (cf. Purdy, 1992), require construal by universal quantification in both their subject and object positions.

4. PRONOUNS

The use of variables in logical language has been considered as an analogue of the use of (anaphoric) pronouns in natural language. The correct semantic analysis of pronominal constructions should therefore be a touchstone for any variable-free approach. Some cases were exam- ined in Bottner (1992a). The sentence

(21) Bill loves his children

can be called true if

(22) { b ) n -dom(C n -L) # 0

holds where L is the relation of loving, C is the relation of being a child of, and

Notice that

i.e., it denotes the set of all x, such that if y is x's child then x loves y. We therefore have the following denotation

(23) [loves his children] = - dom (- [love] n [child] ").

More complicated is it to find a denotation for

(24) Bill loves his children and so does Harry.

It has at least the following two readings:

(25) Bill loves Bill S children and Harry loves Harry's children

(26) Bill loves Bill 's children and Harry loves Bill's children.

30 MICHAEL BOTTNER

A third reading may be provided if the possessive pronoun refers to someone mentioned outside the sentence in question. A paraphrase of th reading (25) of (24) is

(27) Bill and Harry love their children.

This reading can easily be accommodated in RG using the framework developed in a preceding section. There are the following problems: (i) how can the structurally different sentence (24) be mapped onto the structure denoted by (27)?; and (ii) how can the reading (26) of (24) be correctly spelt out in RG? What is needed here is some semantic operation corresponding to the and so does-construction.

A notorious problem for semantics is posed by so-called donkey sentences like, e.g.:

(28) Any man who owns a donkey beats it.

The problem with this type of sentence is that it refers to some object that is mentioned in the embedded relative clause. (28) has man who owns a donkey as a constituent. The semantic structure of this constituent is

where M is the set of men, 0 the relation of owning and D the set of donkeys. But there is no way to embed (29) into a sentence. The structure

does not adequately construe the meaning of (28) but rather of

(3 1) Any man who owns a donkey beats a donkey

which is not equivalent to (28). An interesting type of pronoun is exemplified by the words same and

different as occurring in

(32) John and Mary read the same books

(33) John and Mary read diflerent books.

They have been analyzed as a species of quantifiers by Keenan (1992) and van Benthem (1989). The interesting thing . that they cannot be


construed by standard monadic (or Fregean) quantifiers. The question is whether RG fares any better than quantifier logic in this case. Notice first that there can be set-theoretical structures provided for (32) and (33):

Notice second that (34) and (35) do not admit of a structure parallel to read all/some/no books where read the same books or read difSerent books is a phrase of category VP. (Such an analysis was tentatively given in Bottner (1992a) but not within the confines of RG.) Notice third that no similar structure appears to be available for

(36) All students read the same books

(37) All students read dzflerent books

Whereas the restricted hierarchy of denotations may be appropriate for nouns and noun phrases, it is certainly less appropriate for verb phrases. Verb phrases generally bear a tense specification. Moreover, verb phrase denotations exhibit a wide variety of semantic types like states, events, processes, actions. It is certainly not accidental that the examples of verb phrases of the various English fragments presented so far are in the simple present tense and refer to properties, like are sick, are students, states like, e.g. loves Mary, like some teachers, are in the house, or habits like, for example, bite people, eat some vegetables, date all juniors. The question remains how tensed predicates like, were sick, will be sick or predicates that do not just express a property or a state, e.g. are eating vegetables, open some windows, bake a cake, etc., can be accommodated in RG.

It should be mentioned, though, that Suppes' own position towards RG has not been without reservation. In Suppes (1980) and, in particular, in Suppes (1979), the view is expressed that RG may turn out to be too weak for the purpose envisaged and it is argued for enriching RG

32 MICHAEL BOTTNER

by procedures. In fact Suppes has been arguing for using procedures in natural language semantics from as early as 1973 onwards, cf. Suppes (1973b), and has always emphasized that set theoretical semantics is some useful abstraction from underlying procedures. A first detailed proposal for the enrichment of RG by procedure is given in Crangle and Suppes (1987) and in Suppes and Crangle (1988) under the term of 'context-fixing semantics' (CFS).

CFS is designed for a fragment of arithmetical instruction. The model structures are therefore arrangements of symbols in rows and columns. Since symbols are the entities of the domain D , the denotation of column to the left will be a set of such objects and therefore be higher up in the denotational hierarchy. The denotation of the adjective top as in, e.g. top number or top spot is defined

(38) (XRS) {a I a is the R-last element of S]

where S is some set ordered by some ordering relation R and a is called the R-last element of S iff a E S A ('v'b)(b E S + (b = a v bRa)). (38) denotes a function that takes a pair (R, S ) as its argument where R is an ordering relation on D, S is a subset of D , and returns a subset of D . Since (R, S) E P ( D x D ) x P ( D ) , the domain of (38) is P ( P ( D x D) x P ( D ) ) . Since the domain of this function falls outside RG, the function denoted by (38) falls outside RG, too. Therefore, it appears that a grammar with a CFS cannot be an RG.

A major problem that has not been solved in any of the procedural extensions of RG is the meaning of an action. An action may effect a change of the model structure either in its domain (by bringing new objects into existence) or in its valuation function or in both. The general scheme for a verb phrase a denoting an action would thus be

In order to illustrate this type of verbal denotation let us consider a simple model structure of an arrangement of digits and letters on a line like this:

(40) can be described as a set of objects D ordered by some strict linear relation L, i.e. L is a binary relation on D that is transitive, asymmetric and connected. The action denoted by the verb phrase

(41) remove all digits


applied to (40) leads to the new model structure

(42) a b .

The action denoted by (41) has the effect of changing the model structure (40), since the domain D got changed, the denotation of digit is different, as is the denotation of left and the denotation of every notion that depends on this relation. This change effected on the model structure (40) can be represented as follows:

(43) v v ' D { a , b , 1 , 2 ) { a , bl digits { 1 , 2 } 8 letter { a , b ) { a , b ) left (a , 0, (1, b ) , ( 4 { ( a , b ) }

( a , b ) , ( a , 4, ( 1 , 2).

It is easy to compute output sets from input sets. They may not be affected at all as in the case of the denotation of letter

or may easily be derived as in the case of the denotation for digit

G' = G - G.

More difficult is the derivation of the output relation for L. The procedure that turns L into L' would have to remove from L all pairs that have a digit occurring in it:

As an instance of (39) we propose

(44) [remove] = { ( ( V , x ) , 23') : D' = D - X A

v' = vl D'A L' = L n ( - X x X ) )

But this denotation certainly exceeds the limits of RG.

6. CONCLUDING REMARK

This paper is meant to contribute to a festschrift. A festschrift is some sort of a birthday present, and a birthday present is supposed to come as

34 MICHAEL BOTTNER

a surprise. In this sense, by contribution is not a present, since most of the issues mentioned have been topics of numerous conversations and much correspondence with Pat over the last decade. I would not have brought them up in the present context had I not the strong conviction that Pat will come up with some brilliant and surprising ideas for their solution.

Max Planck Institute for Psycholinguistics, Wundtlaan 1, 6525 XD Nijmegen, The Netherlands

- REFERENCES

van Benthem, J.: 1989, 'Polyadic Quantifiers', Linguistics and Philosophy, 12, 437- 464.

Bottner, M.: 1992a, 'Variable-Free Semantics for Anaphora', Journal of Philosophical Logic, 21, 375-390.

Bottner, M.: 1992b, 'State Transition Semantics', Theoretical Linguistics, 18,239-286. Boole, G. H.: 1854, An Investigation of the Laws of Thought, Cambridge: MacMillan. Brink, C.: 1978, 'On Peirce's Notation for the Logic of Relatives', Transactions of the

Charles S. Peirce Society, 14, 285-304. Brink, C.: 198 1, 'Boolean Modules', Journal of Algebra, 71, 291-3 13. Crangle, C. and Suppes, P.: 1987, 'Context-Fixing Semantics for Instructable Robots',

International Journal of Man-Machine Studies, 27, 37 1-400. Crangle, C. and Suppes, P.: 1989, 'Geometrical Semantics for Spatial Prepositions',

Midwest Studies in Philosophy, XIV, 399-422. Faltz, L. M.: 1989, 'A Role for Inference in Meaning Change', Studies in Language,

13,317-331. Keenan, E. L.: 1983, 'Boolean Algebra for Linguists', in: S. Mordechay (Ed.), UCLA

Working Papers in Semantics, pp. 1-75. Keenan, E. L.: 1992, 'Beyond the Frege Boundary', Linguistics and Philosophy, 15,

199-22 1. Keenan, E. L. and Faltz, L.: 1985, Boolean Semantics forNatural Language, Dordrecht:

Reidel. Link, G.: 1983, 'The Logical Analysis of Plurals and Mass Terms: A Lattice-Theoretical

Approach', in: R. Bauerle, C. Schwarze, and A. von Stechow (Eds.), Meaning, Use, and Interpretation of Language, Berlin: de Gruyter, pp. 302-323.

Purdy, W. C.: 1992, 'A Logic for Natural Language', Notre Dame Journal of Formal Logic, 32,409-425.

Riguet, J.: 1948, 'Relations Binaires, Fermetures, Correspondances de Galois', Bulletin de la Socie'ti Mathimatique de France, LXXXVI, 114-155.


Suppes, P. : 1973a, 'Semantics of Context-Free Fragments of Natural Languages', in: K. J. J. Hintikka, J. M. E. Moravcsik, and P. Suppes (Eds.), Approaches to Natural Language, Dordrecht: Reidel. pp. 370-394.

Suppes, P.: 1973b, 'Theory of Automata and Its Application to Psycho!ogy', in: G. J. Dalenoort (Ed.), Process Models for Psychology, Rotterdam: Rotterdam University Press, pp. 78-123.

Suppes, P.: 1974, 'On the Grammar and Model-Theoretic Semantics of Children's Noun Phrases', Colloques Internationaux du C.N.R.S., Problkmes Actuels en Psycholin- guistique, 206e, 49-60.

Suppes, P.: 1976, 'Elimination of Quantifiers in the Semantics of Natural Language by Use of Extended Relation Algebras', Revue Internationale de Philosophie, 117- 118,243-259.

Suppes, P.: 1979, 'Logical Inference in English: A Preliminary Analysis', Studia Logica, 38, 375-39 1.

Suppes, P.: 1980, 'Procedural Semantics', in: R. Hailer and W. Grass1 (Eds.), Language, Logic, and Philosophy, Proceedings of the 4th International Wittgenstein Syrnpo- sium, Kirchberg anz Wechsel, Austria 1979, Vienna: Holder-Pichler-Tempsky, pp. 27-35.

Suppes, P.: 198 1, 'Direct Inference in English', Teaching Philosophy, 4,405-41 8. Suppes, P.: 1982, 'Variable-Free Semantics with Remarks on Procedural Extensions'

in: T. W. Simon and R. J. Scholes (Eds.). Language, Mind, and Brain, Hillsdale, NJ: Erlbaum, pp. 21-34.

Suppes, P. and Crangle, C.: 1988, 'Context-Fixing Semantics for the Language of Action', in: J. Dancy, J. M. E. Moravcsik, and C. C. W. Taylor (Eds.), Human Agency: Language, Duty, and Value,. Stanford, CA: Stanford University Press, pp. 47-76.

Suppes, P, and Macken, E.: 1978, 'Steps Toward a Variable-Free Semantics of Attribu- tive Adjectives, Possessives, and Intensifying Adverts', in: K. Nelson (Ed.), Chil- dren's Language, Vol. l , New York: Gardner Press, pp. 8 1-1 15.

Suppes, P, and Zanotti, M.: 1977, 'On Using Random Relations to Generate Upper and Lower Probabilities', Synthese, 36, 427-440.

Tarski, A.: 1941, 'On the Calculus of Relations', Journal of Synzbolic Logic, 6, 73-89. Woods, W. A.: 1991, 'Understanding Subsumption and Taxonomy: A Framework for

Progress', in: F. J. Sowa (Ed.), Principles of Semantic Networks: Exploratiorzs in the Representation of Knowledge, San Mateo, CA: Morgan Kaufman, pp. 45-94.


Michael Bottner has given an excellent, detailed survey of problems that remain open for the use of relational grammars in the analysis of the semantics of natural language. Reading his paper is, for me, like having an extended conversation with him. Most of the issues he writes about are ones that we have discussed, in most cases more than once. This is not the right occasion to enter into the necessarily complicated details of

36 MICHAEL BOTTNER

proposing new solutions to the various open problems he poses. What I think is more appropriate is to comment on some of the conceptual matters surrounding the development of relational grammars and their applications.

Skepticism about First-Order Logic as Logical Form. The motivation for my own work in variable-free semantics, as reflected in relational grammars, is skepticism of the tendency of philosophers to claim that logical form should be expressed in the notation of first-order logic. As a psychological thesis about the way we internally compute either the production or comprehension of natural-language utterances, I find this proposal almost certainly the wildest of fairy tales. There is, it would seem to me, no serious evidence of any kind that we process natural language in a way that depends at all on such a logical form. Certainly there can be some usefulness in formal analysis in terms of first-order logic, but not as a serious thesis about the syntactic or semantic structure of natural language. Given our difficulty of learning second languages as adults, it seems likely that a great deal of our individual natural languages are extensively used in our internal storage, generation and analysis of natural language utterances. In other words, the internal computing mechanisms that a speaker of English stores are in detail quite different from those stored by a speaker of Chinese or Japanese, which also differ from each other considerably. This does not mean that the basic mechanisms of language learning in the child are fundamentally different, but it does mean that the end result is different.

Another reason for skepticism about first-order logic is that the analysis of sentences of a natural language in terms of first-order logic only works for what I like to call 'classroom English', and I am sure the situation is similar for other natural languages. Once we go to the robust give-and-take of street talk or any other informal situation of linguistic communication, the resources of first-order logic are much too awk- ward and limited to carry us very far. I also hasten to add that some of these difficulties are also shared by relational grammars, which have as an objective the direct mirroring of the syntactic structure of a natural language. What happens in the case of relational grammars in dealing with rapidly moving dialogue is that a good deal of the syntax can be represented in some form, even if the semantics remains obscure. I tried to bring out the resistance of much ephemeral language use to systematic semantic analysis in my discussion of the Valley Girl phrase grody


to the max (Suppes, 1984, pp. 170-1 72). Those with strong systematic foundational views of language will be, it seems to me, necessarily frustrated in their attempts to give a universal semantic framework for the analysis of natural language. Such universal foundational theses are as hopeless here as in other domains of experience. My ambitions for relational grammars are not so imperialistic. I only hope that they will be able to solve a certain range of problems that will throw light on more systematic uses of ordinary language, if not the cases of grody to the max and its ilk.

My own view, undoubtedly mistaken in all kinds of details, to say the least, is driven by the view that human memory is much more capacious than computations are fast. So I think of natural language as being represented internally by a vast array of shallow trees representing many different kinds of grammatical forms. Much of what is going on can be nicely represented in context-free grammars with a direct semantic interpretation, although more troublesome cases require extensions to generalized phrase structure grammars. But for computational reasons they are not extended to transformational grammars which, unless severely restricted, are undoubtedly computationally too intensive.

Semantics of Actions. Michael rightly remarks that relational grammars, or even their procedural extensions, do not give a very adequate account for the meaning of an action. In fact, in the general approach it is hard to distinguish actions from spatial relations, let us say in the case when they are both represented as binary relations. I am not sure what will be the best way to move toward a deeper and better analysis, but I do have some thoughts and conjectures, especially generated by my recent work on machine-learning of natural language with Michael and Lin Liang. At the top level, in the case of the computer as the natural language user, we have a LISP representation internally, and at this level of abstraction there is nothing special about the representation of actions. At this level of computing, so to speak, an action is not much different from a spatial relation. At the level of LISP, the dominant position of actions in commands shows up only in the structure of the trees, not locally in the nature of the lexical treatment. However, this changes rapidly as we move to the actual details of robotic implementation or analysis of how the human perceptual-motor system works. The overwhelming difference between actions and spatial relations as we move downward in detail is that the actions are connected to the motor

38 MICHAEL BOTTNER

systems as well as the perceptual system. These connections can, of course, be reflected themselves at different abstract levels, but it is surely their existence that is one of the most essential marks of the difference. In the standard treatments of the semantics of commands, perceptual and motor control details do not enter in an explicit and formal way, for the semantic analysis is ordinarily left at a high level of abstraction. Even in the case of quite simple robots, we find that in the programming structure there are at least three or four levels of abstraction above the very concrete control of the robot, before we reach the top level which has the standard LISP representation.

There is no natural bottom to this search for level of detail, and this is surely especially true when we are dealing with the human perceptual and motor systems. The level of detail for semantic or other purposes will need to be fixed by the aims of the investigation, in other words, by the problems that are the focus of attention. In machine-learning of natural language it is obvious that much of the work can take place as far as robotic learning is concerned working only with top-level LISP representations. Obviously this is quite inadequate for other purposes. For example, the learning of concrete concepts in conjunction with language learning. In what we call 'pure language learning' where the concepts used are already assumed available to the robot, such mixing of concept learning and language learning does not occur and a very high level of representation can be adequate.

This is the point with which I want to end. It is at this high level of representation that the relational grammars and the accompanying semantics in terms of extended relation algebras can be quite useful. As we move to greater detail such grammars will be of less use, but there is some reason to hope that the right way to think about perception and motor systems is itself pluralistic in terms of a very considerable degree of decentralization. Communication with the peripheral systems can be conceptualized as being in a stripped-down and relatively meager language. The richness of natural language as we think of it is not used for this communication and is not understood even in an approximate form by the peripheral systems. If this way of looking at things is even roughly correct, it justifies separating a large piece of the linguistic analysis from the details of perception and motor-control, especially in terms of what resides in the peripheral systems which range from the fingertips to the eyeballs, or in the case of a robot, from the grasper to the television camera.


REFERENCE

Suppes, Patrick: 1984, Probabilistic Metaphysics, New York: Blackwell.

WILLIAM C. PURDY

A VARIABLE-FREE LOGIC FOR ANAPHORA

ABSTRACT. In a series of papers from 1973 to 1982, Patrick Suppes developed a model-theoretic semantics for English. Suppes' semantics deviated sharply from the prevailing Montagovian semantics in that it employed neither quantifiers nor variables, had an extremely simple type structure, and was based on the algebra of sets and relations. This paper describes a logic designed to represent reasoning in English. The logic, called C N , was motivated by the work of Montague and his modern-day followers, but departs from that tradition in that it is variable-free, quantifier-free, relational- algebraic, and restricted in its type structure. In this sense, CN is a Suppes-style logic. The objective of its design was to represent reasoning in, or very close to, surface English. To meet this objective, first it was given a phrase structure similar to English. Second, syntactic prominence was given to the 'generalized quantifiers' of English and to their monotonicity properties. LN was presented and developed in several earlier papers. The present paper extends the earlier development to accommodate the treatment of anaphoric pronouns by adding the Hilbert E-operator and a modal operator.

1. INTRODUCTION

This paper presents a theory of anaphoric pronouns that is similar in its objectives to Dynamic Montague Grammar (Groenendijk and Stokhof, 1990j, but in the style of Suppes. That is, the theory provides for an antecedent to bind pronouns that occur beyond its scope to the right (or to the left in certain cases), and does so in a compositional manner. But the formal language in which anaphora are represented is a variable-free, quantifier-free, relational-algebraic language rather than a higher-order quantification language. That is, the language is of the kind developed in Suppes (1973, 1974, 1976, 1979, 1982) to model the semantics of English.

The formal language is C N , presented earlier in Purdy (1 99 1, 1992a, 1992b), extended to accommodate treatment of anaphoric pronouns. The extension adds the Hilbert &-operator and a modal operator. It is by means of these operators that an anaphoric binding is allowed to reach beyond its scope.

The basic notion is that singular pronouns stand for singular expressions or descriptions. Three kinds of description are considered: proper,

I? Humphreys (ed.), Patrick Suppes: Scientific phi lo sop he^ Vol. 3, 4 1-70. @ 1994 Kluwer Academic Publishers. Printed in the Netherlunds.

42 WILLIAM C. PURDY

representative, and generic. A proper description denotes a particular individual. For example, John and John's mother are proper descriptions. A representative description denotes a representative of some property P. For example, that dog which is barking denotes a representative of the property of being a dog and barking. A generic description functions like a representative of a property P , but it is an abstraction such that any property Q that can be truly predicated of it can be truly predicated of any member of the property P . For example, in relation to the sentence a wise man knows his limitations, the description that man who is wise is generic. (Some might prefer the notion that any property Q that can be truly predicated of a generic representative of the property P can be truly predicated of any 'standard' member of the property P; however, this alternative will not be pursued here.)

The theory presupposes a linguistic analysis of English, which guides the translation to C N . The linguistic analysis is assumed to provide anaphoric coindexing and certain other information such as indication of genericness. It is argued that linguistic analysis also must determine anomaly, with the logic restricted to providing suitable representation and deduction of consequences.

The presentation is structured as follows. First the syntax and semantics of C N are defined. Then a complete axiomatization is given. Next, the various forms that anaphoric pronouns can take and their representation in C N are discussed. Based on this discussion, a formal translation to C N is defined for a fragment of English. Lastly, the approach of this paper is compared with that based on the dynamic logics.

2. DEFINITION OF THE LANGUAGE

The syntax and semantics of the language C N are defined in this section. In the following w+ := w - {O).

2.1. Syntax

The vocabulary of C N consists of the following:

1. Ordinary predicate symbols R = U jEw+Rj, where Rj = {Ri : 2 E w ) .

A VARIABLE-FREE LOGIC FOR ANAPHORA 43

2. Singular predicate symbols S = Ujtu+Sj , where Sj = ( ~ j : i E

w l - 3. Selection operators ( ( k l , . . . , I;,) : n E w+, ki E w+, 1 5 i <_ n ) . 4. Boolean operators f l and -. 5. Epsilon operator E .

6. Modal operator 0. 7. Parentheses ( and ).

Let P := R U S, := Rj U Sj , and P! be either R! or s;. LN is partitioned into sets of n-ary expressions for n E w. These

sets are defined to be the smallest satisfying the following conditions.

1. If Pp E Pn then Pr is an n-ary expression. 2. If P," E Pm then ( k l , . . . , k,) P r is an n-ary expression where n = max(ki) lii<m.

3. If X is an n-ary expression then (X) is an n-ary expression. 4. If X is an m-ary expression and Y is an 1-ary expression then

( X n Y) is an n-ary expression where n = max(1, m). 5. If X is an unary expression and Y is an (n + 1)-ary expression

then ( X Y ) is an n-ary expression. 6. If X is an (n + 1)-ary expression then E ( X ) is an (n + 1)-ary

expression. 7. If X is an m-ary expression then (kl , . . . , k,)e(X) is an n-ary

expression where n = m a ~ ( k ~ ) ~ ~ ~ < , . 8. If X is a nullary expression then O(X) is a nullary expression.

2.2. Semantics

An interpretation of LN is a tuple Z = (D, C, 3, {a, : c E C)) where D (the universe of individuals) and C (the set of cases) are nonempty sets, 3 is a mapping defined on R U S satisfying:

1. for each Rf E R,, F ( R f ) G Vn 2. for each Sp E Sn, F(Sp) Vn and for all d l , . . . , dn-1 E V there

exists exactly one d E V such that (d, d l , . . . , d,- 1) E 3(S,R)

and for each c E C : a, is a choice function on 2v such that

44 WILLIAM C. PURDY

Let a = (d l , d2, . . .) E VW and X be an n-ary expression. Then a satisfies X in Z at c, written Z kc X[a] , iff one of the following holds.

1. X = P: and (d l , . . . , dn ) E F ( X ) 2. X = Y and Z kc Y [a] 3. X = Y n Z and ( 1 kc Y[a] andZ kc Z[a] ) 4. X = Y Z and for some d E D ( l kc Y[d] and Z kc Z[da]) 5. X = E(Y) and (D = {d E D : Z kc Y[d,dz , . . . , d,]} and

@c(D) = dl) 6. X = (kl , . . . , k,)Y where Y is Rr, ST or E(Z) for m-ary expres-

sion Z, and Z kc Y [dk,, . . . , dkm] 7. X = O(Y) andforallc' E C ,Z kc/ Y

In the above definition Z kc Y [a] is an abbreviation for not (Z kc Y [a]) , and 1 kc Y [da] is an abbreviation for Z kc Y [d, d l , dz, . . .].

X is true in Z at c (written Z kc X ) iff Z kc X [ a ] for every a E Vw. a satisfies X in Z (written Z k X[a] ) iff Z kc X [ a ] for every c E C. X is true in Z (written Z b X ) iff Z kc X[a] for every a E DW and for every c E C. X is valid (written k X ) iff X is true in every interpretation of LN. A nullary expression of LN is called a sentence. A Set r of sentences is satisfied in Z at c iff each X E r is true in Z at c.

An expression of the form E(Y) will be called a &-expression. If an expression has no occurrences of E , it will be called &-free.

2.3. Metavariables and Abbreviations

Metavariables will be used as follows: k, I , m, n, . . . range over w , c ranges over C, R n ranges over 72,; Sn ranges over Sn; Pn ranges over P n ; Qn ranges over n-ary predicates and &-expressions; X n , Yn, Zn, . . . range over n-ary expressions; X, Y, Z, . . . range over all expressions of LN; En ranges over n-ary &-expressions; E ranges over unary E-

expressions; and S ranges over singular expressions (see Section 2.5). Applying subscripts and primes to these symbols does not change their ranges.

The following abbreviations are introduced for convenience and improved readability.


4. T := (s: s;) 5. AX'Y := x'Y 6. XnXn-I e . . X I Y := (Xn(Xn-I . . . (XIY) . . . ) 7. x1y,2 0 I$-, 0 . . 0 Y: := (. . . (x1Y,2)~;-,). . . Y?) 8. ( k ] , . . . , k,,)*Qrn := ( k ' , . . . , k,)Qrn 9. ( k l , . . . , k,)*(ll,. . . , ln)*Xn := (kl,, . . . , kln)*Xn, where m =

max(li) I si<n

lo. (k j , . . . , krn)*Xm = (k j , . . . , km)*Xm 11. (kl , . . . , k,)*(Xn n Y') := ( (kl , . . . , kn)*Xn n (k,, . . . , k1)*Y1),

where m = max(1, n) 12. ( k l , . . . , k r n ) * ( ~ ' Y m + ' ) := ~ ' ( ( 1 , kl + 1 , . . . , km + 1)*Ym+')

13. xn := (n, . . . , l ) *Xn

It is easy to prove that the abbreviations have the expected semantics, namely:

1. 1 kc (X u Y)[a] iff (Z kc X[a] o r Z kc Y[a]) 2. Z kc ( X Y)[cu] iff (Z kc X[p] implies Z kc Y[a]) 3. Z k ( X = Y)[a] i E ( Z kc X[a] i f f 1 kc Y[a]) 4. Z kc T[a] for every Z, cu and c

5. Z kc nX1 Y[a] iff for all d E V, Z kc X' [dl implies Z kc Y [da] 6. 1 kc X ' Y ~ o . . . o ~:[a] iff for some a l , . . . , a, E V : Z kc

Y:[U~, d l ] and Z kc Y2[a2, all and . . . and Z kc ~:[a,, an-l] and Z kc X' [a,]

7. Z kc ( k l , . . . , k,)*Xm[ol] iff Z kc Xm[dk, , . . . , dk,] 8. 1 kc O(X) iff for some c' E C, 1 kc/ X

These facts, together with the definition of satisfaction, will be referred to as the extended deJnition of satisfaction.

A few examples will make the semantics clear. Z X Y ~ renders 'some X is Y to some 2'; Z A X Y ~ renders 'all X is Y to some Z (independent of the particular X)'; A Z X Y ~ renders 'some .'i is Y to each Z (with the X dependent on the particular 2)'; and Z X Y ~ ~ o Y: renders 'some X is Y; composed with Y: to some 2'.

In the sequel, superscripts and parentheses are dropped whenever no confusion can result.

46 WILLIAM C. PURDY

2.4. Proper C N -Interpretation

Let xm+l, Y N 1 be &-free CN-expressions, where n = max(1, m). Then xm+l and Y1+' are &-independent in Z iff Z k ( A T ) ~ T ( X ~ + I r

YL+l ). An interpretation Z is proper if it satisfies: for all &-free expressions

xmf l , for all d, d l , . . . , dm E D, and for all c E C, there exists c' E C such that:

(i) Z + x m + l [ d , d l , . . . ,dm] implies Z kc/ &(xm+')[d , d l , . . . , dm] and

(ii) for all &-free expressions yl+l such that xm+l and Y1+l are E-

independent in Z, for all d', d;, . . . , d; E D, (Z kc &(Y1+ l ) [d', d; , . . . , di] implies Z +,I E(Y'+') [dl, d; , . . . , dl]).

From this point on, only proper interpretations will be considered.

2.5. Singular Expressions

Certain expressions of L N are singled out for special recognition. They play a role similar to that of functions in predicate logic. They are the singular expressions, defined as follows.

1. Each s,' E SI is a singular expression. If S1, . . . , S, are singular expressions, then:

2. For each SF+' E Sn+l, SI . . . s ~ s ~ " is a singular expression. 3. For each (n + 1)-ary expression X , S1 . . . S ~ E ( X ) is a singular

expression. 4. No other expresion is a singular expression.

Obviously, if S is a singular expression, then Z kc S[d] for exactly one d E D. Let ISIc represent this unique element of 2). When S is &-free, its unique denotation may be written I St.

3. AXIOMATIZATION OF CN

Conventions regarding metavariables given in the previous section are followed in this section. The axiom schemas of C N are the following.

BT. Every schema that can be obtained from a tautologous Boolean wff by uniform substitution of nullary metavariables of CN for sentential variables, n for A and - for 1


C. S n . . . S l ( k l , . . . , km)Qm Sk, . . .SklQm, where n = max

D. S n . . . SI (Xm n Y" G ( S m . . . S I X m n S l . . . SIY", where n = max(1, m)

EG. (SX' n S n . . . s ~ s Y ~ " ) & S n . . . S ~ X ' Y ~ " E l . Sn . . . s~x 'Y~+ ' C - ( E X ~ ~ S , . . . s l ~ Y n + ' ) , where E = E ( x ~ ~

sI . . . s,Y"+') E2. (/iT),(Xm = Y1) (/iT),(&(Xm) &(Y1)), where n = max

( 4 7 4 V

E3. Sn . . . S ~ X ' E ( Y ~ + ' ) Sn-l . . . s ~ x ' E ( s ~ Y ' + ' )

M1. X 0 & o x o , where X0 is &-free M2. oxo & XO

M3. o(xo 2 Yo) c (oxo c oY0) M4. OXo & OOXO M5. OX' & OOXO

P. OX, . . . XI (Ym+l n 2') & OX;, . . . X I (&(yrn+') n z'), where n = max(l,m+ 1), X I , . . . , X n , Y m f l are E-free, and if &(vk+') occurs in 2hhen vk+l is &-free and A T ~ ~ ~ ( ~ ~ ~ ) T (Y m+l = Vk+ ) a

The inference rules of LN are the following.

MP. From X 0 and X0 C Yo infer Yo N. From XO infer OX'

A derivation of a sentence X from a set of sentences is a finite sequence W1, . . . , Wn of sentences such that Wn = X and for all i 5 n, Wi is an axiom or a member of F, or is inferred from Wl, . . . , Wi-l using MP or N. The usual notation r I- X asserts that a derivation of X from r exists, and I- X asserts that X is a theorem of C N .

The following theorem establishes the soundness of this axiomatization.

THEOREM 1 (Soundness). I- X only if+ X. Proox It suffices to prove that the axioms are valid and that validity

is preserved by the inference rules. The following lemma, the proof of which follows directly from the definition of satisfaction, facilitates the necessary arguments.

48 WILLIAM C. PURDY

LEMMA2. Z kc S n . . . SIXn Iff1 kc Xn[lSIIc,. 7 ISnIc].

The validity of axioms C, S, D and EG follow easily from Lemma 2. Proofs for the remaining axioms and inference rules are also straightforward. As an example, details will be given for Axiom E l .

Z kc S n . . . S ~ X ' Y ~ + ' i f f 1 kc X1Yn+'[I~lIc , . . . , ISnIc] (Lemma2) iff 3d E 2, : Z kc X' [dl A Z kc Yn+l[d, IS1l,, . . . ) ISnIc] (definition of satisfaction). Let D = {d E D : Z kc X' [dl A Z kc yn+l[d, (SIIc, . . . , ISnlc]}. Then D = { d E 2, : Z kc (X1 S1 . . . snYn+l) [dl} (extended definition of satisfaction and Lemma 2),

whence Z kc (x' n S1 . . . s~Y"') [@,-(D)], since D # 0 and so

@,(D) E D. But IE(x' n Sl . . . S ~ Y ' " ) ~ ~ = @,(D), and so Z kc E ( X ' ~ S ~ . . . S,Y'+')X'~S~. . . S ~ E ( X ' ~ I S ~ . . . s,Y"')Y"+~ (Lem- ma 2). Since this argument can be made for any Z and c, Axiom El is valid.

Some simple results are given below in the form of lemma and corollary schemas. The proofs are easy and left to the reader.

LEMMA 4. ST.

COROLLARY 5. Sn . . . Sl SY"'~ Sn . . SITyn+l

COROLLARY 6. Sn . . . S~TY"" & S, . . . s1 E Y ~ + ' , where E =

E(S' . . . s~Y '+~) .

LEMMA 7. Sn . . . S ~ X Y ~ " I S n . . . S I T ( X n Yn+').

COROLLARY 8. Sn . . . s l ~ x Y n f Sn . . . SlnT(X 2 Yn+l).


COROLLARY 9. X'Y' r Y'x'.

LEMMA 10. AX'Y' = Y'X1.

LEMMA 11. X n . . . X 1 Y 7 ' = X I - a . X n P n .

COROLLARY 12. AX, 6 . A X ~ Y ~ A X , . . . n x n P n .

Two standard theorems dealing with proof in LN are now given.

THEOREM 13 (Deduction Theorem). If r, YO I- x0, then I' I- YO YO 11 .

Proof: Let W = WI, . . . , Wn be a derivation of XO from r, YO. Hence, r, YO t Wi for 1 5 i 5 n. If Wi is an axiom or an element of F then F t Yo 2 Wi (Axiom BT and Rule MP). If Wi = YO then I? I- YO Wi (Axiom BT). If Wi comes from 1Vj and Wj & Wi by Rule MP, then inductively, I' I- Y O Wj and r t- YO 2 (Wj & Wi). Hence I' I- YO & Wi (Axiom BT). Finally, if Wi = OW, and comes from W j by Rule N , then I- OWj whence I- Y O C OW, (Axiom BT), and so I' I- Yo & q W j . Therefore, r I- Yo & x 0 .

THEOREM 14 (Universal Generalization). If I' I- Sn . . . SI S' xn+l,

where s1 does not occur in SI, . . . , S,, X, or I?, then r I- Sn . . . sl A T X ~ + ~ .

Proof: Let W = W l , . . . , Wn be a derivation of Sn . . . S ~ S ~ X ~ + ' from r . Since S' does not occur in SI, . . . , Sn, X , or r , the derivation W' = W;, . . . WA where W,' is obtained from Wi by uniform

"

substitution of E = &(SI . . . SnXn") for sl, is a derivation of S, . . . S I E X n C 1 from r. By Axioms BT and S, r I- Sn . . . SIEXn+ ' ,

whence by Corollary 6, I' I- S, . . . S1TXn+l, i.e. r t- S, . . . S l ~ ~ X n + ' .

A corollary of Theorem 14 generalizes Axiom BT.

50 WILLIAM C. PURDY

COROLLARY 15. Let Xn be obtained from a Boolean tautology by uniform substitution of expressions of C N for sentential variables, n for A, and- for 1. Then I- ( A T ) ~ X ~ .

The next task is to show that the axiomatization is complete. Let I? C C N be a set of sentences. I' is consistent iff it does not contain XI, . . . , Xn such that t- X1 n . . - n Xn. r is maximal consistent iff for every sentence X @ F, F U {X) is not consistent. I' has witnesses iff it contains SX' and Sn . . . s1syn+' for some singular expression S whenever it contains Sn . . . X' yn+'. Observe that every maximal consistent set has witnesses by virtue of Axiom El . Moreover, according to the next lemma, each such set has witnesses in S1.

LEMMA 16 (Lindenbaum's Lemma). A consistent set of sentences I? can be extended to a maximal consistent set of sentences F+ with witnesses in S1.

The proof of this lemma, using Theorem 14, is standard. It is now shown that an interpretation Z of L N satisfying can be

constructed. Obviously, Z is also a model of I". From these results, it can be concluded that I" is consistent iff it has a model. Completeness of the axiomatization then follows as a corollary.

THEOREM 17 (Satisfiability). Ifr is a consistent set of sentences, then there exists a proper interpretation Z of CN such that Z kc F for some c E C.

Pro05 Since I? is consistent, by Lemma 16, it can be extended to a maximal consistent set r + . It suffices to show that there exists a proper interpretation Z such that Z kc I?+ for some c E C. This will be done in three parts. First, an interpretation Z = (D, C, F, {a, : c E C}) will be defined. Second, it will be shown that Z kc F+ for some c E C. Third, Z will be shown to be proper.

Part 1. Let A be the set of &-free singular expressions and define the binary relation -- on A as follows: SI -- S2 :@ S1S2 E I?+. It follows easily that -- is an equivalence. Let I Si I be the equivalence class containing Si. Define the universe D : = A/--. The denotation function is defined .F(Pr) := {(ISI 1 , . . . , ISn I) : Sn . . . SI Pr E I'+}. Clearly


F satisfies the requirements stated in the definition of the semantics of LN, and for every &-free singular expression S , 1 + S[ISI]. Thus the notation IS1 defined here is consistent with its earlier definition in Section 2.5.

Define rn := {OX : O X E I?+). Let {I?: : c E C) be the set of all maximal consistent extensions of rO. Note that I" is one of them. For each c E C, and for each D E 2", define @, as follows:

1. QC(D) := ISI, if D = {IS1 : S X E I':} # 0 and ( E X ) S E I': 2. @,(0) := QC(D) 3. @,(D) := IS/ for some arbitrary but fixed IS/ E D , otherwise.

In view of Axiom E2, @, is well defined. Clearly, it is a choice function satisfying the requirements of the definition of the semantics of CN.

Intuitively, I' specifies a partial state of affairs or partial case. It may contain contingent (true in some case) as well as noncontingent (true in every case) sentences. I'+ completes the specification of one possible case. It also fixes the set of sentences, denoted r O , that are to be true in every case. Because the truth values of contingent sentences vary from case to case, there is a multiplicity of possible cases. This multiplicity is {r: : c E C). That r0 is exactly the set of noncontingent sentences is shown by the next lemma.

LEMMA 18. lfro I- X then OX E r O . Pro05 Since ru t X , there exists q Wl, . . . , q W, E I" : q WI,

. . . , OWn I- X . By Theorem 13, OWl C (OW2 & . . - C (OWn 5 X ) . . . ) . Hence I- O(OWI & (OW2 & . . . & (OWn 2 X ) . . . ) (Rule N), whence t OOWl 2 ( 0 0 W 2 C . . . & ( n o w n C OX) . . . ) (Axiom M3), and t- OWI C (OOW2 & . . . C (OOWn 5 O X ) . . . ) (Axioms M4 and BT). Finally, q Wl I- 00 W2 C . . . C ( 0 0 W, & OX) . .) (Rule MP). After n such steps, q W,, . . . , q Wn I- OX. That is, t- OX, and so by definition of rO, OX E rO.

COROLLARY 19. X E r,f for all c E C iff OX E rO.

Part 2. Now it will be proved that for all c E C, for all X n E LN, for all IS1 I, . . . , IS,,I E D, Z kc Xn[lSII, . . . , lSnl] iff Sn . . . S I X n E ref. It then follows that in particular there exists c E C such that 1 kc X O

iff X O E I'+. The proof is by induction on the structure of Xn . The

52 WILLIAM C. PURDY

basis step, where X n = Rn or X n = Sn, follows immediately from the definition of F. The induction step involves six cases.

Case 1. X n = F. Z kc Xn[lSII, . . . , /S,I] iff Z kc Yn[lS1l, . . . , I Sn I ] (definition of satisfaction) iff Sn . . . Sl Yn 6 I'f (induction hypothesis) iff Sn . . . Sl Yn E I?: (I?: is maximal consistent) iff Sn . . . s lF E I'f (Axiom S).

Case 2. X n = Ym n z', where n = max(1,m). Z kc Xn[IS1i, , ISnl] i f f Z kc Ym[IS1I, 3 ISml] and kc ~ ' [ [ I s I I , . . ISLI]

(definition of satisfaction) iff Sm . . . Sl Ym E I?; and St . . . SI 2' E I?: (induction hypothesis) iff Sn . . . Sl (Ym n 2 ' ) E I': (Axiom D).

Case 3. X" = YIZn+'. Z kc Xn[lS1l, . . . , lSnl] iff for some IS/ E 27, Z kc Y ' [ I s I ] a n d 1 kc Z n + l [ J ~ I , ISII, . . . , lSnl] (definition of satisfaction) iff SY' E I'f and Sn . . . s I s Z n + ' E I'f (induction hypothesis) only if Sn . . . s1 Y Zn+' E I?{ (Axiom EG). Conversely, if Sn . . . S~Y'Z"+' E I?: then for some S , Sn . . . s ~ s ~ z ~ " , S'Y' E I?: (I?: has witnesses in Sl) .

Case 4. X n = &Yn. Z k Xn[lSll, . . . , ISnI] iff D = {IS1 E V : Z kc S2 . . . s ~ ~ ~ [ ~ s I ] } and a C ( D ) = ISI I (extended definition of satisfaction and Lemma 2) iff &(S2 . . . s n f n ) s 1 E I': (definition of a,)

"

iff SI &(S2 . . . snVn) E I?; (Corollary 9) iff Sn . . . s2s1 &(pn) E I': (Axiom E3) iff Sn . . . S2S1 &Yn E I': (Lemma 3 and Axiom E2).

Case 5. X n = (kl , . . . , km)Ym, where Ym = R r or ST or &Zm and n = max(ki) l<ism. Z kc X n [ISI I , . . . , ISn I ] iff Z kc Ym [ISk, 1 , . . . , I Skm I ] (definition of satisfaction) iff Skm . . . Sk, Ym E I': (induction hypothesis) iff Sn . . . Sl (kl, . . . , km)Ym E I?: (Axiom C).

Case 6. X n = myo. Z kc X n iff for all d E C, Z kc/ YO (definition of satisfaction) iff for all c' E C, YO E I': (induction hypothesis) iff myo E F0 (Corollary 19) iff OY' E I?: (definition of 1 ) .

Part 3. It remains to show that Z is proper. In view of Part 2, it suffices to show that for all &-free xm+l, for all &-free singular expressions S, S l , . . . , Sm, for all c E C, there exists c' E C such that: (i) Sm . . . SI sxm+l E I?: implies Sm . . . Sl SEX^+' E r: and (ii) for all &-free Y"' such that xm+l and Y"+' are &-independent in Z, for all &-free expressions S', Sj , . . . , SI, . . . Si , s'EY'+' E I?: implies S( . . . S~S'EY"~ E I?:. That is, it is to be shown that there exists c'


that 'agrees with c off Sm . . . Sl SEX^+' ' (i.e., c' differs at most from c in that IS1 satisfies &(SI . . . s~x"') in Z at d).

Suppose not. Then there exist &-free Y;" for 1 5 i < n such that xm+l and the q' are mutually &-independent in Z, and &-free singular expressions Si l , . . . , Sij, such that

and

n.. . n Sn,, . . . Sn1&Yzn.

Hence

U S m . . . Sl S(&Xm+l n S1,, . . . Sl I EY;' n . . . n Snj, . . . Snl cYzn)

by Axiom D. Therefore

USm. . . S1S(Xm+l n Sljl . . . SIl&Y;l n . . . i I Snjn . . . Sn1&Yzn

by Axiom P, and so

USm. . . S1SXm+l n Sljl . . . S l I ~ Y / l n . . . n Snjn . . . S n l ~ Y z n

(Axiom D). Repeating this for each of the ~ ~ i e l d s

The following observation shows that this contradicts the consisten-

cy of rD. If Siji . . . Si,TVi E Erf , then Siji . . . Sil EY'' E r: iff Sil ET E r$. Therefore, without loss of generality, it may be assumed that Siji . . . Sil Ci E r:.

COROLLARY 20 (Completeness). k X if X.

4. ANAPHORIC PRONOUNS

Representation of anaphoric pronouns in LN can be introduced with a simple example: A' man passes by. Hel smiles. Hel waves. In

54 WILLIAM C. PURDY

this and subsequent examples, superscripts index antecedent definitions and subscripts link anaphoric pronouns to these antecedents. This is standard notion, apparently motivated by the view that pronouns are similar to bound variables of predicate logic. This view is incomplete. However, for the present only this coindexing will be assumed to be supplied.

An adequate analysis of these sentences must entail that the scope of the indefinite determiner extends beyond M (man) and P (pass by) to include S (smile) and W (wave). The translation of these sentences in LN is the expression M P n E(M n P ) S n E ( M n P ) W , where he is linked to a man that passes by by the &-expression E ( M fl P ) . HOW the scope of the governing subexpression M extends beyond P to S and W can be seen as follows. By Axiom E l , E ( M n P) ( M n P ) n E ( M n P ) S n & ( M n P ) W , andso byaxiomD, E ( M n P ) ( M n P n S n W ) , whence by Corollary 5, T ( M n P n S n W ) . Finally, by Corollary 7, M ( P n S n W ) .

The remainder of this section defines various types of anaphoric pronouns and discusses their representation in C N . The emphasis is on sound representation. Formalization of the translation and comparison with alternative theories (specifically those based on dynamic logic) will be given in subsequent sections.

B-type pronouns (Sommers, 1982) are those which occur 'in construction with' their antecedents, i.e., roughly speaking, in the same sentence. This type pronoun, and only this type, can be treated simply as a bound variable. An example is ~ o h n ' loves hisl mother. Here the pronoun (his) is coreferential with the singular predicate (John) coindexed with it. The translation is J ( J ~ ~ ) E or the equivalent expression J T ( M n i). This treatment can be extended to quantified common nouns, as in ~ v e r ~ ' boy loves hisl mother. This sentence has the translation n BT ( M n t ).

On the traditional view, all pronouns were treated as bound variables. Evans (1977, 1980) argued that while such treatment is appropriate for pronouns which occur in construction with their antecedents, it


is not appropriate when the anaphoric pronoun occurs in a different sentence from its antecedent. Evans called the latter E-type pronouns. He analyzed the semantics of these pronouns at length. Slater (1986, 1988a, 199 1) supports Evans and further argues that &-logic is especially well suited for representation and explication of E-type pronouns. The treatment given here closely follows Evans and Slater.

An example is John owns a' horse and Mary rides itl. Here the pronoun (it) refers to that horse which John owns. This singular indefinite description is precisely rendered by the &-expression E(H n JO) . Therefore a translation is J H ~ n M E R, where E is E(H n JO) . By Axiom E l , one obtains J E ~ n EH f l MER, which is equivalent to E ( H n JO n M R ) , that is, E is a horse which Johns owns and Mary rides.

This treatment is easily extended to sentences such as A' farmer owns a2 horse and hel rides it2. Here the pronoun he refers to that farmer who owns a horse and it refers to that horse which is owned by that farmer. Therefore the sentence is translated F HO n El E2 R, where El is E(F n H O ) and E2 is E(H f~ E1O) . This is equivalent to E 1 ( F n E2(H n 0 n R ) ) .

This treatment of E-type pronouns can be summarized as follows. The existentially quantified antecedents are fully instantiated in accordance with Axiom El , resulting in all governing subexpressions being singular. Each pronoun is then taken to be coreferential with its (now singular) antecedent. Note that the result is not a weakening but an equivalence.

Sommers (1982) pointed out that a pronoun can be ascriptive and not descriptive, denoting something that is antecedently believed (mistak- enly) to exist. These he calls A-type pronouns. For example, There is a' man at the door. No, itl is not a man. Itl is a woman. Here the first sentence, translated MD, is retracted but the reference is preserved. That is, the second sentence denies the first but still uses it to provide the anaphoric reference. It is translated E ( M n D ) M . The third sentence uses the same anaphoric reference to make the correction E ( M n D ) W. Note that the unary expression M n D denotes 0 and so E ( M n D ) denotes an arbitrary choice from 27. This feature of the &-operator is

56 WILLIAM C. PURDY

not shared by similar operators (e.g., the 7-operator). For an extensive discussion of the value of this feature, see Slater (1988a, 1991).

It is not necessary that these pronouns involve mistaken belief. An A-type pronoun can also find its reference in a sentence denied at the outset. For example, It is not the case that a' dog killed the sheep. Itl was a lion. The translation is DSK f l E ( D fl s K) L. Thus negation does not always prevent an antecedent from remaining active.

4.4. Generic Indejinite

The indefinite article can be used generically. For example, A' man in love neglects hisl friends uses A man in love in the sense of a generic representative. The generic indefinite is aptly rendered by the &-expression E ( M n L) coupled with the operator q applied to the translation of the sentence. The &-expression denotes an arbitrary choice from the set denoted by man in love and the modal operator asserts that any such choice validates the sentence. Therefore what is superficially an existentially quantified expression actually has universal force. On this view, translation to LN yields O(E(M n L)A(E(M n L) F) N), or equivalently, q (E(M n L) AT ( p 5 N)) . In turn, by Axiom P, this entails A ( M n L)~T(I" fi). Note however that this sentence also entails TAT(I" & N ) , i.e., a generic sentence also has existential import. In subsequent subsections I argue that the notion of a generic representative plays an important role in the explication of several types of anaphora.

4.5. Conditional Sentences

The indefinite article in a conditional sentence is similar to a generic indefinite. For example, 1f a' man owns a2 horse, hel feeds it2 imparts universal force to both subject and object, to yield: for every man and for every horse, if the former is in the own relation to the latter, then the former is in the feed relation to the latter as well. I argue that, like a sentence with a generic indefinite, such a conditional sentence also should be read as an assertion about generic representatives, but lacking the existential import of generic sentences. Thus the above sentence is similar to the generic assertion A man that owns a horse feeds it, translated O E ~ E ~ F , where El is E ( M n HO) and E2 is E ( H n E1O). On this view, the original conditional sentence is translated ~ ( M H O C


El E2 F ) . Both translations can be shown to entail AMAH(O & F) as expected.

But subject and object do not always acquire universal force. Excep- tions are illustrated by the following sentences. If a' hotel has a2 thirteenth floor, it2 is assigned number fourteen. If a' man has a2 beard, hel trims it2. If a' man has a2 quarter, hel puts it2 in the parking meter. When a phrase does not acquire universal force, it is said to have the 'weak reading'. The first two sentences are easily explained. Each phrase that does not receive universal force has a denotation that is unique if it exists at all. Such phrase might be termed semisingular. While a singular expression S has the property expressed by Axiom S, a semisingular expression Z has only the 'if' half of the property. A useful expression of the semisingular property is given by the schema

That a phrase is semisingular is given by linguistic analysis and is based on common knowledge not logic. Given that a phrase such as thirteenth floor which that hotel has is semisingular, the sentence can be translated just like If a man owns a horse, he feeds it. This translation, in conjunction with the schema defining the semisingular -

property, then yields the desired entailment: AH (F (0 n A) U FO). The explanation for the last sentence is not so clear. Indeed, it is

not without doubt that this sentence is unambiguous. On the reading intended by its author, the quarter is not necessarily unique, but only one quarter is involved. That is, the intended reading is If a' man has a2 quarter, hel puts one2 in the parking meter. Clearly, since this sentence differs from the others in this subsection only in the nonlogical words used, the reading must be determined by linguistic analysis. Given the reading indicated, this sentence is translated to LN as ~ ( ( M Q H El ! l ( Q n E ~ H ) P ) , where El is E(M n QH). (See Purdy 1992a for an extension of LN to the cardinal determiners.) This translation entails A M(QH Q( H n P ) ) .

It should be noted that some hold the view that conditional sentences are ambiguous relative to universal versus existential (or probabilistic) reading. This ambiguity would be reflected in the presence or absence of the operator q in the translation. For a presentation of this position, see Slater (1988b).

58 WILLIAM C. PURDY

Evans (1977) also discussed a variation of the E-type pronoun, which here will be designated EP-type. It is like the E-type, but with the difference that the pronoun is plural. An example is John owns some1 horses and Bill vaccinates theml. The pronoun (them) refers to those horses which John owns. The reference is not, it should be emphasized, some of those horses which John owns, but rather all of those horses which John owns. On this reading, the proper translation is JHO n BA(H n J O ) ~

Again, the possibility of dissent must be recognized (see Slater, 1986). The dissenting view is that the above sentence is ambiguous between some of those horses which John owns and all of those horses which John owns.

4.7. Sentences with Universal QuantiJication

It has been suggested that a universally quantified common noun cannot function as antecedent to an anaphoric pronoun (see Groenendijk and Stokhof, 1991). On this account, for example, the following would be anomalous. Every1 man passes by. Hel smiles. But not all such sentences of this form are anomalous. A counterexample is ~ v e r y ' player chooses a2 marker. Hel puts it2 on square one. Consider the following list of examples.

(1) ~ 1 1 ' men pass by. Hel smiles. (2) ~ v e r ~ ' man passes by. Hel smiles. (3) Each1 man passes by. Hel smiles. (4) Each of the1 men passes by. Hel smiles.

(1) is definitely anomalous. (2) may be anomalous. But (3) and (4) are quite proper. The explanation is linguistic rather than logical. Nida (1975, p. 106) points out that the determiners all, every, each, each of the, each one of the are logically equivalent in that they all possess the meaning components of definiteness and plurality, but they are not linguistically equivalent because they differ in the meaning component called distribution.

When the determiner is low in the distribution component, the phrase in its scope denotes an undifferentiated collection. It is referred to by a plural pronoun, as in ~ 1 1 ' men pass by. Theyl smile. Following the argument given for EP-type pronouns, an appropriate translation is


A M P n A ( M n P)S. However, when the determiner is high in the distribution component, the phrase in its scope denotes a collection of individuals. The collection may be referred to by either a singular or plural pronoun. If the pronoun is plural, the reference is to the undifferentiated collection and the translation is as above. However, if the referring pronoun is singular, as in Each1 man passes by. Hel smiles, the translation should treat the pronoun as a generic representative of that collection of individuals. That is, A M P n U&(M n P ) S .

According to this argument, the example Every1 player chooses a2 marker. Hel puts itz on square one. has the translation /\PMC n 0 ~ ~ ~ ~ 0 , where El is E ( P ~ MC) and E2 is E ( M f l E I C ) . This entails A P M C n A(P n MC)AT((M n C) 0). The previous remarks regarding semisingularity apply here to that marker which is chosen by that player.

Some variations of this example and their translations follow. Every1 player chooses some2 markers. Hel puts them2 on square one. translates to A P M C n EL A ( M n C)O. Every1 player chooses some2 markers. Hel puts some of them2 on square one. translates to A ~ ~ C n O E ~ ( M n E ~ C ) O .

The examples of this subsection again illustrate the point that linguistic considerations contribute to defining the relationship between antecedent and pronoun, and therefore must be used in determining the translation to a logical representation.

4.8. B-Type Revisited

Consider the sentence some boy loves his mother. Using a representative description of boy, viz., E(B), this sentence could be translated E(B)B n E(B)(&(B)M)~, whence E(B)B n E(B)T(M n i ) , and so BT(M n i ) .

Now using a generic description of boy, the sentence every boy loves his mother could be translated (without existential import) q (E(B) (E(B) ~ ) i ) , whence O(&(B)T(M n i ) ) , and so A B T ( M n i). Notice that this is simply an expression of the accepted meaning of a universal sentence, viz., every substitution instance of a ( a ~ ) i , where a ranges over all boys.

Using this approach, it would be possible to render all types of singular pronouns using one of the three kinds of description. However,

60 WILLIAM C. PURDY

this treatment of B-type pronouns is not general, since it can interfere with the treatment of other types.

4.9. Mixed Types

A definition can be linked to several pronouns each of a different type. For example, in the sentence If a ' man loves his: mother, hel phones her2, a man is linked to the B-type pronoun his and also to the E-type pronoun he. The translation is O ( M T ( 0 n i) E ~ E ~ P ) , where El is E ( M n T ( 0 n i)) and E2 is E ( E ~ 0 n El L).

Other combinations are possible; for example, a generic indefinite can be bound to a pronoun, as in A' man who is generous with all hisl friends prospers, which is translated OE(M f l AT ( p E G)) P.

4.10. Explicit Modality

In every case (certainly) and in some case (possibly) can occur explicitly in sentences such as the following. A' man is brought in. Possibly hel is seriously hurt. Certainly hel needs immediate attention. Translation is obvious: M B n OE(M n B)H n U E ( M n B ) N .

'Quantifying in' is not possible in C N as defined in this paper. While this capability could be provided, its usefulness for analysis of anaphora remains a question to be explored. In this connection, cf. Slater (1992).

5. FORMALIZATION OF THE TRANSLATION

To demonstrate that the results of the previous section can be formalized, a translation algorithm is defined. It assumes a prior linguistic analysis which provides not only coindexing of pronouns and their antecedents, but also an indication of those antecedents that are generic (g), those pronouns that are bound (b), and those pronouns that refer to representatives of universally quantified antecedents (r). E-type pronouns and their antecedents are the default and so receive no special indication. Determination of anomaly is delegated to the linguistic analysis; the translation is restricted to constructing appropriate logical representations. It was argued in the previous section that all this is the necessary and proper role of linguistic analysis.

But first a brief summary of representation of anaphoric pronouns in C N will be given. To avoid repetition, let X and Y be sentences with


translations X and Y, J be a proper noun with translation J, V and W be verb phrases with translations V and W, and N and M be common noun phrases with translations N and M. Anaphoric pronouns are singular or plural. A singular pronoun can refer to a proper description, a representative description, or a generic description. A plural pronoun can refer to a part of a collection of individuals or to its totality.

A pronoun that refers to a proper description is simply taken to be coreferential with its antecedent and is therefore represented by the translation of that antecedent. Hence J' V. Hel W is translated J V n J W . A pronoun that refers to a representative description, as in a' N V. Hel W, is represented by the &-expression E(N n V). A generic sentence, a19 N V. Hel W, translates to o(E(N)V) n o(E(N)W). E ( N ) is a generic description. A pronoun that refers to it is coreferential with it and so also represented by E(N) . A conditional sentence is explained as a variety of generic sentence. If X then Y is translated O ( X 2 Y). In a conditional sentence, singular pronouns that refer to proper or representative descriptions are represented as explained above. A sentence in which an antecedent is universally quantified also is considered to be a kind of generic sentence, Here a singular pronoun referring to a universally quantified antecedent is represented as a generic representative of the collection denoted by that antecedent. ~ v e r ~ ' N V. Hel, W is translated A N V n o(E(N n V) W).

A plural pronoun that refers to the totality of a collection, as in some1 N V and theyI W, is translated N V n n ( N f l V) W. Similarly, a plural pronoun that refers to a part of a collection, as in some1 N V and some of theml W, is translated N V n ( N n V)W.

The translation is specified by an attribute grammar. To keep the grammar brief, morphological rules necessary to achieve proper noun and verb forms are omitted; the grammar is allowed to be syntactically ambiguous; and some grammatical combinations are omitted. The value of the attribute T is a translation of its argument. The value of the attribute 8 is a substitution. A substitution is a string of atomic substitutions of the form ( X / p i ) . concat is a function which concatenates substitutions. subst is a function which applies the substitution that is its second argument to the LN-expression that is its first argument. Following accepted convention, 'copy rules' such as T(SS) + r(S) are omitted. A good reference for attribute grammars is Waite and Goos (1 984).

62 WILLIAM C. PURDY

ss+ sss T(SSI) +- (7(SS2)) n ( su~s~ (T( s ) , Q(ss2))) Q(SSl) t ~oncat(Q(SS,), Q(S))

IS S --t S and S ~ ( s l ) sub~ t ( (~ (S2 ) ) n (7(S3), ~ ( S I ) ) )

Q ( s l ) t ~ortcat(Q(S2)~ Q(S3))

lit is not the case that S T ( S ~ ) t (T(S2)) Iin every case S ~ ( s l ) + o ( ~ ( S 2 ) ) Iin some case S O ( ~ ( s 2 ) ) lif S then S ~ ( s i ) +- S U ~ S ~ ( ~ ( ( T ( ~ Z ) ) L (~ (S3 ) ) ) l ~ ( S I ) )

Q(S1) + concat(@(S2), Q(S3)) ID sup(i) CNP VP T (S) +- T (D) T (CNP) T (VP)

e(s) + concat((((.r (CNP)) n (7 (VP))) /pi),

subst(Q(VP)l ( P ~ / P ) ) ) la sup(ig) CNP GVP T (S) + O(E(T(CNP))T (GVP))

0(S) t concat(((7CNP)) /pi) e(GVP))) IPN sup(i) VP T (S) + T (PN) T (VP)

Q(S) +- concat((.r(PN)/p,),

subst(Q(VP) 1 (pi 1 ~ ) ) IPRO VP T(S) t T(PRO)T (VP)

Q(S) +- subst(Q(VP), (T (PRO)/p)) ISPRO sub(ir) VP ~ ( s ) n ( & p , ~ ( V p ) )

Q ( s ) + subst(Q (VP), (&pi /p) ) ID sup(i) CNP TVP PRO T (S) + T (D)T(CNP)T (PRO)T ( ~ v P )

B(S) + ((T(cNP)) n (T (PRO)T(~VP) ) ) /P~ )

IPN sup(i) TVP PRO T (S) t T (PN) T (PRO) T ( ~ v P )

e(S) + ( ~ ( P N ) l p i ) IPRO TVP PRO T(S) t T ( P R O ~ ) T ( P R O ~ ) T ( T ~ P )

8(S) t 0 ISPRO sub(ir) TVP PRO T(S) + o (E~~T(PRo)T(~vP) )

e (s ) + s PRO -+ SPRO sub(i) T(PRO) + &pi

ID of IPPRO sub(i) T(PRO) + ~ ( D ) p i IPPRO sub(i) T (PRO) +- all pi

CNP -+ CNA who VP T (CNP) +- (T (CNA)) n (T (VP)) Q(CNP) t subst(B(VP),

(&((T(CNA)) n (T(VP))/P)) ICNA


CNA --t A C N A

ICN

VP + TVP D sup(i) CNP

ITVP POSSP sub(ib)

supu) CNP

ITVP PN sup(i)

IIVP

GVP -+ TVP a sup(ig) CNP

T(VP) +- T(D)T(cNP)T(TvP)

Q(VP) +- ((r(CNP)) n p ~ ( T V P ) l p i ) T(VP) + T((T(CNP)) n ~ ( Y v P )

0 (VP) + concat((pi (T ( ~ N P ) )

~ P T ( T V P ) I P ~ 1, Q(CNP)) T(VP) +- T(PN)T(YvP)

Q (VP) +- (T(PN) I pi

TVP -+ donot TV ITV

IVP --t donot IV IIV

The lexicon is assumed to contain determiners some, all, no, a, every, each as well as supply of adjectives (A), common nouns (CN), proper nouns (PN), intransitive verbs (IV), transitive verbs (TV), singular pronouns (SPRO), plural pronouns (PPRO), possessive pronouns (POSSP), and indices (i). a translates to some; every and each translate to all; otherwise T is the identity function on the lexicon. 8 is the constant function 8 on the lexicon.

The determiners some, all, and no appear in LN-expressions as abbreviations defined: some X ~ Y := X'Y, all X'Y := AX'Y, and no X ' Y := X1 Y. The determiners could be extended to include one, two, . . . , most as well. See Purdy (1992a) for an extension of LN to these determiners.

6. DISCUSSION

This section further characterizes the representation of anaphora in LN by comparison with the dynamic logics of Groenendijk and Stokhof and their colleagues (e.g., Groenendijk and Stokhof, 1990, 199 1 ; Chierchia, 1990, 1992; van Eijck and de Vries, 1992). For this purpose, Dynamic Montague Grammar (DMG) will be taken as representative. In DMG, existential quantification is 'dynamic' in that it can extend its scope

64 WILLIAM C. PURDY

to include sentences conjoined on the right. The notion of 'possible continuations' of a formula is central. The possible continuations of a formula 4, denoted Tq5, is defined to be the set of formulas, or equivalently the set of sets of cases, that are possible given the truth of that formula. Thus Tq5 is upper closed in the power set of the set of cases. If + is a possible continuation of 4, then the dynamic conjunction Tq5; T$ will have a nonempty set of possible continuations. Dynamic conjunction provides for anaphoric pronouns (as to-be-bound variables) to be brought into the scope of their antecedents (as quantifiers) without sacrificing compositionality. Other operations (universal quantification, negation, disjunction, implication) are 'static' in that they do not allow continuations.

Creation of possible continuations extending the scopes of antecedents to include coindexed pronouns also exists in CN, but by virtue of a different mechanism. Recall the example A' man passes by. Hel smiles. Hel waves. The translation of these sentences in LN is the expression M P n E ( M n P ) S f l E ( M n P ) W . The set of possible continuations of M P is the set of sentences consistent with M P , or with the equivalent expression E ( M n P) ( M n P ) . Since a sentence is true at some set of cases, the set of possible continuations is a set of sets of cases. E ( M n P ) S is asserted by the discourse to be one possible continuation. The set of possible continuations of M P n E ( M n P ) S is the set of sentences consistent with E ( M n P ) ( M n P n S ) , or equivalently, the set of sets of cases corresponding to this set of sentences.

It is seen that CN provides the basic capability of DMG and does so in a manner that closely parallels that of DMG. However, the parallel is not exact. Specifically, E ( M n P ) ( M n P n S n W ) implies T ( M n P n S fl W ) , but this is not an equivalence since T ( M n P n S n W ) implies only the weaker E ( M n P n S n W ) ( M n P n S n W ) . This distinction is not found in DMG, where pronouns are treated as bound variables. The increased expressiveness afforded by this distinction in CN results in a representation that is more faithful to the semantics of English. The sentences John liked a girl. He asked her for a date. are rendered in CN as j ~ i n j&(G n ~ L ) D , which is faithful to the meaning 'John liked a girl and John asked that girl which he liked for a date'. Contrast the sentences John asked a girl for a date. He liked her. Here the LN rendition is ~ G D n ~ E ( G n j D ) L , corresponding to the nonequivalent meaning 'John asked a girl for a date and John liked


that girl which he asked for a date'. In DMG, the former sentences are not distinguished from the latter.

The primary motivation underlying DMG was to develop a fully compositional semantics of natural language discourse. With respect to singular pronouns, DMG achieves this goal with admirable elegance. Translation from English to LN is also compositional, with the place holders (pi) playing a role similar to that of the discourse markers in DMG. Compositionality is reflected in the fact that the attribute grammar of the previous section is an S-attribute grammar, i.e., one employing only synthesized attributes. More precisely, the semantic value attached to a given node is determined by the semantic values attached to its daughter nodes and the rule by which they were combined.

However, with respect to plural pronouns, DMG falls short of total compositionality. The following is a simple illustration of this. John owns some horses and Bill vaccinates some of them would be translated compositionally to 3 x ( H ( x ) A O ( j , x ) ) A V ( b , x ) , which according to the semantics of dynamic logic entails 3 x ( ( H ( x ) A O ( j , x)) A

V ( b , x ) ) . But then John owns some horses and Bill vaccinates (all of) them must, if compositionality is to be enforced, translate to 3 x ( H ( x ) A O ( j , x ) ) A 4, where 4 translates Bill vaccinates (all of) them. Clearly there is no translation 4 that yields the proper entailment V x ( ( H ( x ) A O ( j , a ) ) -+ V(b, a ) ) . On the other hand, translation to LN is uniformly compositional for both singular and plural pronouns.

It might seem that more is assumed of a prior linguistic analysis than is the case with DMG. Translation to Llv depends on the linguistic analysis to provide not only coindexing of pronoun and antecedent; but also to indicate those antecedents that receive a generic reading or a semisingular reading, and to indicate those pronouns that are read as generic representatives of a distributed collection. Further, C N dele- gates judgments of anomaly to the linguistic analysis.

DMG seems not to impose these requirements. However, this may be illusory. Unless one defines both dynamic and static versions of the operators of DMG, there are many 'exceptional' constructions in English to be dealt with. For example, DMG proscribes coindexing a plural (universal) antecedent and a singular pronoun. But English sentences in which this occurs come easily to mind (Section 4.7). Addi- tion of a dynamic universal quantifier can remove this proscription, but now linguistic analysis is needed to decide which universal quantifier to

66 WILLIAM C. PURDY

use. It is possible then that what DMG would require of the linguistic analysis is similar to what C N requires.

Delegation of greater responsibility to the linguistic analysis leaves C N free to provide representation to a broader range of anaphoric phenomena. Negation does not necessarily prohibit possible continuations, nor does universal quantification. It is even possible to bind pronouns to the lefr of the antecedent, as in If hel enters without knocking, ~ o h n ' will surprise Mary (see attribute grammar).

L N has a strong linguistic motivation. Its SOV (subject-object- verb) structure is transparently similar to the SVO structure of English. This structure directly dictates formation of the descriptions and set- terms that represent pronouns. Representation of singular pronouns as descriptions and plural pronouns as set-terms is faithful to the semantics of English and also facilitates surface reasoning. It permits uniform treatment of singular and plural pronouns in a compositional manner. The important role given to generic descriptions is also linguistically motivated. This kind of description is well suited to represent pronouns that occur in certain conditional sentences and sentences in which a collection is given the 'distributed' reading, as well as patently generic sentences.

C N can claim simplicity relative to DMG. L N is a first-order logic with a complete axiomatization and powerful and natural derived inference rules, such as the monotonicity rules (Purdy, 1991). C N has the simple type structure of Suppes-style languages. By contrast, DMG is a higher-order logic with a rich type structure. With this comes greater expressiveness and capability, but also greater complexity.

This paper has a limited objective, viz., to show that it is possible to provide the general capability of DMG in a Suppes-style deductive system. Only simple English constructions were treated. Many issues, such as the context of utterance and the speaker's intention, were not considered. It remains to be determined whether the approach described here can be extended and whether it will offer useful insights.

ACKNOWLEDGMENT

I wish to thank Patrick Suppes for the interest he has shown in my work. This has been a source of encouragement and inspiration. I also wish to thank Hartley Slater, who introduced me to the uses of the E-calculus


in connection with natural language, for a number of helpful comments on an earlier version of this paper.

School of C o m p u t e r and Information Science, S y r a c u s e University, S y r a c u s e , NY 13244-41 00, U.S.A.

REFERENCES

Chierchia, Gennaro: 1990, Anaphora and Dynamic Logic, Prepublication LP-90-07, ITLI Prepublication Series, University of Amsterdam.

Chierchia, Gennaro: 1992, 'Anaphora and Dynamic Binding', Linguistics and Philos- ophy, 15, 11 1-183.

van Eijck, Jan and de Vries, Fer-Jan: 1992, 'Dynamic Interpretation and Hoare Deduc- tion', Journal of Logic, Language, and Irzformation, 1, 1-44.

Evans, Gareth: 1977, 'Pronouns, Quantifiers, and Relative Clauses (I)', Canadian Journal of Philosophy, VII, 467-536.

Evans, Gareth: 1980, 'Pronouns', Linguistic Inquiry, 11, 337-362. Groenendijk, J, and Stokhof, M.: 1990, 'Dynamic Montague Grammar', in L. Kalman

and L. Polos (Eds.), Papers from the Second Symposium on Logic and Language, Budapest: Akademiai Kiado.

Groenendijk, J, and Stokhof, M.: 1991, 'Dynamic Predicate Logic', Linguistics and Philosophy, 14, 39-100.

Nida, Eugene A.: 1975, Compositional Analysis of Meaning, The Hague: Mouton. Purdy, William C.: 1991, 'A Logic for Natural Language', Notre Dame Journal of

Formal Logic, 32,409-425. Purdy, William C.: 1992a, 'Surface Reasoning', Notre Dame Journal of Formal Logic,

33, 13-36. Purdy, William C.: 1992b, 'A Variable-Free Logic for Mass Terms', Notre Dame

Journal of Formal Logic, 33, 348-358. Slater, B. H.: 1986, 'E-Type Pronouns and &-Terms', Canadian Jourizal of Philosophy,

16,27-38. Slater, B. H.: 1988a, Prolegomena to Formal Logic, Aldershot: Avebury. Slater, B. H.: 1988b, 'Subjunctives', Critica, XX, 97-106. Slater, B. H.: 1991, 'The Epsilon Calculus and Its Applications', Grazer Philosophische

Studien, 41, 175-205. Slater, B. H.: 1992, 'Routley's Formulation of Transparency', History and Philosophy

of Logic, 13, 215-224. Sommers, Fred: 1982, The Logic of Natural Language, Oxford: Clarendon Press. Suppes, Patrick: 1973, 'Semantics of Context-Free Fragments of Natural Languages',

in: J. Hintikka, J. M. E. Moravcsik, and P. Suppes (Eds.), Approaches to Natural Language, Dordrecht: D. Reidel, pp. 370-394.

Suppes, Patrick: 1974, 'Model Theoretic Semantics for Natural Language', in: C. H. Heidrich (Ed.), Semantics and Communication, Amsterdam: North-Holland, pp. 284-313.

68 WILLIAM C. PURDY

Suppes, Patrick: 1976, 'Elimination of Quantifiers in the Semantics of Natural Language by Use of Extended Relation Algebras', Revue Internationale de Philosophie, 30, 243-259.

Suppes, patrick: 1979, 'Variable-Free Semantics for Negations with Prosodic Varia- tion', in: E. Saarinen, R. Hilpinen, I. Niiniluoto, and M. P. Hintikka (Eds.), Essays in Honor of Jaakko Hintikka, Dordrecht: D. Reidel, pp. 49-59.

Suppes, Patrick: 1982, 'Variable-Free Semantics with Remarks on Procedural Exten- sions', in: T. W. Simon and R. J. Scholes (Eds.), Language, Mind, and Brain, Hillsdale, NJ: Erlbaum Associates, pp. 21-34.

Waite, William M. and Goos, Gerhard: 1984, Compiler Construction, New York: Springer-Verlag.


It is very satisfying to see how well Bill Purdy's analysis of anaphora in a variable-free logic can treat so many instances of English usage. His introduction of the Hilbert E-operator to facilitate the analysis of anaphora is clearly and carefully set forth. At the time of writing this commentary, I do not see a better alternative. It may be that his approach will be the right one for the future. Since the €-operator is a way of eliminating quantifiers, it is not surprising that it does well with anaphoric pronouns. However, the uses of anaphora are many and wonderful. The analysis of quite different cases, for example, that of the anaphoric use of same and diflerent by Bottner (1992) shows that other methods and other devices can be just as useful. It is not clear to me exactly how Purdy would analyze the many cases in Bottner's article in his language LN. Moreover, even in the cases of something closer to ordinary pronominal reference, it is not clear to me that his current use of the €-operator would satisfactorily represent some men in love neglect their friends, others do not, although some extensions of the notation to permit selection of distinct objects from the same set would make it relatively straightforward. Presumably Purdy could adopt the devices used within the framework of extended relation algebras by Bottner to represent such anaphoric subtleties as those involved in the meaning of same and digerent, so that if we put Purdy's and Bottner's articles together we could certainly say that we are coming closer to a reasonable analysis of anaphora within the framework of variable-free semantics, not that any of the three of us would want to claim that all problems have been solved.


One point of methodology on which Bill and I differ is in the use of an intensional logic, in his case LN, into which to translate the expressions of ordinary language. This follows the style of Montague and his later followers. As Purdy notes, of course he is deviating from the Montague tradition in using a variable-free algebraic notation. In my own view, it is better to let the syntax be that of natural language and attach the variable-free semantics directly to the surface of the natural-language utterances. This approach is hardly feasible with the Montague-style intensional logic, and it will not always work with variable-free semantics. But my whole concept of a potentially denoting grammar was aimed exactly at such use and it also goes with my psychological intuitions that this is the right route for a thoroughly realistic theory of how language is actually produced and comprehended by us. Purdy stresses the importance of inference, which we may take as the keystone of a logic. I have tried to support the idea that inference, as much as possible, should be directly in English, using variable-free semantics as the back- bone of such informational structure (Suppes, 1979). What this means in terms of the nuts and bolts of actual usage is that the computational aspects of inferences are very different when they are done directly in English as opposed to first being translated into an intensional logic, the inferences made in the intensional logic, and then translated back into the natural language. Fortunately for my standpoint, most of the virtues of Purdy's analysis are independent of taking such a stance about translation into an intensional logic.

Set Theory without Variables. In spite of my differences about the methodology of using an intensional logic for representing the meanings of natural language expressions, there are virtues to a language like Purdy's L N that need to be stressed. It is easy to compare its strength with various results in the literature about variable-free systems. As is stressed in Bottner's paper in this volume there is a long history of such variable-free approaches, beginning at least with Peirce in the nineteenth century and considerably extended by Schroder at the end of that century. Further work in the same direction done early in the twentieth century by Lowenheim and Schonfinkel, and somewhat later by Curry and his colleagues on combinatory logic, showed clearly enough the power of variable-free methods. One of the most elegant formulations of set theory is the restricted one, close to relation algebras found in Tarski's last work (Tarski and Givant, 1987) on a formulation of set

70 WILLIAM C. PURDY

theory without variables. It would, I think, be instructive to compare Purdy's language LN with the language L x of Tarski and Givant. More generally, I think we have by no means exhausted the methods of analysis and what can be accomplished in an elegant way without variables. What is especially important about Tarski's approach to variable-free semantics is that the system Is weaker than standard formalizations of set theory in first-order logic, but essentially all of the standard mathematical theorems can be formulated without difficulty. Tarski and Givant show that their system is equivalent to a first-order language restricted to three distinct variables. It would be interesting indeed to see if there are reasonable natural-language sentences that cannot be translated into their variable-free set-theoretical notation.

REFERENCES

Bottner, Michael: 1992, 'Variable-Free Semantics for Anaphora', Journal of Philo- sophical Logic, 21, 375-390.

Suppes, Patrick: 1979, 'Logical Inference in English: A Preliminary Analysis', Studia Logica, 38, 375-39 1.

Tarski, Alfred and Givant, Steven: 1987, A Formalization of Set Theory without Vari- ables, Providence, RI: American Mathematical Society.

J. MORAVCSIK

IS SNOW WHITE?

ABSTRACT. Taking the well known sentence 'snow is white' the paper shows that a proper analysis of elements of English of this kind demonstrates that truth-conditions cannot be given as the semantic analysis of such units. Since there are different applications of 'snow' and of 'white', the sentence cannot be judged as true or false, only uses of it. It is shown that these uses are indeterminate in number and should not be regarded as a case of ambiguity. A semantics is sketched within which the core meanings of the constituents as well as their multireferentiality can be represented.

It is explained why natural languages cannot be regarded as formal languages in Tarski's sense, and why this is an advantage for natural languages rather than a handicap.

The paper does not support a conclusion that the semantics of natural languages does not admit of rigorous analysis. It says only that such an analysis will have to use tools other than the standard set-theoretic semantics known since Tarski as 'formal semantics'. Obviously other formal tools must be available to capture the data of the semantics of a language like English.

Much recent work on the semantics of natural languages has been guided by the following assumptions (e.g. Davidson, 1967; Montague, 1974):

(i) A natural language like English is, or can be represented as, a formal language in Tarski's sense.

(ii) In particular, the sentences of a natural language like English can be given truth-conditions in Tarski's technical sense of that notion.

(iii) Giving the truth-conditions of the sentences of a natural language contributes to specifying what it is to understand that language.

Claims (i) and (ii) are about the characterization of the expressive power of a language. Claim (iii) links a certain semantic representation to the explanation of one of our cognitive capacities. One could hold (i) and (ii) without holding (iii). Below, however, we will consider definitions that one might give for 'natural language', and reflections on these suggest that restricting oneself to (i) and (ii) is unlikely to yield theoretically interesting results.

The purpose of this paper is to show that all three of these claims are false. Thus alternative approaches to the semantics of natural languages

P Humphreys (ed.), Patrick Suppes: Scient& Philosopher, Vol. 3, 71-87. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

J. MORAVCSIK

are needed. One of these will be briefly sketched later. In the first, negative part of the paper my strategy will be to take a well known example from the literature, and to back my conclusions by a detailed examination of the semantics of that sentence. The sentence is:

(1) Snow is white

Hence the related truth-conditional formula:

(2) 'Snow is white' is true if and only if snow is white.

As a separate proposal, not the only way of thinking of the semantics of language understanding is:

(3) 'Snow is white' means that snow is white.

I shall present considerations below to show that claim (3) too needs to be rejected. It is worth noting that the example originated with Tarski. But he used it only as an informal illustration of technical matters. He himself did not believe that the techniques he developed could be applied to natural languages (see Tarski, 1936). In the following I will bypass the problem of the paradoxes of truth that Tarski discovered, and will concentrate on other considerations that deal more specifically with features of natural languages as such.

I . NEGATIVE ARGUMENTS

Let us consider the constituents of the sentence: 'snow is white'. The subject expression 'snow' is a mass term. Thus its most natural interpretation would be in mereological terms (see Pelletier, 1979). The extension of 'snow' is, then, the sum of all entities that are parts of one scattered particular, snow. We need not worry about maximal parts. The semantics - if not what we know about physics - would allow snow to dominate all of reality. There are minimal parts, however, Not all parts of snow are themselves snow. If one uses some of the set-theoretical proposals that have been made about mass terms (Gabbay and Moravc- sik, 1973) one would still have to come up with a characterization of the extension that would coincide with what the mereological analysis yields; i.e. all of the snow in the world.

The differences between the mereological and set-theoretical approaches represent a conflict to which we alluded before. For those interested only in a representation of expressive power, uniformity of

IS SNOW WHITE? 73

technical representation is of paramount importance. Hence the insistence on set-theoretic representation, regardless of what it does or does not show about understanding. For those who think that any interesting analysis of natural languages should also have some relevance to how the mind understands language, uniformity is less important. It is more important to offer a representation that can be used to study how the human grasp of quantitative concepts differs from our grasp of concepts that contain qualitative individuating features.

Let us consider now the full extension of 'snow'. Even if we restrict ourselves to the actual world and not raise problems about the modalities, the obvious falseness of the sentence under consideration strikes us immediately. Of course not all snow is white. Much of it is grey, or even black, and some of it is colored in a variety of other ways. Yet, there must be a reason why to the initially unreflecting mind the sentence seems true. (Similar considerations apply to 'grass is green'; this too is obviously false when taken literally, since much grass is brown, yellow, etc., and yet in some ways we think that the sentence says something that is true.)

We begin making headway with this puzzle when we consider: what are the constituents of snow? There are the flakes before they fall to the ground, there are the flakes together as a snow cover on the ground, there is snow turning into slush, there is the melting snow, etc, Not all of these are white. Are we to restrict the extension to some privileged items among the ones mentioned? It seems odd to restrict snow to be just the falling flakes. The fallen snow is for many contexts the snow we most often talk about. But the snow on the ground is often not white. Furthermore even if we restricted the extension to falling flakes, it still would not be true that all snow is white. For on account of pollution in the atmosphere many of the flakes falling in different parts of the world are already black or grey before becoming visible to us.

One might also try to exclude melting snow and slush by talking only about dry snow. But this raises the question: how dry? Some snow is very good as material for snowballs, but would be regarded as not good for skiing. But even when listening to ski reports we hear about wet snow, and the announcer does not mean to confront us with a contradiction. These considerations show that the problem is not merely one of taking dry snow and snow melted into water, and ponder where on this continuum one wants to draw the line, and how arbitrary such a line would be. The truth of the matter is, rather, that for different uses

74 J. MORAVCSIK

and in different linguistic contexts items with different constituencies count as snow.

Perhaps the situation can be remedied by bringing in purity. Snow is white when it is pure snow. Thus we think of snow in the Alps or the Himalayas as pure snow, not affected by pollution, and think that the sentence under consideration applies to that. Other kinds of snow can then be called snow in derivative senses. Is this not like describing something under idealized conditions? Such an analogy fails on many counts. First, we do not attach meaning to words in a natural language with our view only on items in the extension that qualify only under idealized condition. Pure water in a scientific sense may be H20, but pure water in the ordinary sense is drinkable water, and thus water that contains several ingredients besides H20. Pure snow is more like pure potable water. Just like healthy grass is green in certain seasons, and pure water is drinkable, so pure snow is unpolluted snow, and such snow is white. But what if the source of whatever makes some flakes already in the sky grey or black were not to have human intervention as its cause? Snow flakes could be grey or black on account of changes in the atmosphere that have natural causes. So black snow need not be such because of pollution.

To say "this is not snow any more" and "this snow is not white" is not to say the same thing. Thus the statement that snow is white cannot be taken as expressing partly the nature of snow. At most it is an empirical generalization, which we are now rephrasing so as to be about pure snow. But since we do not know what the source of coloring is in all cases, and there are cases of snow falling already grey and black, we cannot even accept this statement as true.

One might say: "snow is white; in different contexts this may not be the same degree of whiteness. What we mean in different contexts is that the snow is white enough." Such thoughts bring us to the question: what counts as white?

'White' too is a mass term. In its extension, mereologically interpreted, we find the sum of all the white items that are parts of the one scattered particular that white is. Meaning in this case is difficult to define, since white is a simple basic perceptual quality. The O.E.D. seems to acknowledge this, since in addition to saying that it is produced by, e.g., reflection of light and is without hue, it adds that it is the color of snow and milk. One can safely assume that these are given

IS SNOW WHITE? 75

merely as examples that are paradigmatic in many ordinary contexts, rather than as substances that must exhibit the color in question,

More informative specifications of the meaning of 'white' rely on contrasts. White contrasts with yellow, grey, light brown, etc. Of course when we juxtapose these colors we must admit that the boundaries are not sharp, and thus a color word like 'white' suffers from vagueness. We shall leave, however, this point aside, because though Tarski's definition of a formal language, and hence of a language over which we define truth-conditions, cannot suffer from vagueness, various technical ways have been developed to handle this semantic phenomenon. In any case, this is not what seems to me essential to natural languages in such a way as to make them not suitable to be treated as a formal language.

Contrast-dependency by itself does not pose a threat to formalizations in terms of truth-conditions as long as the contrasts stay fixed. This, however, is not the case. In some contexts we contrast 'white' with 'yellow', and in others with, e.g., 'off-white'. In some cases 'white' contrasts with 'grey', and in others even the slightest deviation of whiteness from pure white to a shade of grey.

Furthermore, in different contexts we invoke different criteria as to what counts as white. E.g. a white shirt is white even if it has spots on it. In fact, this will be true even if the spots were already made in the factory. Again, how much white snow has to cover a landscape for us to say that the fields are white? Surely many patches of brown are permissible. Or for that matter, a white suit need not be pure white; it is a white suit as long as it contrasts with a light brown or yellow suit.

This last point shows that the variations in the extension of 'white' are not random, but depend on the variety of human interactions with the environment and the resulting utility of drawing the boundaries in many ways. We also draw the distinctions in different ways depending on the artifacts we produce. E.g., on US highways there are white and yellow lines. Everything counts as white that is not distinctively yellow, even if it is worn, dirty, etc. On the other hand, when a painter consults a customer on the color in which the walls of a room should be painted, every shade becomes crucial. So in the cases of shirts, walls, books, roads, etc. different extensions are involved. Once, however, we fix the appropriate context, the extension remains fixed also.

This variation of extension and its link to contexts created by the variety of human interactions with the environment that we find in the cases of 'white' can be seen also from the evidence we surveyed

76 J . MORAVCSIK

concerning 'snow'. In the case of a snowfall the exact shade of white seems irrelevant. In the case of a snow-storm we count also 'heavy', i.e. wet snow. In the case of "there is enough snow in the Sierras" as issued in the context of a ski report, only relatively dry snow counts. Here, too, the linguistic variety is derivable and predictable from the variety of possible interactions with the environment.

In view of these considerations we can come to appreciate the complexity of extensional variations in the uses of the compound NP 'white snow'. We need to consider which use of 'snow' is being utilized, and which use of 'white'. Yet this is not a case of typical lexical ambiguity. It is not like the case of 'bank', where two unrelated meanings happen to be attached to the same word. 'White snow' is not ambiguous, but is subject to a variety of predictable extensional variations.

Thus even if we regiment English by assigning separate words to different word meanings so as to avoid ambiguities like that of 'bank', we still have problems with the schema:

'Snow is white' is true (in L) if and only if snow is white.

For, e.g. the extension on the left side could be interpreted as covering all of the snow that is on the streets and is not blackened completely by traffic conditions and pollution, while the right side is taken to be applying only to snow that fell in the high mountains and other places away from the effects of human interactions. So the left side covers a lot more than the right side. Hence it is not true that the truth of the left side depends in all cases on the right side expressing what is the case.

11. FORMAL REPAIRS?

Formal semantics assumed, since Frege, that the paradigmatic case of a sentence with semantic analysis is one for which we can specify an intensional content that stays fixed as long as the language to which the sentence belongs does not change. Furthermore, this intensional content depends on the intensions of the parts, and it determines truth or falsity. But already Frege knew that this will not hold for certain cases such as those in which indexicals like 'here' or 'now' occur. Modern formal semantics invented a formal device, namely that of restricting interpretations according to indices that constrain the extensions of the parts of

IS SNOW WHITE? 77

the sentence, and hence the conditions of application for the whole sentence. This enables us to give formal representations of sentences with spatial, temporal, or speaker and hearer relative indices. This original list has been extended over the past decade to cover communicational factors and perceptual situational factors. Thus one might ask: why not add another index, call it the quality-dependent contextual index, and continue with formal semantics as before?

In spite of its surface attractiveness this proposal has several flaws. In order to place these into proper perspective let us reflect on what indices really are. Formally - one might say, mechanically - they simply limit or relativize truth to a frame of application within which variations in extension and truth do not arise. It is crucial, then, that all legitimate indices take what is intuitively a large but well defined extension and restrict, or constrain this. Furthermore, for most of the standard indices we can find individual lexical items to which these are linked.

The phenomenon that we uncovered is not linked to a specific sub- vocabulary of English. Practically the whole descriptive vocabulary is affected with the possible exception of the language of mathematics (Moravcsik, 1989). We saw, however, that not all recently introduced indices are linked to specific words. But there is another key difference between our multireferentiality and the other phenomena mentioned. The others are cases of restricting application. Not all spaces, not all speakers, count. But in our cases the various practical contexts create rather than restrict extension. It is not as if we could survey all of the possible future extension-creating contexts that our future and possible interactions with the world can create. Thus the intuitive idea that underlies the practice of relativizing truth is missing in our type of case. The expanding ranges of application and contexts cannot be described in a rigorous way, while this is possible in the cases in which the introduction of indices is legitimate.

All of this does not mean that we have a case of linguistic anarchy. We can describe in general terms what leads to the emergence of the variety of reference fixing contexts. We can also explain why in cases of natural languages this is needed. But the emerging contexts do not fit the.mould of taking space, time, perceptual information, etc. and then relativizing to specific parts of clearly described general notions. Thus the device in our case would be a mere ad hoc technical device without any explanatory power. This does not mean that there will be no precise rigorous treatments of semantics in natural languages,

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but that - as Tarski himself saw - these will not be centered on truth- conditions. If there are no conceptual constraints on relativizing truth to indices, then the device becomes an ad hoc way of treating anything, and the indices introduced end up as merely contents of a conceptual wastebasket. Within such a framework the claim that natural languages can be given truth-conditional analyses becomes a trivial remark instead of an interesting and informative remark that would lead, whether true or false, to an increase in our understanding of the nature of natural languages. Indexing should be a clearly constrained practice, to follow the dictum Dr. Suppes often enunciated in his seminars: "better to be clearly wrong than to be vaguely right."

111. FORMAL SEMANTICS AND THE MIND

Why should resolutions about questions concerning formalisms lead to interesting theses about language understanding and cognition? Let us consider what a natural language is. A natural language is a language that can be learned by a human or sufficiently humanlike creature as his or her first language under normal circumstances. This characterization, though not precise, allows us to view the question of whether a given invented 'artificial' language is or is not a natural language as an open empirical issue. The alternative of construing a natural language as one that we come upon in anthropological research and a nonnatural language as one that happened to be invented in an A.I. center is a theoretically uninteresting characterization. Thus, right from the start, the very notion of a natural language is linked to a characterization that involves cognitive capacities. It leaves open the question whether all natural languages have either in their syntax or in their semantics interesting common formal characteristics. Thus since the very notion of a natural language is a partly psychological concept it would be odd not to seek characterizations of the semantics that shed light on aspects of cognition.

Even if someone is interested only in representing expressive power, introducing arbitrarily indices is not illuminating. Furthermore, it is yet to be shown that apart from relations to cognition the class of natural languages constitutes in terms of purely formal properties a 'natural' class. Those of us interested in exploring the nature of natural languages, as defined above, will look for alternative approaches.

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The evidence considered so far shows that there will be a difference in levels between where we specify the meaning of what a sentence expresses and even in strictly qualitative terms the proposition that it expresses. Of course, we know that such a distinction is drawn in the case of standard indices of space, time, etc., but in this case we are drawing this distinction within the domain of the purely qualitative, and not in the realm of external factors such as speaker, spatial position, etc. Now ambiguity and the kind of multireferentiality that we explored are seen as blemishes in a language from the standard logical point of view. Let us, however, consider reasons for maintaining that for a natural language ambiguity and multireferentiality are advantages.

First, a natural language is a diachronic phenomenon. Thus optimally it should have the kind of semantic structure that facilitates changes in extension caused by conceptual changes rather than a formalism that posits rigidly different meanings for every change in extension. If every new conceptual wrinkle requires the introduction of new words, thinking and communication will be hindered, especially in situations in which meanings and concepts are fluid with no sharp criteria of application or clearly marked extensions.

This phenomenon can be illustrated by considering the type of case in which - either in science or in philosophy - we start with an ordinary expression like 'having a part of something', or 'force', turning it into a metaphor as we try to forge a technical concept, and eventually give the word a technical meaning. In such cases it is very helpful if we can look at layers of intensional content and can say: "this much remains constant, this is what is changing." In sum the diachronic and dynamic dimensions of natural languages demand that there be possibilities of gradual meaning and extension change. The multireferentiality illustrated above enables a language to fulfil this function.

Secondly, a natural language is often used for person-to-person communication. In many such contexts both speaker and hearer prefer that some indeterminacy of meaning remain in the semantics of the words used. For example, "you left the door open" uttered in the presence of people affected by noise, warmth, etc. in the same context as the speaker, can be a report, or a request, or a command to close the door. Furthermore, what counts as open depends on whether we are considering cold air coming in, or light seeping in when none is wanted, or whether we worry about being overheard. Both speaker and hearer

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might want a piece of language that can be interpreted flexibly with regard to these alternatives.

Thirdly, natural languages are learned at various stages of development. This learning is facilitated by words having many senses and by multireferentiality. For example, terms like 'father' and 'mother' are learned by a child first in purely social, functional ways (source of warmth, care, etc.) and with extension fixed for only conceptually simple and statistically dominant types of cases. Meanings are enriched later on, and thus more contexts are generated in which reference and extension need be fixed. Even at a mature stage we need to be able to project meaning in order to understand sentences like 'he was like a father to me'. Other examples requiring a flexible intensional content include words like 'sick' or 'healthy'. In these cases we first associate the word with observable circumstances and only later grasp the essential underlying conditions (typically unobservable) that govern application. This, in turn, enables us to understand varieties of new contexts in which the extensions of these words need to be fixed.

Reflection on the semantics of the vocabulary of a language like English shows that most of the lexical items are affected by the kind of multireferentiality that we have been considering. This is hardly surprising, since most of the vocabulary involves items that come to be used either in the context of humans interacting with the environment or in contexts in which humans interact with each other. Both of these types of contexts are flexible, and within them new contexts requiring the fixing of extension open up all the time. There is no a priori way of delineating for all times all of the ways in which humans will interact with the environment or with each other.

We cannot build into the meaning of a word all of the contexts in which extension has to be fixed (Moravcsik, 1990, Ch. 6). There will always be a 'slack' between what can be legislated now and what will - semantically - challenge us in the future. Who knows in which ways humans will interact with snow? Or in which ways the color white will assume great importance for some practical task? Who could have predicted the extensions of legal meanings that needed to be constructed in order to deal with laws required for activities in outer space? The same situation arises in the case of human interactions. Who could predict all the different ways in which the extension of 'family' is being fixed and will be fixed? Instead of manufacturing constantly new words that show

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no semantic relatedness to each other, we can rely on multireferentiality to ease transitions.

Before sketching the outlines of a semantlc theory that accounts for the layers and structures of meaning required, let us summarize what was shown about truth in this essay. Of course, it is agreed by all that sentences of English and other natural languages can be true or false. Our issue was whether these sentences have truth-conditions in Tarski's sense. What has been shown is that most English sentences can express a variety of truths, even when we consider only intensional content and ignore relativity to space, time, and speaker. Hence the mechanism that Tarski invented for formal languages will not be suitable for the dominant parts of natural languages. We need a semantics within which we can distinguish sentential meaning from the proposition expressed, even within the strictly intensional content. We need also a theory within which layers of meaning are distinguished, both for words and sentences, to represent the required structures.

IV. MEANING AND DENOTATION RECONSIDERED

One way of accounting for the salient facts uncovered with regard to meaning and reference would be to divorce meaning from reference. Such a view has been sketched in some writings of Putnam. Such an approach, however, seems to face difficulty in a theory such as ours that attempts to account for a variety of language use, other than purely contemplative and descriptive ones. Language is also used to give direction and guidance for action, and to formulate answers to questions involving considerations of denotation. The key example for language use combining meaning and denotation is that of explanation. We need intensionality to formulate explanations, and we need denotation guided by the intension to arrive at the explanadum.

One solution involves the following key theses: (i) meanings are explanatory schemata, giving salient necessary conditions for lexical application. (ii) There are three layers of semantic content, apart from standard indexicality. First, meaning specified as explanatory necessary conditions, then a variety of contexts within which denotation is to be fixed, and then the denotations and truth that emerge within the contexts specified on the second levels (Moravcsik, 1990, Ch. 6).

Explaining what something is has typically two stages. First, to give a general framework of how to locate the items under consideration

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on a conceptual map. We can do this, for example, also for the much discussed concept of a game. Then we focus on the area within which the items are that the investigation is focussing on. For example, ballgames practiced by many people in the U.S.A. We then distinguish within this area games such as baseball, football, volleyball, basketball, etc. This is clearly the explanation we need in everyday discourse, whether for the actual or potential player or the spectator. Neither of these constituencies demand that we distinguish a given game from all actual and possible games of all types; an impossible task in any case.

This two-stage process proposal corresponds to our semantic intuitions. Take, for example, the expression 'equal treatment'. One would first specify in general terms what equal treatment is (e.g. treating others regardless of differences in worth, status, character, and not placing them lower on a preference scale with respect to desired goods than others, including oneself). People may understand equal treatment on that level, and agree that it should be extended to a certain set of humans. But then we need to take the second semantic step, namely to come to an understanding and agreement as to what counts as equal treatment in this context.

Brief reflection should convince us that the meanings of words like 'walk', 'snow', 'white', and indeed all or most of the descriptive English vocabulary have structures that requires two-step application procedures like the one just outlined. This is because their uses reflect a variety of interactions between humans and between humans and the environment. This variety and the ensuing dynamics of denotational contexts must be reflected in the lexical semantics. This means that we should be able to 'read off' the ways in which contexts for denotation emerge within the framework of sentences within which these words occur. The aspects along which contexts emerge are predictable, the total emerging set is not. Let us now apply this theory to a typical sentence such as: 'the child walks'.

The meaning of 'child' is, roughly: "a human, with appropriate conditions of individuation, and persistence, less than full age as determined by the conditions governing various human interactions." This meaning requires generating contexts for fixing reference, because the extensions will be different for 'child labor', 'child vs. adult (as in legal matters)', or 'she is still a child', as used in discussions of maturity.

The same considerations apply to 'walk'. The meaning is: "locomotion with legs by putting one foot in front of the other with one foot on

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the ground at all times, covering appropriate ground." The criteria for how much ground need be covered vary depending on various types of agents; child in various senses, sick person, healthy adult, etc. At this stage, then, we have an incomplete proposition, roughly the meaning of: "human less than full appropriate age is moving legs one after another always touching ground with one, covering appropriate ground." This incomplete proposition constrains but does not fully determine extensions and truth. To arrive at those, we need to consider the different denotational contexts for both lexical items, and then see which combination is expressed in a text or by a speaker. At this stage only, do we have fixed denotations and thus truth or falsity.

The analysis of 'snow is white' has the same structure. The meanings of 'snow' and 'white' are specified in terms of necessary properties adding up to an explanatory scheme. Then, as we saw earlier, on the basis of the explanatory schemes extension fixing contexts are generated, depending on the variety of interactions between humans and between humans and the environment. For each of these contexts determinate extensions can be assigned, unless standard indexical relativization is needed as an additional level. The conclusion is: only some 'kinds' of snow have some 'kinds' of whiteness. The Tarski apparatus would require that we should be able to answer our original question: "is snow white?" with a determinate reply. But this we cannot do. Too many snows, too many whitenesses, too many combinations.

There is also psychological evidence for this way of cutting up the conceptual pie. For the kind of knowledge required to understand the meanings of 'the child walks', and 'snow is white' seems distinct from the one required for fixing extension within a given qualitatively prescribed context. The first kind of knowledge is just general comprehension of syntactic and semantic conditions. The second type involves knowing more detail about what can or cannot be expected from children in various contexts, how people interact with snow, etc. A person could be very good at comprehension on the first, more general level, and be bad at the second level. Conversely, some people can become good at denotation fixing once the general conditions are clarified for them.

This way of looking at comprehension has important consequences that should be utilized in educational projects. For example, people might come to understand that a good physician should care for his or her patients. It is, however, a second step to spell out all of the different

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contexts for caring, from bedside interactions to decisions reached in a hospital about allocating resources. Finally, it takes further training to help people understand what counts as caring in the various contexts. Even in very different fields we can detect an analogous distinction. For example, it is one thing to understand a mathematical claim in general, and a distinct matter to determine whether it is true or false. An adequate semantics should account for that. Of course, in that case, the various contexts will have more to do with types of proofs and calculations than with humans interacting with the physical environment.

In view of these considerations we can return to sentence (3) listed in the Introduction, and see that it is false. 'Snow is white' does not mean that snow is white, because without further context specification for extensions there is no proposition that snow is white. Rather, we need to grasp an incomplete proposition, look at the relevant lexical items, such as 'appropriate', that show why and how contexts for extension and truth need to be generated, and then move to the specification of complete propositions. At this point the task of linguistic comprehension is completed, and questions of verification take over. In this way, our analysis keeps truth, leaves meaning and extension linked, but without the apparatus of truth-conditions in Tarski's sense, and takes note of certain cognitive facts concerning language processing. This is as it should be, since as we saw the concept of a natural language is essentially tied to cognitive facts.

ACKNOWLEDGMENT

I wish to acknowledge many helpful suggestions from Professor Edgar Morscher.

Department of Philosophy, Stanford University, Stanford, CA 94305, U.S.A.

REFERENCES

Davidson, D.: 1967, 'Truth and Meaning', Synthese, 17, 304-333. Gabbay, D. and Moravcsik, J.: 1973, 'Sameness and Individuation', Journal of Philos-

ophy, 70,513-526.

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Montague, R.: 1974, Formal Philosophy, R. Thomason (Ed.), New Haven: Yale University Press.

Moravcsik, J.: 1989, 'Between Reference and Meaning', Midwest Studies in Philoso- phy, XIV, Notre Dame: University of Notre Dame Press, pp. 68-83.

Moravcsik, J.: 1990, Thought and Language, London: Routledge. Pelletier, F. J. (Ed.): 1979, Mass Terms: Some Philosophical Problems, Dordrecht:

Reidel. Tarski, A.: 1936, 'The Concept of Truth in Formalized Languages', reprinted in Tarski,

A.: 1956, Logic, Semantics, and Meta-Mathematics, Oxford: The Clarendon Press, pp. 152-278.


I agree with much of what Julius has to say about the highly context- dependent nature of the meaning of much natural language. I do want to remark in detail on various specific issues, and here and there map out some areas of disagreement between us. Still, much of what of he says is in agreement with the work Colleen Crangle and I have done on context-fixing semantics (for example, Suppes and Crangle, 1988).

Propositions. To my surprise, Julius in several instances stands by what is more or less the standard concept of proposition. Thus in Section 3 he talks about "the proposition that it expresses", referring to a sentence. It is exactly this pernicious tendency to think that a sentence expresses some unique proposition that I have argued strenuously against in my various papers on congruence of meaning. Just as in geometry there is no one notion of congruence, so in the theory of meaning there is no one fixed and definite concept of synonymy. Given the relativization of context of much else that he has to say, I suspect that this usage on the part of Julius is just a hangover from the bad old days when he might have believed in such fanciful objects as propositions.

Meanings as Explanatory Schemata. I like this general idea of Julius's but I find it hard to fill in the details of how we actually operate mentally with such schemata. Until we can think through how such schemata are actually used in either producing or comprehending language, which means a rather developed theory that separates meanings from syntactic considerations, it will be hard to get beyond the generalities.

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It is easy to start out by saying that model-theoretic semantics will not do and to turn to a discussion of procedural semantics to start to specify what one might mean by 'explanatory schemata'. Then one begins to think about all the mysteries of memory and perception as well as the puzzles about the mechanisms or computations for paraphrasing that are part of our language ability. The more one reflects on them, the deeper the mystery becomes. Even if we have a baptismal christening, the mysteries remain. It is a touchstone of the realism and power of theories of cognition to test them against their capabilities for dealing with the kind of problems just mentioned. As yet, what we have is unfortunately far from satisfactory.

Language Learning. Julius also discusses language learning but does not pursue the topic very far. I want to mention that the mysteries alluded to in the previous paragraph are present in extra strength when we turn to language learning. How does a child extend in a natural way what he has learned in 18 months to the vastly greater amount he knows and is able to handle at 36 months? How is the grammar and semantics put together in such a way that by the age of 5 years most of the important features of language he or she will command have been learned? There are now literally thousands of developmental psycholinguistic studies full of useful information about this process, but we are as yet very far from having anything like a satisfactory theory. One conclusion does seem to be true and that is that children put meaning ahead of syntax. There is also some kind of case to be made for their learning meanings only in very restricted contexts initially and then gradually widening out. This seems like a natural simplifying hypothesis but how the internal machinery is put together and is able to compute so wonderfully, either in terms of production or comprehension, is still very much out of reach. Recently I have been working on machine-learning of natural language. What success we have had has been by essentially assuming away most of the tough problems we must ultimately face in language learning. Even if we have begun with nothing about the grammar of the language assumed, we have not adequately dealt in our research on machine-learning with the simultaneous intertwining of the learning of language and concepts. One thing that is evident is that there is an endless amount of practice involved. By the time the child reaches kindergarten, the child has heard millions of sentences and has uttered a very large number. It takes a lot of practice to get any good at language.

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How are we to think about these mechanisms of practice? Obviously there is in learning a kind of automaticity, but our grasp of the details of these automatic or semiautomatic mechanisms of language handling are quite imperfect as yet.

REFERENCE

Suppes, P. and Crangle, C.: 1988, 'Context-Fixing Semantics for the Language of Action', in J. Moravcsik and G . Taylor (Eds.), Human Agency: Language, Duty and Value, Stanford: Stanford University Press, pp. 47-76, 288-290.

PAUL WEINGARTNER

CAN THERE BE REASONS FOR PUTTING LIMITATIONS ON

CLASSICAL LOGIC?

ABSTRACT. The question in the title is answered with a yes, albeit with a special proviso. In the first place the answer agrees with Pat Suppes' words "there are a number of reasons for moving very slowly to the adoption of a new logic." Secondly, the strategy proposed is not to move to a new logic but to put limitations or filters on Classical Logic when needed. This is based on the conviction that those questions where the application of classical logic leads to difficulties - like that of probability assignments to conjunctions, redundant elements of the consequence class, non-invariance of results against language frames - are beyond questions of validity (of semantics in the narrow sense) and therefore cannot be solved by changing classical validity. A more appropriate way seems therefore to leave classical logic (and validity) unchanged and distinguish it from new properties which are obtained by putting special limitations (filters) on it. The procedure is as follows: after classifying certain difficulties in Section 2 problems are given in detail in Section 3 which are special examples for these difficulties. In Section 4 limitative criteria for classical logic are offered which give solutions to the problems of Section 3. Moreover, special formal properties of these limitative criteria are discussed and some theorems about them are proved.

1. INTRODUCTION

It is a great honour and a great pleasure for me to contribute to the volume in honour of Patrick Suppes. I have profited a lot by reading Pat's books and articles and by several discussions with him. Pat's attitude towards logic is characterized by an important feature: logic has to be applicable to science and it has also to be taught in a way as to fulfil this purpose. In this respect his books Introduction to Logic and Axiomatic Set Theory are masterpieces. It certainly is in agreement with his view that logicians can learn from scientists. But what about the question of changing (classical) logic because of its inapplicability in certain fields of science or of paradoxical results when it is applied in certain areas when investigating the structure of science? His answer is very careful here (as the subsequent passage will show) and is neither a rough 'yes' nor a rough 'no'; i.e. there is no general permission that classical logic may be changed (weakened, restricted) in order to be

t? Humphreys (ed.), Patrick Suppes: Scientijic Philosopher, Vol. 3, 89-124. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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adequately applicable in some field of empirical science. Because the question of which changes can be made if some workable basic logic is to be preserved is very difficult. And, moreover, such a workable basic logic may not be the same in all applications. Therefore, any such move has to be done with great care, as Patrick Suppes emphasizes with regard to Quantum Logic:

There are, however, a number of reasons for moving very slowly to the adoption of a new logic, especially a logic that is clearly weaker than classical logic. Those who have rushed pell-mell into the advocacy of a non-classical logic for quantum mechanics of the kind just described have been guilty of passing rapidly through the zone of uncertainty to a new kind of certainty, namely, a certainty of not being able to make measurements at all. ' A more liberal view is taken by Dalla Chiara:

As a logician, I must say that I do not feel any particular allergy to a situation of mixture of logics. On the contrary, it seems to me that, generally, a form of plurality of logics cannot be avoided in modern ~ c i e n c e . ~

This view seems too liberal since deviation from classical logic has a lot of other consequences which seem intolerable in science. A special one concerning measurement is mentioned by Suppes above; the very general one underlying it was emphasized strongly by Popper on several occasions: using weaker logics weakens testability strategies in general.

For these reasons, and several others which are in some sense analogous, my general view is to preserve classical logic (usually First Order Predicate Logic) with its concept of validity (since it is complete: a sentence is valid iff it is provable) but make the changes as restrictions with some kind of filter (limitative criterion). Thus, with respect to Quantum Logic, the filter separates valid (derivable) and realizable conjunctions from valid and non-realizable conjunctions (in the case of events such as position and momentum). In regard to Relevance Logic the filter separates valid and relevant (non-redundant) formulas or derivations from valid and non-relevant (redundant) ones. Where Epistemic and Deontic Logic are concerned, the filter separates valid unparadoxical from valid paradoxical statements . . . etc. The main point of this view is to leave classical logic as a basis that is unchanged concerning its concept of validity but to restrict it by filters in an appropriate way when applied to certain fields of empirical science (natural sciences or humanities).

Observe that the program of Anderson-Belnap and their followers is in general to really change logic; i.e. to change the validity concept. The view defended here, however, is based on the conviction that

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questions like that of probability assignment to conjunctions (cf. Quan- tum Logic), redundant elements in the consequence class (cf. Relevant Logics), or invariance of logical and mathematical results against a language framework and coding (cf. 3.3.1 and 3.3.2) go beyond validity questions (i.e. beyond questions of semantics in the narrow sense) of First Order Logic. And therefore they cannot be solved by manipulating (weakening) validity in some way or other. Leaving validity unchanged and distinguishing it from additional properties like limitations for conjunctions, redundant parts, etc. which are handled by additional filters seems to be more appropriate.

Before coming to the question of what kinds of phenomena make up a threshold sufficiently strong for applying such a restrictive filter, I want to mention some examples of principles which we interpret as rather invariant against such changes (restrictions). One example is certainly the principle of non-contradiction if expressed in a tolerant way, i.e. as not implying bivalence: a proposition p and its negation non-p can never both be true (or have a designated value). Observe that this version is tolerant with respect to many-valued logic, since it holds also if p and non-p both have an indefinite (undefined) value. Another is the principle of logical consequence which says that logical deduction is truth-preserving, i.e. if all the premisses are true the conclusion must also be true and if the conclusion is false some of the premisses must be false.

I shall not continue with such general principles since the paper is concerned with more special problems and their solution. In Section 2 I shall give very roughly different reasons for putting limitations (filters) on Classical Logic. In Section 3 some important examples are given in detail which are classified into four groups. They show problems for which some restriction of Classical Logic seems suitable. Section 4 offers possible solutions for such restrictions (filters). Two different (simple) criteria, the A and K criteria, are discussed in detail and new (so far unpublished) theorems about them are proved. I want to mention here that the parts on formal properties of the A criterion (first interpretation) and the K criteria (4.122-4.123 and 4.212-4.216) are the result of joint work with Professor Andrzej Wronski (Krakow) and the proofs of Theorems 1-3 are due to him. Two further more general criteria are offered (Section 4.4) which seem more suitable as candidates for a limitation put on Classical Logic if it is applied to different areas. My feeling is that there probably will not be one criterion (filter) which can

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solve all the different problems but there might be a few which solve relatively many problems.

2. REASONS FOR RESTRICTING CLASSICAL LOGIC

The following facts seem to be stronger or weaker reasons for changing (restricting, limiting) logic in some way or other:

2.1. Classical logic, when applied to empirical states of affairs, dictates something which cannot be realized. For instance, if it dictates an evaluation of a conjunction whenever conjuncts are evaluated. This cannot be realized for certain quantum states and also not for certain epistemic states: knowledge or belief of certain propositions does not imply evaluation of all their conjunctions, let alone knowledge or belief of it (cf. 3.1.4).

2.2. Classical logic, when applied to the theory of explanation, confirmation, prediction, law-statements, disposition predicates, verisimilitude etc., leads inevitably to paradoxes which seem to have a common source in certain redundancies permitted by deductive inference.

2.3. Classical logic, when applied to epistemic states, value states and deontic states leads to paradoxes which seem to have essentially the same common source as those mentioned in 2.2 above.

2.4. Classical logic, when used for translation of one language framework into another (even when using logical equivalence) leads to paradoxical results.

3. EXAMPLES OF PROBLEMS

Subsequently I shall give some important examples which show in detail how the application of classical logic leads to difficulties and paradoxical situations. Since I do not take up the problems of Quantum Logic here (for some connections cf. 3.1.4 (1) and 4.1.4.8 below) the following examples will be taken from the areas mentioned in 2.2, 2.3 and 2.4 above.

PUTTING LIMITATIONS ON CLASSICAL LOGIC

3.1. Selected Examples in the Area of 2.2

3.1.1. Hesse 's Confirmation paradox3

Hesse defines deductive theory confirmation by the following three conditions: S confirms T iff ( I ) S is not tautologous, (2) T is consistent and (3) T I- S. Hesse adds the further assumption of transitivity of confirmation and then shows that the following paradox is derivable: every synthetic T is confirmed by every synthetic and true S. This result can also be obtained with the help of a weaker assumption than that of transitivity, i.e.: if A confirms T and A' is consistent, true and logically implies A, then also A' confirms T. The proof uses the validity of T t- T V S and S I- T V S. By the definition T V S confirms T and by strengthening the confirmans from T V S to S by the above criterion it follows that (any arbitrary true synthetic) S confirms any (synthetic) T.

As can easily be seen, the essential step in the proof which leads to the paradox is the valid principle of classical logic (called principle of addition) p I- p V q. The problematic point is that q may be any proposition (well-formed formula) whatsoever, even one not connected with the premisses. It will be seen subsequently that this principle is the culprit of a lot of difficulties in different areas (but not the principle of disjunctive syllogism: p V q, l q I- p which is ruled out by Anderson-. Belnap-style Entailment and Relevance systems).

3.1.2. Carnap 's Paradox of Disposition predicates4

Assume Carnap's definition of disposition predicates:

Dx :* (Vt)(Cxt + Rxt) .

The idea is that x has disposition D means that x has a certain property B if certain conditions C hold for x. The paradox results from the fact that ('v't)(Cxt -+ Bxt ) is derivable from ('v't)lCxt (i.e. from the fact that the conditions do not obtain).

The principle of classical logic responsible for the paradoxical consequence is a form of the ex falso quodlibet: proving any implication (or inference) from the negation of the antecedent (premiss). This amounts - by the Deduction Theorem - to having a contradiction in the antecedent (premiss).

94 PAUL WEINGARTNER

3.1.3. Goodman 's paradox5

Making use of the simpler version due to Hempel, this paradox can be stated thus: assume t f to be a future point of time such that t l . . . t , < t f . The hypothesis H = ('v'x, 'v't)(Fxt + Gxt) is then confirmed by the observational data B = {Fa l t l A G a l t l , . . . , Fa,,t, A Ga,t,). Assuming a pathological predicate G* defined as G*xt :- [( t 5 t f A Gxt ) v ( l t 5 t f A ~ G x t ) ] leads to the valid equivalence of B to B* = {Fa l t l A G*al t l , . . . , Fa,t, A G*a,t,). The paradox is that B* confirms H* = (Vx, 'v't)(Fxt + G*xt) which is inconsistent with H (since it predicts that the F s have to be non-Gs for all t > t ), while B (which is logically equivalent to B*) confirms H. On a closer look (just using the definiens of G*) one can see that one direction of the equivalence between B and B* contains a disjunctive weakening, i.e. an instance of the principle of addition:

3.1.4. The Paradox of Verisimilitude

According to an intuitive and very plausible idea of popper6 a theory A is nearer to the truth than a theory B iff the true consequences of A (AT, also called truth-content of A) exceed those of B ( B T ) and the false consequences of A (AF, also called the falsity content of A) are included in those of B ( B F ) . More accurately: A is nearer to the truth than B (A > T B) iff:

(i) BT C AT and AF 5 BF or (ii) BT C AT and AF c BF

Popper's aim was to show that even if we are confronted with two false theories - the normal situation in science according to him - we can choose the better one, the one nearer to the truth. Strangely enough, one can apply classical logic to prove that it follows from this definition that no two false theories can stand in this relation to each other.'

I shall give Tichy's proof (p. 156f.) which is very transparent: Assume A to be false. Proof part (i). Assume BT c AT. Hence for some true sentence p:

p E AT and p $ BT. Since A is false there is some false sentence f such that f E AF. Now consider the conjunction (p A f ) which is false.

PUTTING LIMITATIONS ON CLASSICAL LOGIC 95

Then (p A f ) E AF. On the other hand: (p A f ) $! BF (since otherwise p E BT, which contradicts the assumption). Therefore AF g BF7 and hence A bT B.

Proofpart (ii). Assume AF C BF. Hence for some false sentence q: q E BF and q # AF. Since A is false there is some false sentence f such that f E AF. Now consider the disjunction (1 f V q), which is true. Then (1 f V q) E BT. On the other hand (1 f V q) # AT (since otherwise q E AF - because of f E Cn(A) - which contradicts the assumption). Therefore BT g AT, and hence A )iT B .

Looking at the proof steps one can see two properties of the consequence relations (in classical logic).

(1) If proposition p, q, 1- . . . , etc., are elements of the consequence class of some theory A then also all the conjunctions built up from these propositions are elements of the consequence class of A. This is reasonable in the sense of the truth-functional relation, but not in the sense of criteria which are beyond questions of truth: a mathematician having proved theorems p, q, r . . . , etc., will not claim that by this he has also proved the further theorems p A q, p A r . . . , etc. And a computer is even forbidden by a restriction in the program to derive all the conjunctions of those separate theorems it has already proved; i.e. it '

is forbidden to do superfluous or redundant work, it is restricted in order to avoid redundancies. This point has also a bearing on Quantum Logic: evaluation of events p, q, r does not necessarily give an evaluation of all the possible conjunctions.

(2) If a proposition q is an element of the consequence class of A then so is any implication of the form r -+ q (or equivalently l r V q) or any disjunction q V r where r is completely arbitrary. Also, this property is rather trivial if judged by truth conditions but strange if we think of logical consequences drawn in science from assumptions or hypotheses. It is an important observation that scientists, if they speak of consequences of scientific theories, do have in mind something much more restricted than that what logic permits to be an element of the consequence class. Philosophers of science, on the other hand, when they describe what scientists do -explaining and confirming hypotheses, establishing laws, etc. - allow all the consequences which logic permits. My claim is that this is the main reason for most of the well-known paradoxes in the theory of explanation, confirmation, law statements, disposition predicates, etc. discussed by philosophers of science and

96 PAUL WEINGARTNER

logicians. It will be shown later that such paradoxes including the difficulty with verisimilitude can be solved in a straightforward way. Before showing this, however, it is important to concentrate on the kinds of proof steps and definitions which have been used.

3.2. Selected Examples from the Area of 2.3

3.2.1. Paradoxes in Epistemic Logic Have Dlflerent Sources

Insofar as they are connected with assumptions of the underlying system that are too strong, I have criticized them el~ewhere .~ Such assumptions are usually that of the modal system S4 can be interpreted as the epistemic operator 'knows that'. A simple consideration shows that this is mistaken: though logical consequences of necessary propositions are necessary, logical consequences of propositions which are known are not in general known too. The respective epistemic systems suffer from deductive infallibility and logical omniscience (another consequence of taking '0' as 'knows that'). And these are not properties of human knowledge. But there are also other difficulties, which are independent of stronger or weaker epistemic assumptions, since they obtain from propositional logic as soon as it is extended with an operator for 'a knows that' (attached to wffs of propositional logic): since -p -+ (p -+ q ) is a theorem we have immediately l p -+ (p --+ aKq) as a theorem of epistemic logic: if the Serbians do not stop the war then: if they stop it president Bush knows that he will win the election. This is a sentence which is true by logic. The culprit here is the ex falso quodlibet version, which is logically equivalent (by exportation) to the statement that anything follows from a contradiction.

3.2.2. Paradoxes in Deontic Logic

Such paradoxes usually arise in the following way. Most Deontic Logics satisfy the following three principles:

( I ) The theorems of classical two-valued propositional logic. (2) If a is a theorem of propositional logic then O a is a theorem. ( 'Oa'

for 'it is obligatory that a ' . 'a ' , 'P' represent arbitrary formulas). (3 ) I- O ( a + P) -+ ( O a -+ OD).

Assuming these three principles a number of paradoxes can be derived:

PUTTING LIMITATIONS ON CLASSICAL LOGIC

Ross paradox9

The paradox is that if it is obligatory that p obtains (say that pollution should be reduced) then it is obligatory that either p obtains or (an arbitrary) q obtains (say that water is poisoned by chemicals). An analogous paradox is obtained if '0' is replaced by 'P' ('permitted'). When we look at the proof we see again that the principle of addition, p I- p V q, plays the decisive role. This is remarkable because, as has been shown already in 3.1.1-3.1.4, this principle is the culprit not only in paradoxes of confirmation, disposition predicates, Goodman's paradox and verisimilitude (one part), but also in such a different area as that of deontic logic.

Other deontic paradoxes are those of Derived Obligation: O l p -+ 0 ( p -+ q), of Commitment: l p -+ (p -+ Oq) and of the Good Samaritan: O l p -+ 0- (p A q). lo The first and the second derive from a form of the ex falso quodlibet principle (cf. 3.2.1 above) the last from the principle of addition, which can be seen if ~ ( p A q) is transferred to l p V l q by DeMorgan's law. Instances in the above order are:

(1) If it is obligatory that something p is not the case (say that some nation will begin a war) then it is obligatory that if this nation '

begins a war then q (say that this nation completely destroys the other one).

(2) If p (say that a certain contract is sealed by two nations) is not the case then: if p is the case (i.e., if the contract is sealed) then (some arbitrary) q (say that these two nations commit war) is obligatory.

(3) If it is obligatory that war is avoided then it is obligatory that it is not the case that both: war obtains and the Red Cross helps the injured.

3.2.3. Paradoxes of Value Theory

Assuming the three principles, mentioned in 3.2.2 for value judgments and using 'WT' as a value operator applied to sentences of propositional logic one can derive analogous paradoxes. Its three main types can be obtained just by replacing the operator '0' ('it is obligatory that') in the above mentioned Paradoxes of Deontic Logic by 'WT' ('it is a value that'). Examples can be obtained by replacing '0' by 'WT' in the instances (1)-(3) above.

PAUL WEINGARTNER

3.3. Selected Examples from the Area 2.4

3.3.1. DeJinitions of Verisimilitude

These are not invariant with respect to logically equivalent languages. This curious result is prima facie independent of the paradoxical consequences of Popper's plausible definition of verisimilitude (discussed in 3.1.4). Moreover the non-invariance has been shown not only for Pop- per's definition but also for others, like Niniiluoto's. The first person who discovered this kind of language dependency was Miller." The language dependency can be shown as follows:

Let the first language L1 have the predicates Vl, PI and the second L2, the predicates V2, U2 where the following equivalences hold:

Assume that the true theory represented in L1 is Tl = Vl a Pl a and in L2: T2 = V2a A U2a. The respective false theories are the following:

Then the truth content of A and B and the falsity content of A and B are as follows:

Applying the definition of verisimilitude in 3.1.4 it follows: in L1 proposition A1 is nearer to the truth than B 1 . However, in L2 (which is logically equivalent to L1) the respective logically equivalent translation B2 (of B 1 ) is nearer to the truth than A2 (which is the translation of Al ) .

The same result of language dependency of definitions of verisimilitude can also be shown with more complicated hypotheses (many-place predicates and quantifiers and qualitative laws). A similar result can also be obtained for quantitative hypotheses.

PUITING LIMITATIONS ON CLASSICAL LOGIC 99

3.3.2. Language and Coding-Dependency of Some Special Results in Logic

In 1952 enk kin" posed the following problem: if C is any standard formal system adequate for recursive number theory, a formula G, (having a certain integer q as its Godel number) can be constructed which expresses the proposition that the formula with the Godel number q is provable in C. Is this formula provable or independent in C? Kreisel has shown14 that the answer to this question depends on which formula is used to express the notion of provability in C. He gives two different constructions of 'provable in C' and shows that, according to the first construction, the formula in question is provable in C, whereas it is independent according to the second.

Since it is a general desideratum to make results as independent of the language framework (the coding) as possible a natural question is this: is there an appropriate or designated way for constructing the notion 'provable', a kind of canonical representation which can be distinguished from others by reasonable conditions, such that results are invariant in respect to codings satisfying these conditions? This is not an easy question. Since, for instance, for changes from a full (usual) analysis to a cut-free analysis the following unexpected result holds: Godel sentences (one simple form is: "I am not provable") for cut-free analysis are just those for full analysis. But Henkin sentences (one simple form is: "I am provable") are all equivalent for full analysis but not all equivalent for cut-free analysis. And more curiously: all literal Henkin sentences for cut-free analysis are refutable in cut-free analysis.15 This is even a much stronger case, as being independent, as in the result above.

For such cases the proposed limitations in Section 4 do not offer a solution.

In the following case of coding dependency a method was discovered to make the result again invariant.

Cain and Damnjanovic have shown16 that, for a language of arithmetic plus a one-place predicate T ( x ) (partially interpreted), the existence of non-paradoxical ungrounded sentences - if the weak Kleene (3-valued) logic is the base - depends on the way the Godel numbering is chosen. In fact many more things depend on it: the number of fixed points, definability of sets in the minimal fixed point, etc. On the other hand, on the strong Kleene (3-valued) logic the results are invariant with respect to different Godel numberings: there exist sentences that

100 PAUL WEINGARTNER

are neither paradoxical nor grounded. There are 2N() fixed points, etc. 'Paradoxical' and 'grounded' are understood in the sense of Kripke. l 7

The weak Kleene (3-valued) logic (values: T (true), U (undefined), F (false)) has classical outputs whenever there are only classical (T, F ) inputs, but outputs U whenever at least one input is U. The strong Kleene (3-valued) logic has the maximal truth values for V, the minimal ones for f\ and p -+ q is defined by l p V q . In both, the negation of U is l i . . 1 8

In order to receive invariance in respect to the choice of Godel numbering (when the weak Kleene logic is taken) the authors considered finding conditions for a 'natural' or 'designated' Godel numbering but abandoned this since a particular Godel numbering seems much like a particular coordinate system. However, the authors showed that by extending the language in a suitable way the results become the same as on the basis of the strong Kleene logic and are invariant with respect to Godel numbering. The slight enrichment of the language consists in including a function symbol for a particular primitive recursive function.

Thus in this case to obtain invariance, enriching the relatively restricted language based on the weak Kleene (3-valued) logic was sufficient. It will be seen subsequently that for solving the problems of the paradoxes shown in examples 3.1.1-3.1.4, 3.2.1-3.2.3 and that of the language dependence of verisimilitude (3.3.1) restricting the relatively tolerant language of classical logic with its rich consequence class is sufficient.

4. STRATEGIES FOR SOLUTIONS

4.1. The Aristotelean Idea for Applied Logic

Aristotles' syllogistics was designed to be applied in science (and philosophy) and in everyday argumentation. Compared to First Order Pred- icate Logic it has more than one restriction: one is that it is monadic, which is in general a disadvantage, although it does give decidability. Another is that all the arguments have exactly two premisses and one conclusion - again a disadvantageous limitation though profitable to transparent argumentation. The one which interests us is the following: the conclusion must not contain predicates which do not already occur in one of the premisses. More accurately, the conclusion contains only subject-term and predicate-term, whereas the premisses contain in addition the middle-term which is eliminated by the syllogistic infer-

PUlTING LlMITATIONS ON CLASSICAL LOGIC 101

ence. This property of syllogistics might have been the reason why Kant called logic (syllogistics for him: he was unaware of the developments in Scholastics and in Leibniz) analytic, i.e., the conclusion is not richer than the premisses. If we apply this idea to Propositional Logic we may say that the conclusion must not contain propositional variables which do not already occur in the premisses. This property I have called A- Relevance (where for the expression 'relevance' one could take also the broader notions 'restriction', 'limitation').

Before I show that this Aristotelean limitation already solves a lot of problems discussed and described with the examples in Section 3, I want to discuss more formal properties of the Aristotelean limitation applied to Propositional Logic.

4.1.1. Terminology

By IF we denote the set of all formulas built up in the usual way by means of propositional variables from an infinite set V and the well- known connectives A (conjunction), V (disjunction), + (implication) and 1 (negation). We write a, D,?, . . . for formulas and p, q, r , . . . for propositional variables. The notion of a subformula as well as that of an occurrence of a subformula in a formula are as usual.19 By V ( a ) we denote the set of all propositional variables occurring in a and by cv[p : P] the formula which results by substituting P in place of each occurrence of the variable p in a . The letter C is reserved for the familiar 2-element matrix of classical propositional logic. If M is an arbitrary matrix similar to IF and a E IF then we say that a is valid in M , symbolically M + a, iff every valuation of propositional variables in M gives the formula a a distinguished (designated) value. If C a, then we say simply that a is valid and we abbreviate 'not M /= a' by MY a .

Since the Aristotelean limitative idea is concerned only with inference it will be natural to restrict its applicability to valid implicational formulas (of Propositional Calculus) of the form a + P, where a , ,O E IF. From this understanding it follows that the filter (restriction) is posed only on the main connective + but not on other arrows (4) which might occur in a or in ,O (or in both). If, however, the filter (restriction) is put on all arrows (in all the subformulas) the resulting system will be different. This means that we will get two different systems out of Classical Propositional Calculus if we apply the Aristotelean limitation either only to the main connective or to all the arrows of a formula


of the form a -+ p. The first application, though, seems to me to be closer to the Aristotelean idea, but both systems will be seen having interesting properties. The second is closer to the ideas of those logics which aim at strengthening implication like those logics of Relevance which originate from Anderson and Belnap. Since the term 'relevance' is used with different meanings I shall speak of the A-criterion and of A-restricted (first interpretation of Aristotles' imitation)^' and of the A*-criterion and A*-restricted (second interpretation).

4.1.2. Formal Properties of the Aristotelean Limitation: First Interpretation: A -Criterion

DEFINITION 1. A formula a + P is A-restricted iff a - P is valid (in Classical Propositional Calculus) and V ( P ) & V ( a ) .

Examples. A-restricted valid formulas are: p --+ p, p + lip, laws of commutation, association, distribution, DeMorgan, modus ponens, modus tollens, disjunctive syllogism, hypothetical syllogism in the form: [(p -+ q) A (q -+ r ) ] -+ (p -+ r ) , ( p A q) + p (simplification), (p A P) * P, [(p V q) + r ] -+ (P + 4, [P V (P A q)] + p, (P ' 4) * ('Q --+ 'P), [(P + r) A (q -+ 791 -+ [(P V q) - TI, [(P + q) ' p] - p, . . . , etc.

Valid formulas which are not A-restricted are: (p A l p ) + q, -I, - (1, + Q), Q -+ (P -+ q), P + (P V q), P - [(P A q) V (P A T ) ] , P + (q - (11. (P A TP) -+

(Q A l q ) , [P + (q A l q ) ] * (P ' Q) + [(P A r) + ql, ( P -+ q) + [(P A r ) + (q A 41, . . - 9

etc.

4.1.2.1. Important general properties are the following: (I) Closure under Substitution: it is easily seen that the set of all implicational formulas obeying the A-criterion is closed under substitution: the essential point here is to remember condition V(P) & V ( a ) , which says that the set of propositional variables occurring in the consequence part of an implicational formula is a subset of the set of propositional variables occurring in the antecedent part. So if we substitute any (arbitrarily complicated) formula a for a propositional variable p occurring in the consequent part we have to substitute the same formula a for the occurrences of the same variable in the antecedent-part. From this it is seen that by substitution condition V(P) & V ( a ) cannot be violated.

PU'ITING LIMITATIONS ON CLASSICAL LOGIC 103

Since the A-criterion is closed under substitution we can apply the well-known method of Adolf Lindenbaum to produce an infinite matrix M such that for every a , /? E IF', M a -+ ,8 iff a -+ ,8 is A-restricted.

(2) Closure under Transitivity of Implication: A-restricted implication is closed under transitivity, i.e., if a -+ ,8 and /? --+ y are both A-restricted, then so is a -+ y.

(3) No Closure under Modus Ponens: the set of A-restricted formulas is not closed under Modus Ponens. This can be seen from a simple counterexample: though ( p A q ) -+ p and ( ( p A q ) -+ p ) -+ ( l p -+

1 ( p A q ) ) are both A-restricted, l p -+ - ( p A q ) is not. This is important since one important difference between the A-criterion and the A*- criterion is that only the latter is closed under Modus Ponens.

(4) From the counterexample concerning Modus Ponens closure one can see immediately that the A-criterion is also not closed under contraposition.

( 5 ) A-restricted implication is not preserving A-restriction. This is seen from the example in (3).

(6) If a/ -+ /? is A-restricted then all four cases are possible concerning the A-restriction of a and p (provided a and /? are implicational formulas). Both, neither or none of both may be A-restricted.

In 1 9 8 5 ~ ~ I found a matrix semantics which characterizes the set of all valid formulas of Classical Propositional Calculus which obey the A-criterion. But at that time I did not realize that the matrices which define the negation and the connectives A, V, -+ and the restrictive (relevant) iA (for short, matrix semantics Mw) can in fact be used for describing both systems, the one which results from applying the A-criterion and the other which results from applying the A*-criterion. For applying the A-criterion -+* (the matrix of it) must only be used for the main connective whereas --+ (the matrix of it) has to be used in all other cases (implicational subformulas). This was also the intended system in that essay. For applying the A*-criterion (cf. 4.1.3) every implication in a formula has to be interpreted with iA. The difference was first described by ~ c h u r z . ~ ~

The following proofs show some general features of A-restriction (A-relevance).

4.1.2.2. Representation by a jinite matrix. Above it was shown that a representation of the set of A-restricted formulas is possible with infinite


matrices. A concrete example of this is given in my (1985). However, one can show that a characterization with finite matrices is possible, too.

Let a , b, c be pairwise distinct objects and let A be the matrix in which { a , b, c) is the base set, {a, b) is the distinguished subset and functions corresponding to connectives are given in the following tables: ~~~ b b b b b b b b b b

b b b b b b

L E M M A . For every y E F - V, A F y iff there exists a, P E F such that a+ 4 = y and V(4) V(a.

Prooj Suppose first that a, P E F and p E V(4) - V(a). We shall prove that A F a -+ 4. Indeed, take a valuation v which gives the value c to p and the value a to all remaining variables. Then v(a) = n and v(4) E {b, c) which implies that v(a -+ 4) = c.

Suppose now that y E IF - V and u is a refuting valuation for y in A. Then y must be an implicational formula and thus y = a -+ P for some a, p E IF. Since v(a -+ 4) = c then v(a) = a and it follows that v(p) = a for every p E V(cu). Hence the inclusion V(P) V(a) would yield v(P) = a and finally v(a -+ P) = a, which is a contradiction.

4.1.2.3. Further Result on the A-Criterion. Next let us form the Carte- sian product A x C . Recall that the base set of the matrix A x C consists of all ordered pairs (x, y) where x E {a, b, c) and y E {0,1}, the distinguished (designated) subset is {(a, 1)) (b, 1)) and the functions corresponding to connectives are defined componentwise, i.e.

We are now in a position to prove

THEOREM 1 . For every a, ,B E IF, A x C a -+ P i f f a -+ P is A-restricted (A-relevant) (i.e. iffV(P) & V(a).)


ProoJ: Follows directly from the Lemma and the well-known theorem of Stanislaw JaSkowski according to which A x C k a -+ P iff A k a - , p a n d C k a - , p .

Remark. The argument of the proof of Theorem 1 can be used to establish the following general result:

If M is an arbitrary matrix similar to IF then for every a, /3 E IF, A x M k a - piff M k a --+ p a n d V ( P ) V(cu).

4.1.3. Formal Properties of the Aristotelean Limitation: Second Interpretation: A* -Criterion

DEFINITION 2. (A*) A formula a -+ ,O is A*-restricted (or: A*- relevant) iff a -, p is A-restricted (A-relevant) and every implicational subformula of it is A-restricted (A-relevant).

The set of all valid formulas of Classical Propositional Logic which obey the A*-criterion can also be described by the matrix semantics M~~~ mentioned in 4.1.2. But in this case all arrows in a formula have to be interpreted by -+* (whereas for the A-criterion only the main connective -+ of a formula has to be interpreted by -,*). As mentioned, the two resulting systems are entirely different. Here the second system resulting from putting the A*-criterion on Propositional Calculus will be briefly discussed.

(1) Schurz has proved24 that the matrix Mw interpreted according to Definition 2, i.e. characterizing the A*-restricted Propositional Cal- culus is a special matrix of the matrix class proposed by M. ~ u n n ~ ~ to reconstruct Parry's axiom system26 for analytic implication.

(2) From (I) and Dunn's completeness proof it follows further that the class of all A*-restricted (A*-relevant) formulas is axiomatizable by Dunn's axiom system (1972).

(3) Modus Ponens Closure: the set of A*-restricted (A*-relevant) formulas is closed under Modus Ponens. This was not the case, as we saw (cf. 4.1.2.1 (3)) for the set of A-restricted formulas.

(4) The A*-criterion also satisfies the two other closure conditions: closure under substitutivity and closure under transitivity of implication.


(5) A*-restricted (A*-relevant) implication preserves A*-restriction (A*-relevance).

(6) If a -+ p is A*-restricted (A*-relevant) then provided cr and P are implicational formulas they are both A*-restricted (A* -relevant).

(7) A*-restriction (A*-relevance) is not closed under contraposition. Thus though (p A q ) -+ p is A*-restricted, l p -+ ~ ( p q ) is not.

4.1.4. Application of the Aristotelean Limitation of Classical Logic to the Problems

It is easy to show that all problems (paradoxes) which are stated in the Examples 3.1.1-3.3.1 - with the partial exception of the one of verisimilitude (3.1.4) - can be solved (avoided) by applying the Aristotelean Limitation of Classical Logic (A-criteria, i.e. A or A*) to it. This can be seen more accurately as follows.

4.1.4.1. Hesse 's Conjirmation Puradox. The proof uses the principle of addition p -+ (p V q ) which (though valid) does not satisfy either the A- or the A*-criterion.

4.1.4.2. Carnap's paradox concerning his definition of the disposition predicate uses the (valid) principle l p -+ (p -+ q ) ( ' l p ' for not obtaining the antecedents) which also does not satisfy the ~ - c r i t e r i a . ~ ~

4.1.4.3. In Goodman 5 paradox one essential step in the proof (recall 3.1.3) again uses the principle of addition (as in 4.1.4.1) which does not satisfy the A-criteria.

4.1.4.4. The epistemic paradox of 3.2.1 uses the same principle as in 4.1.4.2. So do two of the deontic paradoxes listed in 3.2.2.

4.1.4.5. The Ross Paradox again uses the principle of addition (a very frequent culprit in different paradoxes) and the remaining deontic paradox listed in 3.2.2 uses the valid though not A-restricted principle l p -+ ~ ( p A q ) which, by DeMorgan, is equivalent to the principle of addition.


4.1.4.6. Verisimilitude: The second part of Tichy 's inadequacy proof of Popper's original definition (given in 3.1.4) can be handled by the A-criteria (A or A*). The proof step is from q to f -+ q or from q to 1 f V q; i.e. from arguing that if q is a consequence of some hypothesis h then so is f -+ q (or 7 f V q). But this amounts to classically valid principles p + (q -+ p) and p -+ ( l q V p) neither of which obey the A-criteria.

For the first part of Tichy's proof a further restriction has to be made which accepts as (restricted, relevant) consequences only the smallest consequence-elements but not in addition everything what can be built up of them by the connectives, even not by A. That Popper's definition of verisimilitude can be rehabilitated if both a stronger restriction than the A-criteria and a splitting up in consequence elements is used has been shown elsewhere.28

4.1.4.7. But the A-criteria (A and A*) can also be used to avoid the difficulty described in Example 3.3.1 (the definition of verisimilitude is not invariant against transformation into logically equivalent languages). The essential point is this: in the construction of equivalent languages L1 and L2, predicates are defined by disjunctions of other predicates. Such a construction seems rather artificial: in the sciences predicates (primitive or defined) are never split up into disjunctive predicates. Also in the definitions of predicates with the help of primitive predicates conjunctions and intersections are used rather than disjunctions. Since moreover the newly introduced predicates that are constructed by splitting into disjunctions are arbitrary, the whole construction obtains its validity from the principle of addition P x -+ ( P x V Qx) or from a similar principle: P x -+ (Vx -+ P x ) . This latter is an intrinsic part of the definitions Vl - 1/2 and U2 ++ (Vl +-+ P I ) and Pl - (V2 ++ U2) (cf. 3.3.1). Both principles are ruled out by the A-criteria as non- restricted (non-relevant).

4.1.4.8. This leads to mentioning explicitly an important point which was already implicitly clear from the properties of the A and A*-criterion: both criteria are not invariant against logically equivalent transformations.


This reminds us of an important fact. Transformations in science are usually more refined than logically equivalent transformations; 1.e. they go beyond questions of validity or guarantee of the same truth value.

The following example has to do with Quantum Logic. There the valid equivalence of classical logic

A ++ [(A A B) V (AA -B)]

plays a decisive role. According to Mittelstaedt it leads, with a probability interpretation, to contradictions in quantum mechanics; i.e. only p(A) 2 p(A A B) + p(A A -B) ('p' for 'probability') can hold and also only a weak implication from right to the left instead of the classical equivalence above. 29

Now it is interesting that the above equivalence is ruled out by the A- criteria because the implication from left to right is not A-(A*)- restricted; i.e. the A-criteria permit only that direction which is permitted by Quantum Logic.

Furthermore, if the conditions for commensurability are expressed by the following definitions

the definiens parts which are logically valid are not A-(A*-)restricted. This again shows that the A-restrictions are in accordance with restrictions needed in Quantum ~ o ~ i c . ~ '

I have some further results in this direction but the field seems worth further investigation^.^ l a

4.2. The Limitation Proposed by Korner

Korner's original proposals3'b were not consistent and led to consequences which he certainly did not intend. The main point was that he did not distinguish between a subformula and the single occurrence of that subformula. The idea behind this is not difficult to grasp and was first formulated by

What is the main idea in Korner's limitation? It is the idea that if we can break out a part of a valid formula, exchange it by its negation (by an arbitrary part), put it in and the formula remains still valid, then this part must be inessential, redundant, irrelevant with respect to that formula.


DEFINITION 3. (K) A formula a is K-restricted (K-relevant) iff n/ is valid and no single occurrence of a subformula in a can be replaced by its own negation salva valididate of a .

As can be seen, the K-criterion can be applied to arbitrary formulas. Moreover one can generalize the criterion by replacing 'by its negation' by 'by an arbitrary formula'.

Examples. The valid formulas p -+ p, l ( p ~ l p ) , p v l p , p - 1-p, laws of commutation, association, distribution, DeMorgan, modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, . . . , etc. are K-restricted (K-relevant).

The following valid formulas are not K-restricted: (p A q) -4 p (simplification), p + (p V q) (addition), (p A l p ) -+ q (ex falso quodlibet), (I ' (P + (I), 'P + (P -+ 4) . (P ' Q) " [ ( P A T ) + ql, (P + q) V (q -+ PI, [ (P V q) -+ rl ' (P + 4, P ' (p -+ p) . . etc.

4.2.1. Formal Properties of Korner 5 Limitation

4.2.1.1. Many observations about the K-criterion have already been described by Cleave: ( I ) Every valid subformula of a K-restricted formula is K-restricted. (2) If a -+ ,B is K-restricted (for short: a + K /3) then none of a, p, l a , 14 are valid. (3 ) a A ,O is K-restricted iff a and 4 are such. (4 ) If a +K ,8 and a iK y then a iK ( p A y). If a iK p and y iK 4 then ( a V y) -+K p. a iK p iff 14 +K 10.

( 5 ) K-restriction is preserved under commutation, association, double negation, and DeMorgan's

4.2.1.2. All the facts above can be easily derived from the following statement characterizing the paradigm of non-K-restriction (non-K- relevance):

THEOREM 2. A valid formula a E IF is not K-restricted ifthere exist 4 , y E IF and p E V such that a = P[p : y] , 4 is valid and p occurs in p exactly once.

Proof: Suppose that a formula a E IF is valid and not K-restricted. Let y be a subformula of ct whose certain occurrence in a can be replaced by l y salva validitate of a (such y exists by (K)). Let 4 be the formula


which results from a by removing the subformula y at that particular occurrence and putting in its place a new variable p which does not occur in a. Then a = P [ p : y] and p occurs in P exactly once. The fact that p is valid follows immediately from the observation that so are P [ p : y] , P [ p : ly] and P [ p : y] -+ ( P [ p : 1 y] -+ P ) (the last formula is valid for arbitrary P E F which can easily be proved by induction).

Now suppose that P, y E IF, p E V , a = P [ p : y], /3 is valid and p occurs in /3 exactly once. Consider the occurrence of y in a which corresponds to the unique occurrence of p in P. Then y can be replaced by ly at this particular occurrence salva validitate of a. Indeed, since p occurs in ,B exactly once then the above replacement results in P [ p : ly] which is a substitution instance of a valid formula P. This shows that a does not obey (K).

4.2.1.3. It is worth noting that Theorem 2 implies the following:

COROLLARY 1. The notion of K-restriction remains unchanged if instead of K one takes the following condition, which is apparently weaker:

DEFINITION 4. (K') No single occurrence of a propositional variable in a can be replaced by its own negation (by an arbitrary propositional variable) salva validitate of a.

Pro05 Suppose that a valid a E IF fails to obey (K). Then by The- orem 2 one can find $, y E F and p € V such that a = P [ p : y], P is valid and p occurs in P exactly once. Pick a concrete occurrence of a variable q in y and consider its corresponding occurrence in a which lays within that particular occurrence of y in a which corresponds to the unique occurrence of p in p. Then replacing q by l q in a at the considered occurrence one gets a formula being a substitution instance of ,6 which in turn is assumed to be valid. This shows that a, does not obey (K').

Warning. The fact that we are working with the language IF, which does not contain propositional constants, is essential for the above proof. The presence of constants in the language would have made it impossible to infer (K) from ( K ' ) . ~ ~


4.2.1.4. The second corollary to Theorem 2, which in itself is rather trivial, has some unexpected consequences.

COROLLARY 2. The set of valid formulas from IF which do not obey (K) is closed under substitution, but for K-restricted formulas this is not the case.

Pro05 The first claim follows directly from Theorem 2 and the second by observing that for example ( p V l q ) -+ ( l q V p ) is K-restricted but ( p v l p ) + ( l p V p ) is not.

In view of Corollary 2 it is impossible to grasp the concept of K- restriction by means of logical matrices. Strangely enough, we can apply Lindenbaum's technique to produce an infinite matrix M such that for every a E F, M + a iff a is valid and not K-restricted. How- ever, we have:

4.2.1.5. THEOREM 3. Let M be a matrix such that for every a E IF, M a iff a is valid and not K-restricted. Then the number of elements of M cannot be finite.

Prooj Take arbitrary n _> 2 and pairwise distinct variables po, . . . , p,. For every 2, j E 10, . . . , n), i # j we put yij = (p i V l p j ) -+

( i p J V pi) and we define yn as the left-associated conjunction of all y q standing in an arbitrary but fixed succession. It is easy to check that:

(i) yn as well as all yij are K-restricted.

Let us say that a substitution E is fine if & ( p i ) = ~ ( p j ) for some i, j E 10, . . . , n}, i # j . Then we have:

(ii) ~ ( 7 , ) is not K-restricted whenever E is a fine substitution.

Indeed, if i, j E {0, . . . , n) , i # j , a E IF and & ( p i ) = € ( p i ) = a then ~ ( 7 , ) cannot be K-restricted because it contains c ( y i j = (a v l a ) -+

( l a V a ) as a subformula. Suppose now that we are given a matrix M such that M + a

whenever a is valid and not K-restricted. We shall prove the following:

(iii) If M F yn then M has more than n elements.

Indeed, suppose that v is a refuting valuation of y , in a matrix M whose number of elements does not exceed n. Then for some i, j E (0, . . . , n},


i # j we must have v ( p i ) = v ( p j ) . Define a fine substitution E by putting & ( p i ) = ~ ( p j ) = pi and ~ ( q ) = q for every q E V - { p i , pj } . Now for every q E V , v ( q ) = , u ( ~ ( q ) ) and thus we can easily show by induction that for every a E F, v ( a ) = v ( ~ ( a ) ) . Consequently v ( Y ~ ) = v ( E ( T ~ ) ) and SO M F ~ ( 7 , ~ ) which contradicts (ii). Having proved (iii) one can now infer the desired result by combining it with (0.

4.2.1.6. Finally observe that K-restriction has the following further properties: (1) K-restricted implication is not transitive (already shown by Cleave). Observe that [ ( p A q) v ( p A r ) ] -+ [ p A ( q v r ) ] and [ p A ( q v r )] -+ [ ( p A q ) v r ] are both K-restricted but [ ( p A q ) v ( p A r )] -+

[ ( p A q ) V r ] is not, because the second occurrence of p can be replaced by l p salva validitate.

(2) K-restriction is closed under modus ponens, but trivially so, because a and a -+ b cannot be both K-restricted. This follows from 4.2.1.1 (2), since if a is not valid it cannot be K-restricted (or K- restriction is not applicable).

(3) K-restriction does not imply A- or A* -restriction, for p -+ [PA ( q v l q ) ] is K-restricted but not A-(A*)-restricted. Of course A-restriction does not imply K-restriction. Thus ( p A q ) -+ p is A-(A*-)restricted but not K-restricted.

4.2.2. Application of the Korner Limitation of Classical Logic to the Problems 3.1.1-3.3.1

As in 4.1.4 it is also easy to show that all problems (paradoxes) stated in the examples 3.1.1-3.3.1 (with a partial exception of the one of verisimilitude) can be avoided by applying the Korner limitation. This is seen as follows:

4.2.2.1 As with the A-criteria (A and A*) the valid principles which lead to paradoxical consequences in 4.1.4.1-4.1.4.5 and the second part of Tichy's inadequacy proof (4.1.4.6) do not satisfy the K-criterion either. Also the principles mentioned in 4.1.4.7 are not K-restricted.

4.2.2.2. Also both K-criteria (K and K') are not invariant against logically equivalent transformations. But in respect to the equivalence in


4.1.4.8 the K-criteria rule out the implication from right to left (just the opposite of the A-criteria and not in accordance with Quantum Logic).

4.3. Shortcomings of the A- and K-Criteria

(I) The criteria cannot be applied to arbitrary formulas. (2) Formulas which have been called paradoxical frequently are not

ruled out such as: ( p -. q ) V ( q -+ p ) , ( p A q ) -+ ( p -+ q ) . (3) By adding a tautology in the antecedent with appropriate variables

any implication can be made A-(A*-)restricted.

4.3.2. K-Criteria

( I ) The K-idea, replacing a component (single occurrence of a subformula) by its negation salva validitate does not specify where the replacement takes place (or should take place). This means that also in the antecedent which corresponds to the premisses restriction is required by the criteria. On the one hand this leads to further distinctions between premiss restriction (relevance) and conclusion restriction (relevance). However, it is my experience with applied logic that what is really needed in the application of logic to science is conclusion restriction or conclusion relevance. That means one should not violate the common understanding (also in science) that premisses are allowed to be richer (contain more information) than the conclusion. But exactly this is violated to some extend by the K-criteria. A simple consequence is the instance that (p A q ) -+ p is not K-restricted since q can be replaced by l q salva validitate.

(2) Formulas which have been called paradoxical frequently are not ruled out like: p -+ [p A ( q V l q ) ] , in general: many instances of adding tautologies with A in the conclusion.

4.4. Combinations of A- and K-criteria

Because of the shortcomings mentioned above my colleague Schurz and I have developed several criteria which have the merits of both A- and K-criteria but not their shortcomings. Not so many formal properties are known for these new criteria compared to the theorems stated for the elementary criteria A and K.


A most suitable combination of A- and K-restrictions was proposed in Schurz and Weingartner (1987). With this criterion (abbreviated Rl) and an additional restriction in respect to smallest consequence elements we showed that Popper's original idea of verisimilitude can be rehabilitated (where in Popper's definition has to be replaced by -I (inverse inference), since the class of restricted (relevant) consequence elements is not necessarily deductively closed). Further it is shown that the R l -criterion can be adequately applied to simple physical theories like that of the ideal gas law. Moreover, R l is applicable to both Propositional and Predicate Logic.

The R1 -criterion does not apply to arbitrary formulas but to valid inferences and valid implications! formulas.

DEFINITION 5. I- a is Rl-restricted (relevant) iff it is valid and there is no propositional variable (or predicate) in the conclusion which may be uniformly replaced on some of its occurrences by any arbitrary propositional variable (or predicate of the same arity) salva validitate of I? t- a.

The R l -criterion implies the A-criterion. It has also some features of the K-criterion but it does not have the disadvantage described in 4.3.2 (1). R1 has an interesting connection with substitution via the A-criterion.

Let us call an inference r I- a a substitution-generalization of I?* I- a* iff I?* t a* is a substitution instance of I' t- a. Then one can prove the following theorem:

THEOREM 4. r t- a is R l -restricted (relevant) ifS it is A-restricted (relevant) and every valid substitution generalization of r I- a is also A -restricted (relevant). 35

Thus R l is an interesting and appropriate strengthening of the Aristote- lean Limitation. It solves (avoids) those paradoxical cases which are solved (avoided) by the A-criterion and a lot more. Moreover for R l Schurz was also able to give a semantics which is based on an algebra of articulated propositions. The main point is that all the restrictive ideas


for Classical Logic go beyond normal semantic considerations, i.e. truth values. That means in this special case of R1 that the position of the occurrence of the proposition is essential in addition to its truth value. Therefore the semantics had to be built on such additional condition^.^^

4.4.2. Generalizing the Rl -Restriction to Arbitrary Formulas

Since 1988 I have made proposals for generalizing the R1-criterion. There were two main objectives: to make the criterion applicable to arbitrary formulas (not just to inferences and implicational formulas) and to avoid a few shortcomings, like tautologies in the premisses. The first was achieved by applying R1 to positive parts of a formula, instead of to the conclusion only. This makes it more general and thus is able to also rule out 'classical paradoxes of implication' like (p -+ q ) V ( q -+ p) or (p + q ) V ( p -+ l q ) . They cannot be ruled out by R1.

The second problem was solved by the following condition for a: no proper subformula of a formula tr is a tautology or contradiction.

This is a rather strong condition which also rules out conjunctions of valid formulas, which may be interpreted as a disadvantage. On the other hand, the condition has the advantage of not allowing tautologies in the premisses. Observe that contradictions in the premisses or tautologies in the conclusion are ruled out already by R l (because in this case some occurrences of propositional variables or predicates are replaceable by arbitrary ones in the conclusion salva validitate of the inference).

A third condition which should help to rule out further redundancies like repetitions and tautologies in the premisses was this: the formula in question has to be maximally general in respect to substitution; i.e. it is such that it is the maximal substitution generalization, it has only substitution instances but is not itself a substitution instance of any formula. Usually axioms (of formal systems, for instance Propositional or Predi- cate Calculus) have that property. I have tried to show elsewha-e37 that axioms, principles and important theorems in mathematics and empirical sciences also have that property: they are formulated in a form which is maximally general in respect to substitution (relative to some field of investigation and to some degree of abstraction). This notion is worth further investigation, though it seems to me now that it is too strong a limitative criterion for Classical Logic.

In the following Sections 4.4.2.1 and 4.4.2.2 I propose two different, rather sophisticated criteria (limitative filters) for Classical Logic, which both contain the R1 criterion. They go beyond it in different


directions, however; both having some advantages and disadvantages over the other. They both are applicable to arbitrary formulas.

DEFINITION 6. a is R 1 G-restricted (relevant) iff

(I) It is not the case that a propositional variable (or predicate) in a is replaceable on some of its occurrences by an arbitrary propositional variable (or predicate of same arity) in some positive proper part(s) (subformulas) of a, salva validitate of a.

(2) No proper part (subformula) of a can be a tautology or contradiction.

A positive part of a formula is defined according to ~ c h i i t t e ~ ~ :

(a) F is a positive part of F . (b) If 7 A is a positive (negative) part of F then A is a negative

(positive) part of F . (c) If ( A v B ) is a positive part of F, then A and B are both positive

parts of F . (d) If ( A A B ) is a negative part of F , then A and B are both negative

parts of F . (e) If (A + B ) is a positive part of F, then A is a negative part and

B is a positive part of F . (f) If ( A + I) is a negative part of F , then A is a positive part of F .

This criterion has the merits of Rl . Beyond Rl it rules out paradoxical propositions of Classical Logic like (p + q ) V ( p -+ l q ) , ( p -+

q ) V ( q + p ) . . . and similar ones. It also rules out tautologies in the premisses (condition (2)). However it is not only conclusion restrictive but also partially premiss restrictive in the following sense: if there is a positive part in the premiss (antecedent) which is redundant in the sense of being replaceable by any arbitrary part then the respective formula is not RIG-restricted. More accurately: if there is a propositional variable (or predicate) in some positive part of the premiss (antecedent) of n which is replaceable by any arbitrary propositional variable (or predicate) salva validitate of a then a is not RIG restricted.


This leads to the disadvantageous consequence that though ( p A q) -+

p is R1 G-restricted (relevant) ( p A l q ) -+ p is not because q is a positive part of the last formula.39 That means that RIG is restrictive in a similar sense (though certainly not as strong) as we know it from computer assisted deduction. For instance Horn Clauses not only forbid any negations (thus also negations in the antecedent) but also disjunctions, letting through only conjunctions in the antecedent and atomic structure of the consequent. Because RIG is partially premiss restrictive in the above sense the following Rlg-criterion might be preferable in this respect.

DEFINITION 7. a is Rlg-restricted (relevant) iff the following conditions are satisfied:

(I) a -+ p is R l g-restricted iff a -+ is R1 -restricted. (2) If a A p is Rlg-restricted both a and P are. (3) If either l a -+ p or l p -+ a is Rlg-restricted then a V ,8 is

R 1 g-restricted. (4 ) ~ ( a p) is Rlg-restricted iff l a V 1/3 is so. (5) ?(a V ,8) is Rlg-restricted iff l a A 1 P is so. (6) ~ ( a -+ p) is Rlg-restricted iff a A 1 , O is so. (7) l l a is Rlg-restricted iff a is so.

This criterion also has the merits of R1. Beyond R1 it gives like RIG greater generality (applicability to arbitrary formulas) though the clauses (2) and (3) are not equivalences. R lg does not solve the problem with the paradoxes ( p + q ) V ( q + p ) . . . etc. because clause (3) does not give an answer for a V p if l a -+ /3 and 1/3 --+ a are not Rlg-restricted. Thus these paradoxes are not ruled out. Further Rlg does not forbid tautologies in the premisses. In these two respects it is less preferable than RIG. On the other hand one can conjoin it also with clause (2) of RIG to solve the last mentioned problem.

If clause (2) is strengthened to an equivalence one can prove a fixed point theorem: let T*(a) be a set of valid implications logically equivalent to a. (T*(a) is not unique since disjunctions may be transformed into implication in two ways). Then the following holds:


THEOREM 5. a is Rlg-restricted (relevant) ifSthere exists afixedpoint T*(A) which consists exclusively of R l -restricted implications.41

The reason why I weakened clause (2) is that there are important cases in the area of application of the criterion where restriction (relevance, definiteness, evaluation) of the parts of a conjunction do not transmit this property to the conjunction (cf. 2.1 and 4.1.4.8).

Institut f i r Philosophie, Universitat Salzburg, Franziskanergasse I , A-5020 Salzburg, Austria

NOTES

P. Suppes (1984, p. 91). M. L. Dalla Chiara (1 98 1, p. 347).

"f. Hesse (1970, p. 50). Carnap (1936, p. 440).

"oodman (1955, p. 730; Hempel (1965, p. 70). Popper (1963, Appendix); (1973, p. 330f). This was shown by Tichy (1974) and independently by Miller (1974). ' Weingartner (1982). Cf. Dalla Chiara (1991), Gochet-Gillet (1991). "ass (1 944, p. 38). "' All three are due to Prior. Cf. Prior (1954, p. 64) and (1958). " Miller (1975). Miller showed the language dependency for other definitions in his (1978). l 2 For the sake of simplicity individual variables and quantifiers are skipped. This simple example is due to Schurz (1990, pp. 316ff). l3 Henkin (1952, p. 160). l 4 Kreisel (1953). '"f. Kreisel-Takeuti (1974). l6 Cain-Damnjanovic (1991). l7 Kripke (1975). '' Cf. Kleene (1962, p. 334). '"he concepts 'subformula of a formula' and 'occurrence of a subformula in a formula' can be formalized in different ways. For a simple possibility see Leblanc (1968). "' The A-criterion has been applied by myself since 1980 and investigated in different forms (strengthenings) in Weingartner and Schurz (1986) where we spoke of different kinds of A-relevance.

Cf. Weingartner (1985). The main matrix MW is given on pp. 573 and 574. A class of matrices which can be equivalently described by MW was found independently of Dunn (1972) whose work was known to the author only in 1990.

PUTTING LlMITATIONS ON CLASSICAL LOGIC 119

22 Cf. Schurz (1991, p. 65). However, Schurz interprets my matrices in Weingartner (1985) as only characterizing A*-restriction and proves that this system cannot be characterized by finite matrices. This is true (see 4.1.3) for A*-restriction but does not hold for the system which results from A-restriction and not for the one of Wronski to which he refers (p. 68). See the following lemma and theorem. 23 Cf. Weingartner (1985, p. 573f.). 24 Schurz (1991, pp. 67ff.). 25 Dunn (1972). 26 Parry (1933). 27 Since the A-criterion is given only for Propositional Logic here it is not directly applicable to problems formulated in Predicate Logic. The necessary complication (define the propositional counterpart of a formula of Predicate Logic . . . etc.) I skip here; several such strengthenings and extensions have been done in Weingartner and Schurz (1986). However, this inaccuracy which is for the sake of simplicity is complemented by the simple advice to apply criterion R1 (Section 4.4.1) directly. R1 is in any case the strengthening of the A-criterion but a better one than our earlier proposals. 28 Schurz and Weingartner (1984, Section 4.4). 2"f. Mittelstaedt (1991, p. 210f.). ") c f . Mittelstaedt (1989, pp. 200ff.). 31a Cf. Weingartner (1993). 3'b Korner (1 959, pp. 24f. and 66f.) and Korner (1 979, p. 378). 32 Cf. Cleave (1 973174). 33 Cleave (1973174, pp. 120ff.). " In Czermak and Weingartner (1983, p. 48f.) we have given six different possibilities of interpreting sentential symbols. According to these six interpretations, we can say that K follows from K' when the language satisfies interpretations (1) or (2) or (6). But it does not follow from K' when it satisfies conditions (3), (4) or (5). 35 The proof is due to Schurz. Cf. Schurz (1991, p. 71). 36 For details cf. Schurz (1991, pp. 77ff.). " Cf. Weingartner (1989). The condition was proposed as a part of limitative criterion (relevance criterion) which is otherwise like RIG in my (1990, p. 328). 38 Schiitte (1977, p. 11). 3"his example was correctly pointed out as a disadvantageous consequence of RIG (RlW in his terminology) by Schurz. Cf. Schurz (1991, p. 75f.). However his other criticism is mistaken: [(p -+ q) A ( r -+ s ) ] -+ ( p + q ) is not ruled out, because r is not a positive proper part. On the other hand + (p V q)] A jr -+ ( r V s ) ] (p. 76) is in fact ruled out by the first condition and by the second condition that no proper part can be a tautology or contradiction. 40 The R l g criterion was proposed by Schurz (1991) with an equivalence in clause (2). 4 1 For the proof see Schurz (1991, p. 76).

PAUL WEINGARTNER

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Semantics', in: G. Dorn and P. Weingartner (Eds.), pp. 563-575. Weingartner, P.: 1989, 'A Proposal to Define a Special Type of Proposition', in: P.

Weingartner, and G. Schurz (Eds.), Philosophy of the Natural Sciences. proceedings of the 13th International Wittgenstein Symposium, Kirchberg/Wechsel, Austria, 1988, Vienna: Holder-Pichler-Tempsky, pp. 348-353.

Weingartner, P.: 1990, 'Antinomies and Paradoxes and Their Solution', Studies in Soviet Thought, 39,3 13-33 1.

Weingartner, P.: 1993, 'A Logic for QM Based on Classical Logic', in: A. Luz Garcia Alonso, E. Moutsopoulos, and G. See1 (Eds.), L'Art, la Science et la Me'taphysique, Bern: P. Lang, pp. 439-458.

Weingartner, P. and Schurz, G. : 1986, 'Paradoxes Solved by Simple Relevance Criteria', Logique et Analyse, 113, 3-40.


Paul's paper certainly convinces me that there are positive reasons for putting limitations on classical logic. I already believed that the answer

PAUL WEINGARTNER

was affirmative, but the detailed data and theorems he reviews put the case in very strong terms. It would now be nice to have additional pieces of genuinely empirical data in some extended systems that are easy to formalize, as for example Zermelo-Frankel set theory. It would be interesting even if tedious to put together data on which principles of classical logic are actually used in most proofs of the first six hundred to seven hundred theorems proved. (In my own book on set theory there are between six hundred and seven hundred theorems.)

In proposing such data, I do not mean to suggest that the evidence about the paradoxes that Paul presents is not persuasive. It is. It is just that we also seek other kinds of data in making any final decision about a system of logic to adopt in practice. We need to know, and need to know with some clear attention to what is actually needed, the data on practice - data that are necessarily in some cases implicit because of the way that proofs are ordinarily given, once a subject is developed to any extent. I mean by this that the sentential inferences are not laid out explicitly and a glance at proofs even in a subject so close to logic as Zermelo-Frankel set theory will not make at all clear what is the frequency of use, if any, of various principles of classical logic. Similar investigations in computer science led to extremely interesting changes in the basic design of the instruction set for microprocessors. I am referring here of course to RISC machines, where the acronym RISC stands for 'reduced instruction set computer'. As we think about automating elementary inference on computers, some similar data would, I suspect, be very suggestive.

Congruence and Meaning. My second point concerns the relation between the results on restricting classical logic or introducing new logics, to less-developed work of my own on congruence of meaning. (My papers on congruence are collected together in Suppes (1 99 I).) The intuitive idea is to replace the fixed notion of synonymy by weaker and stronger notions of congruence and meaning. The use of the concept of congruence and the way in which I formulated the program is obviously suggested by the use of weaker or stronger notions of congruence in geometry, and I have therefore baptised the program 'The Development of a Geometrical Theory of Meaning'. Just as in the case of geometry, when a concept of invariance is introduced or a concept of congruence, there is the corresponding problem of axiomatizing the geometry, so in


the case of congruence of meaning there is the problem of axiomatizing the corresponding logic.

One thing that comes out of the many different kinds of congruence that I have discussed and the concrete linguistic examples surrounding them is that any general theory of meaning must be partly syntactic in character. Purely model-theoretic considerations will not permit strong senses of congruence to be developed. The typical trivial example is the addition by conjunction of any tautology to a premise. One natural way of justifying syntactic considerations playing a part in meaning is that they have a direct influence on conditions of computation, which should be part of any constructive theory of meaning. In fact, a natural notion of computation is to add to standard conditions the additional condition that two expressions are strongly congruent when they satisfy certain standard conditions such as logical equivalence, and also require the same computation time to determine their truth, to put in standard form, or to make a canonical inference from, etc.

Let me end by mentioning one kind of congruence that is of great importance in the actual use of language but whose accompanying logic has scarcely been studied at all in a systematic way. What I have in mind is congruence of paraphrases where much information can be dropped in paraphrasing an utterance. The paraphrasing we do in memory is obviously quick, natural and unconscious. It is impossible for anyone under standard conditions to remember even a significant part of the detailed utterances heard on a given day or even in many cases, in a given hour. What an individual can do, however, is often paraphrase with great success the essential content of what was said. Obviously when we talk about 'essential content' we have in mind a context of background knowledge that is assumed for the paraphrase, and we also have in mind several judgments about what is essential and important in the message that was heard or read. The computation process by which we automatically do such paraphrasing is still theoretically inaccessible.

On the surface a requirement for such a computational theory seems very far removed from classical logic, and certainly the details are, but the underlying objective is similar. We have an extremely familiar phenomenon present in the use of all natural languages and yet we have very little study of its technical details even though its universality of use is greater than that of classical logic. Moreover, the theory of computation involved has more than a single part. The way in which paraphrases are stored in memory is surely different from the


way they are produced when asked for a paraphrase, that is, when they are generated as natural language utterances. So, we really have two parts: first, what is the computation with reduced storage in memory, and secondly, what is the mechanism for generating utterances from that storage. It is this common theme of the computational devices used that especially marks the similarity to the problems of classical logic discussed so thoroughly by Paul.

REFERENCE

Suppes, P.: 1991, Language for Humans and Robots, Cambridge, M A : Blackwell.

JAAKKO HINTIKKA AND ILPO HALONEN

QUANTUM LOGIC AS A LOGIC OF IDENTIFICATION

1. BACKGROUND

The tremendous richness of Pat Suppes' work has duly been noted and praised in many of the papers in this collection. Yet it seems to us that the multiplicity of his interests and approaches has not been done full justice. We will try to fill one omission in this paper.

It has been noted that one of the characteristics of Pat's work is the use of probabilistic concepts rather than old-fashioned logic. This observation is of course correct, but it should not lead us to forget that Pat has also evinced a great deal of interest in more purely logical methods, both those offered by ordinary logic and those provided by set theory. He has written excellent expositions of both, books that are still unique in their usefulness for applications to the philosophy of science. Moreover, earlier Pat showed a keen interest in one particular application of classical logic concepts to the foundations of scientific theories, viz. the application of the concepts of definition and definability to scientific theories.

This paper is written in the spirit of this facet of Pat's work. We will try to show that there are ways of developing further the applicability of logical concepts to the philosophy of science so as to enable us to approach from a new angle a problem which Pat has been keenly interested in, viz. in the problem of quantum logic and perhaps more generally in the conceptual foundations of quantum theory. Pat has discussed this problem area from the vantage point of probabilistic concepts. The fact that we are not in this paper using probabilistic concepts is not due to any animus against them. We are merely trying to see how far another approach which Pat has also used can be employed in the interesting field of quantum theory.

Quantum logic is usually studied von oben her. That is to say, quantum theory is viewed in all its conceptual richness and, starting

P Humphreys (ed.), Patrick Suppes: Scientijic Philosopher; Vol. 3, 125-145. @ 1994 Kluwer Acudernic Publishers. Printed in the Netherlands.

126 JAAKKO HINTIKKA AND ILPO HALONEN

from it, some aspect (substructure) of it is located that can reason- ably (or unreasonably) be called logic. Finally, the properties of the resulting structures are compared with various logic-related structures, such as Boolean algebras (which they are not) or orthomodular lattices (which they usually are). Typically, most of the attention is paid to propositional-level structures.

In this paper, a converse approach is being used. We start from the familiar logic, with its well-established model theory. This model theory is not really interesting on the propositional level. Hence we will be considering mostly first-order theories, which have a rich and well understood model theory. On the level of first-order theories, we will construct, as it were, scale models of certain quantum-theoretical phenomena. No claim is made that they capture all the conceptually interesting quantum-theoretical phenomena.

One advantage of this procedure is that we can show that certain quantum-theoretical phenomena are conceptually independent of certain other phenomena. This can be hoped to demystify some of the philosophical problems in this area. Often, a single unifying explanatory principle is sought which would account for all (or most of) the different conceptual problems of quantum theory, be this leading idea complementarity or discontinuity or whatever. We suspect strongly, and will argue by reference to specific examples, that different quantum- theoretical phenomena are fragments which are conceptually independent of each other.

In turns out that our familiar first-order logic, together with its model theory, is almost but not quite adequate for our purposes. We have to develop it further in certain directions. But what directions?

Here we obtain a useful clue from Pat's work. One of the unique features of his introductory logic text is an exposition of the theory of definability. Suppes has also been engaged in discussions and even disputes concerning the definability of specific, scientifically important concepts, for instance the definability of the basic concepts of classical mechanics.

But even the current logical theory of dsfinability is not quite what we need, even after the intensive development it has experienced after Suppes wrote his textbook. But what more is needed? In order to find an answer, let us have a look at quantum theory.

When a logician approaches quantum theory, the most striking aspect that he or she encounters is likely to be a preoccupation with questions

QUANTUM LOGIC AS A LOGIC OF IDENTIFICATION 127

of observability and unobservability, measurability and unmeasurabil- ity. These are closely related to questions of definability, but they are not precisely questions of definability. They do not concern what can be said of a variable on the basis of a theory alone. They concern whether a concept can be determined on the basis of a theory plus empirical observations. They are questions of identijiability rather than dejinability.

But what is the logic of identifiability as distinguished from definability? Some people have assimilated these two concepts to each other, but this has only contributed to conceptual confusion without answer- ing our question. It might even seem that the notion of observation is not a logical one, and that a logical theory of identifiability is therefore impossible.

It is nevertheless possible because it is actual. The interrogative model of inquiry which has been developed by Jaakko Hintikka and his associates provides a basis for developing such a theory.

2. THE INTERROGATIVE MODEL O F INQUIRY

The interrogative model can be formulated in the form of a game. This game can be called a game against nature, in that it is a two-person game in which only one player ('the Inquirer') is active whereas the other ('Nature') comes in only (or mainly) as a source of answers to the Inquirer's questions.

At each stage of the game, the Inquirer has a choice between two kinds of moves, (i) a logical move (also called an inference move or tableau-building move), and (ii) an interrogative move. The former consists essentially in the Inquirer's drawing a deductive inference from earlier results. The latter consists essentially in the Inquirer's putting a question to Nature.

In order to formulate the rules for these moves, we can use as our bookkeeping method - or as the Inquirer's method - a variant of E. W. Beth's method of semantical tableaux. Very briefly, a tableau which begins with T in the left column and C in the right column is the record of a thought-experiment, viz. an attempt to think of T as true and C as false in the same model ('world'). If such an attempt is frustrated in all directions (i.e. if the tableau closes), C cannot fail to be true if T is.

If the Inquirer can close the game tableau, no matter what Nature does, it is said that she or he can derive C from T interrogatively. If


C can be derived interrogatively from T in M , we will express this relation in the following way:

The interrogative derivability of C in a model M of T is in an interesting way intermediate between a logical implication from T to C and the truth of C in M .

For the purpose of this paper, we will make a simplifying assumption. It oversimplifies the situation in certain respects. Hence it cannot be made in general. Fortunately, the oversimplification does not matter for our purposes in this paper. This assumption allows the inquirer to raise any yes-or-no question. Hence for any answer W which Nature can (and must, if queried) give, the inquirer can ask 'W or not W?' and obtain W as an answer. What this means is that the relation ( I ) of interrogatively derivability reduces to the ordinary logical consequence relation

where A is the totality of all answers which Nature will yield.

3. COVERING LAW THEOREM

Without discussing the details of the interrogative model, we will only mention one important and interesting metatheoretical result concerning the logic of interrogative inquiry because we will return to it later in this paper. It is called the Covering Law Theorem. We formulate it on the basis of the simplification mentioned in the preceding paragraph.

THE COVERING LAW THEOREM. Assume that

(a) T U A k P(b) for a given individual constant b (b) T U A is consistent (c) not T I- P(b) (d) P does not occur in A (e) b does not occur in T .

Then there is a formula H [XI with x as its only free variable such that


(ii) T t (Vx ) (H[x] > P ( x ) ) (iii) P does not occur in H [x] (iv) All the constants of H [ X I are shared by T and A.

The Covering Law Theorem has interesting philosophical suggestions. It can be compared with Hempel's well-known 'covering law model' of scientific explanation. Clearly, in the approximate analogy between the two the sentence H[b] expresses the obtaining of suitable 'initial conditions' while the clause (ii) expresses the holding of a suitable 'covering law'. The connection is constituted of the tentative idea that interrogative derivability from a theory might serve as an explication of scientific explanation.

Here T is to be thought of as the set of all the initial theoretical premises the inquirer has at her or his disposal, while A is the set of all the available answers. What the assumptions of the Covering Law Theorem then say is essentially that one can interrogatively derive from T and A together the applicability of P to a case (individual) not already contemplated in the 'theory' T , and that one can do this without asking any questions directly about P. The conclusion says that there then exists a covering law which is derivable from T alone and whose applicability to b is derivable from the answers A alone.

This special case of the Covering Law Theorem can be generalized:

Let T and A be sets of closed jirst-order formulas and F = F [bl , b2, . . . , b,] a formula. Assume the following:

(a)' T U A I- F (b)' T U A is consistent (c)' not T t F (d)' None of the constants of F other than bl, b2, . . . , b, occur in A (e)' F is closed (f)' b l , b2 , . . . , b, do not occur in T .

Then there is a jirst-order formula H [x 1 , x2, . . . , x,] such that

(i)' A t H [ b l , b2,. . . , b,] (ii)' T t (Yx1) (Vx2) . . . ( V X ~ ) ( H [ X ~ , X ~ , . . . , xn]>F[x1 ,x2 , . . . ,x,]) (iii)' None of the constants of F [XI, 2 2 , . . . , x,] occur in H [ x i , ~ 2 , . . . ,

xnl (iv)' All the constants of H [x 1 , 2 2 , . . . , x,] are shared by T and A.


4. IDENTIFIABILITY

Without going any further, we will next mention a few words about identifiability in general. What is the connection between the interrogative model of inquiry and identifiability? Again, very briefly, the fundamental idea is very clear. It can be said that in the same way as interrogative derivability is a generalization of deductive derivability, identifiability is a generalization of definability. There exists a rich logical theory of definability, developed among others by Veikko Rantala. That theory has a great deal of potential philosophical interest. Now, because of the analogy mentioned above, a general theory of identifiability can be developed. The interrogative model of inquiry suggests a number of definitions which are roughly parallel with the corresponding definitions of different kinds of definability.

5. THEORY OF IDENTIFIABILITY AND QUANTUM LOGIC

What can this theory of identifiability do to elucidate problems of quantum theory? In this paper, we will present a few case studies rather than anything like a unified theory.

The first case pertains to the very structure that is usually called quantum logic. It is the structure of closed subspaces of a Hilbert space, which is the same as the structure of projection operators in such a space. But what are projection operators? Essentially, they are the observables according to quantum theory. A projection operator is associated with the set of possible observations concerning some quantity. Hence quantum logic is not the structure of different quantum- theoretical quantities, but of their observational manifestations.

Now what is the natural rational reconstruction and generalization of quantum logic? Its nature as a logic of observables, sketched in the preceding paragraph, gives us our first clue. It is the idea of using as the semantical value of an expression, for instance, of a (simple or complex) one-place predicate P ( x ) , in a given model M, something different from its normal semantical value 1 P ( x ) 1, which of course is its extension

More specifically, we propose to consider as the nonstandard semantical value I I P ( x ) 1 1 of P ( x ) , not the set of individuals that have that predicate,


but the set of those individuals which can interrogatively be shown (on the basis of a given theory T) to have it. In other words, the proposal is to stipulate that

This is a natural proposal, but it does not yield a structure that can use- fully be compared with quantum logic. The reason is that nonstandard logics of the kind instantiated by quantum logic codify in the first place new interpretations of logical constants rather than reinterpretations of nonlogical concepts. In the case of the so-called quantum logic, these logical constants are the propositional connectives & and V. Negation might also enter into the picture, but in quantum logic it is obviously left alone.

This observation, combined with the general idea that the desired nonstandard interpretation focuses on the observable aspects of the relevant model, motivates the following stipulations:

The resulting interpretation of the logic of monadic predicates will be called an identification interpretation of the first kind. In discussing it, we will often omit the indications of the arguments of the one-place predicates under scrutiny, with the understanding that it is the same for all the predicates in question.

The rules defining the identification interpretation are assumed to be applied from inside out, that is to say, applied first to the inmost disjunctions and conjunctions. Clearly, the rules are intended to be in a sense idempotent, i.e.

II llpll & IIQII 1 1 = Ilpll n IIQII II IIPll v IlQII Il = llpll u IlQII.

6. WHAT IS THE LOGIC OF IDENTIFICATION LIKE?

What is the structure resulting from the identification interpretation? The following are among the metatheoretical results that can be estab-


lished here:

l lP& QII C llPll l lP&QII C IIQII

llpll L I l P V QII IlQll C IIPVQII

IIP QII = llPll n IIQII Here it is easy to see that

11-7' & QII G l lpll n IIQII- The converse can be proved by the Covering Law Theorem. For if b E 1 1 P(x) 1 1 n 1 1 Q (x) 11, there are by the Covering Law Theorem Hp[x, dl, d2, . . .] and HQ [x, el, e2, . . .] such that

But then we have


In virtue of the Covering Law Theorem, this implies that b E 11 P(x) & Q(x) 11, which suffices for the proof.

These results show that the structure yielded by the identification interpretation is very close to a lattice structure. We will come back to its relation to lattices later.

As a corollary we obtain

The most interesting property of the identifiability interpretation is that it obeys the same distribution laws as an orthomodular lattice. For instance, the usual distribution law

is not valid. A counterexample is offered'by any pair P , Q where

llp & QII

is a proper subset of

For ther, we can put R =-- Q, which results in

On the other hand,

ll ( P & Q) v ( P & R) ll = II llP & QII v llp & Rll II = II llP & Qll v llp & - QII II = llp & QII U IIP & - QII

= (IIPII n IIQll) u (llpll n II - QII).

This is strictly smaller than 1 1 P 11 as soon as either 1 1 Pll n 1 1 Q 11 is strictly smaller than IlPll n IQI or IlPll (7 1 1 - QII is strictly smaller than 11 PI1 n I -- QI. This is illustrated in Figure 1, where the shaded areas are J J P & QII and J J P & QII, respectively.


L v - #

Q - Q Fig. 1.

Consider, likewise, the dual distributive law

We can obtain a counterexample by putting R =- Q. Then

IIP v (Q & R)ll = II IlPll v llQ & - QII ! I = I1 IlPll v IlQ & - QII II = IIPII.

On the other hand,

l l (p v Q) & ( p v R)ll = l l (p v Q) & ( P v QII = llp v QII n llPv - QII.

If IlPll u IIQII is a strict subset of IIP V QII or llPll U 1 1 - QII is a strict subset of ( 1 PV - Q ( 1 , then this is a strict superset of ( 1 P(I.


b v

Q - Q Fig. 2.

This can be illustrated by Figure 2, where the shaded area is the complement of I I P V Q I I.

On the other hand, consider the orthomodular law

whenever IlRll 5 llpll and IlQll 2 II - Pll. This law is valid, for then, on the other hand

ll(P & Q ) v ( P & R)ll = II llP & QII v llP 8'5 Rll II = llP & QII lJ llp & Rll = (Ilpll n IIQll) u (Ilpll n IlRII) = fl u IlRII = 11R11.


I l P l l I l -P I1 f l - e

- \

L

P - P

Fig. 3.

On the other hand,

From the Covering Law Theorem it is seen that this equals

The diagram in Figure 3 illustrates the situation.


This does not suffice to show that the logic of identification is an orthomodular lattice, for we have not shown yet that it is a lattice in the first place. What is missing is the associative law, which can be seen to fail sometimes in on the identification interpretation.

7. WHAT IS OUR QUANTUM LOGIC LIKE?

This does not close the issue, however. First, it is easily seen that the (first) identification interpretation is the wrong side of the coin if our purpose is to understand quantum logic. On the first identification interpretation, the crucial fact is that a disjunction (P v Q) is more easily identified than its disjuncts, i.e. that normally

llp v Q I I has as its proper subset

In quantum logic, the interesting fact is that a conjunction (P & Q) is more difficult to identify than its conjuncts, i.e. that

llP & Q I I is a proper subset of

This can be corrected by choosing the duals of the main interpretation rules formulated above. The result is what will be called the second identification interpretation. The crucial part of its definition can be formulated as follows:

lip & Q11 = {b E do(M) :N ( M : T I-- ( P & Q))} llP v QII = { b E do(M) :- ( M : T I-- ( P V Q ) ) } .

Being the dual of the first identification interpretation, the same laws hold mutatis mutandis, which here means the changes caused by the duality.

Now there are only two laws that remain to be shown to be valid in order to show that the second identification interpretation yields


an orthomodular lattice. They are the two laws that are essentially calculated to enforce associativity, viz. the following ones:

if IIPII G IIQII and

llRll C IIQII 7 then llp v Rll L 11Q11

and

if IIPII C IIQII and

IIPII C llRIl7 then

llpll G IIQ & Rll.

Both can be verified. From the laws established for the first identification interpretation, by duality, we obtain

By means of this observation the former law is easily verified. In order to deal with the latter law, we must recognize the peculiar

situation in quantum theory. The concepts to be identified there are not monadic predicates but values of time-dependent variables. (The models of the underlying theory T are then the temporary states of the system.) That means that the different concepts to be considered can be taken never to share the same values, that is ((P & QII is always empty. On this basis the latter law is seen to be valid in the special case corresponding to the situation in quantum theory.

Thus, in the sense indicated, quantum logic is structurally identical with the logic of the second identification interpretation. Because of the motivational near-identity of the two, it is not much of an exaggeration to say that the so-called quantum logic is a branch of the general logic of identijication.

8. CONSEQUENCES

This result has a number of philosophical consequences. For one thing, quantum logic is not a rival of ordinary logic, but a special application and further development of logic. Also, the need of quantum logic cannot be used to support any far-reaching metaphysical or other philosophical conclusions. It is a structure that so to speak might arise in any suitable first-order theory.


This conclusion is worth spelling out more fully. Our argument showing that the 'logic' of quantum mechanics has the structure of an orthomodular lattice is predicated on extremely weak assumptions. All we had to assume is that the so-called quantum logic deals with the observables of quantum theory and that the relevant concepts are single-valued functions. No other nontrivial nonlogical results were needed.

Hence the identification of the logic of quantum theory as an orthomodular lattice tells precious little about the interesting realities which

quantum theory is dealing with. All it really accomplishes is to call our attention to the fact that the concepts that the so-called quantum logic deals with are as it were the observable components of the relevant concepts. The phenomena that is supposed to be captured by means of a special quantum logic are independent of other quantum-theoretical phenomena.

9. IDENTIFICATION AND DISCONTINUITY

Thus our results so far might seem to be somewhat bland, at least far removed from the far more exciting-looking realities of actual quantum theory. Be this as it may, our results can in any case be related to certain other interesting features of quantum theory.

One such feature can be seen from the Covering Law Theorem. It shows that for each b E do(M) for which M : T t P[b], there is a kind of generalization codified in the formula H [x, a1 , a2, . . .] . Why cannot these generalizations be combined so as to form a single identificatory formula for P? The reason is that the generalizations might be infinite in number, which is possible only if they can be arbitrarily small. Admittedly, there is no measure defined on do(M), but it is easy to introduce suitable measures defined on the subsets of do(M). Then the only thing we need to assume here is that the set of individuals satisfying are given formula without P satisfied in M receives a constant nonzero measure.

One can nevertheless formulate sufficient and necessary conditions for the different 'generalization' Hb [x, a1 , a2, . . .] (for different b E do(M)) to be possible to combine in a single formula. Likewise, the second identification interpretation IlPll of P may or may not be captured by any explicit formula (with a finite number of parameter individuals).


Now a characteristic feature of quantum theory is that the identification range 1 1 Pll of the concepts considered in quantum logic can in fact be represented explicitly in the theory.

Let us now assume such a representability for a suitable first-order theory T. Consider now a step from a model M I of T to another one M2. Insofar as the identification interpretation 1 1 PI1 of a given predicate P changes, it must change at least by a certain minimal amount, which cannot be made smaller by choosing M I and M2 differently.

What this means is clear. If the models Ml, M2 are states of a system at a time, so that identification really means identification a t a time, then, insofar as the identification interpretation 1 1 Plj of P changes, it changes discontinuously. Thus the discontinuity phenomena of quantum theory can be considered simply as consequences of the representability of the observability range of the relevant quantum-theoretical concepts in the formalism of quantum theory itself.

Even though we cannot launch an extensive investigation here, it is clear that we are dealing with extremely interesting matters. One of the main conceptual problems in quantum theory has always been to understand quantization, that is to say to understand how irreducible discontinuities can result from theories which prima facie deal with continuous variables. (This is the philosophically interesting aspect of the so-called wave-particle dualism.) For instance, Heisenberg's recent biographer writes: "The objective for Heisenberg and his allies, then, was to find an irrefutable way to incorporate discontinuity into Dirac and Jordan's formalism" (David S. Cassidy in Scientijic American, May 1992 p. 110). Now we can see that the door is opened for discontinuity as soon as the conditions of observability for a quantity are explicitly expressible in a theory itself.

10. ON THE HIDDEN VARIABLE PROBLEM

As a further illustration of our approach, we will offer a few remarks on the hidden variable problem. What is this problem? One way of looking at it is this: given a first-order theory T[P, Q] containing two constants P, Q which we can think of as being of the kind we have been discussing, one can ask whether the Heisenbergian uncertainty can be eliminated by adding new theoretical concepts ('hidden variables') to it.


In this paper, we will not discuss in any depth the nature of quantum- theoretical ('Heisenbergian') uncertainty. For our purposes, it suffices to assume that such an uncertainty manifests itself in the failure of a concept to be identifiable through a quasi-definition of the form

where bl , b2, . . . , bk E do(M). Given this problem, the precise form which it assumes depends on

the particular theory TIP, Q]. Hence a full answer to the quantum- theoretical problem of hidden variables will undoubtedly depend on the actual mathematical formalism of quantum theory.

Some light on the conceptual situation can nevertheless be thrown by our approach. For instance, we can establish general results as to what hidden variables can or cannot do. For this purpose, we can recall that since we are dealing (by assumption) with first-order theories, the usual Craigian interpolation theorem holds. It can easily be seen to imply the following

HIDDEN VARIABLE THEOREM. Assume

(a) ( T [ H , P ] u A ) t- ( v x ) ( P ( x ) ++ D ( ~ I , b2, . . . b k , ~ ] ) (b) not T [ H , P ] (Vx) ( P ( x ) t-- D [ b ~ , 62, . . . , b,t, x ] ) (c) M I= T [ H , PI (d) H, P do not occur in those members of A which were used to prove

(a) (e) H does not occur in D[b l , bz , . . . , bk].

Then there is a formula I such that

(i) ( T [ H , P] U A) I- I (ii) I t- (b'x)(P(x) - D[bl , b2, . . . , b k , x ] )

(iii) The same concepts are used in answers in deriving (i) as were used in deriving (a). (Hence H and P are not used in deriving (i).)

(iv) I does not contain H, but it may contain P. (v) All other predicates of I occur both in T [ H , P ] and in D[bl, b2,

. . ., bk, x].

We have used here the same simplifying assumptions as in the formulation of the Covering Law Theorem above.


What does this metatheorem tell us? We propose to interpret P as the predicate to be measured (identified) and H as a 'hidden variable'. Then (d) says that neither H nor P is observable, as is intended.

The crucial assumption (a) says that the hidden-variable theory T [ H , PI is successful in the sense that it enables us to identify or 'measure' P.

What the conclusion then says is that one can empirically (interrogatively) derive from the hidden-variable theory a conclusion I. (See (i) and (ii).) This conclusion does not contain the 'hidden variable' H. (See (iv).) It will therefore be a truth in the poorer language which does not contain H , i.e., in the language of the theory which does not contain the 'hidden variable'. And it logically (deductively) implies a reference-point definition of the concept to be measured, as shown by (ii).

Hence the work that the hidden variable was hired to do can be done without its help. If a hypothetical hidden-variable theory enables one to measure P in M , then there is a proposition without H true in M which logically implies a definition of P and which can even be established empirically on the basis of the hidden-variable theory.

Moreover, this truth I, even though it is free of the hidden variable H, obviously is radically different in content from any theory (in the same language) according to which P is not measurable in M . The hidden-variable theory, if successful and true, would thus lead to a radically different observational, i.e. hidden-variable-free, theory, too.

This result concerned in the first place on particular model M of T [H, PI. It can nevertheless be made independent of M.

Of course, the question of what the changes are which a hidden- variable-free theory T [ H , P ] would necessitate in the original hidden- variable theory TIP] has to be discussed case by case. Likewise, there is no way of deciding whether these changes are in principle acceptable or whether they can be ruled out by general considerations without examining the particular theory in question (including its experimental backing). However, the need of radical changes in T [ P ] is independent of the details of T [PI if T [H , P ] is true and if it enables one to measure P.

This is very much in line with von Neumann's original intentions in tackling the hidden-variable problem. He writes:


We shall show later (IV.2.) that an introduction of hidden parameters is certainly not possible without a basic change in the present theory (Mathematical Foundations of Quantum Mechanics, Princeton U.P., 1952, p. 210; emphasis added).

What we have shown is that the same can be said already on the level of first-order theories. In general, it seems to us that the real message of von Neumann's result has been misunderstood. He did not prove that hidden-variable theories are impossible, but that they are misguided. Instead for looking for hidden variables, one should look for a different no-hidden-variables theory - if you do not like the present quantum theory. Accordingly, the postulates of von Neumann's proof were not supposed by him to be a priori truths, but empirically establishable ingredients of our usual quantum theory.

Once again, it turns out that a major conceptual issue in quantum theory is firmly rooted in the basic logical properties of theories in general.

Jaakko Hintikka, Department of Philosophy, Boston University, Boston, MA 02215, U.S.A.

Ilpo Halonen, Department of Philosophy, University of Helsinki, Unioninkatu 40B, Helsinki, Finland


My association with Jaakko Hintikka goes back at least 30 years and I recall with pleasure the many years during that period when he was a part-time colleague at Stanford. Jaakko and I have given joint seminars on many different topics and have continued our conversations and correspondence during the years since he left Stanford for full-time commitments elsewhere.

The paper of Hintikka and Halonen performs a useful service in showing how quantum logic can be derived and thought about in a context that makes little if any use of quantum mechanics as such. This is not surprising, for the pure logic, as such, of quantum mechanics is


necessarily a very high abstraction from the details of' any quantum- mechanical calculations or experiments. On the other hand, uses or interpretations of quantum logic outside of quantum mechanics have not been much developed. It is also not surprising that their results use the concept of observable and that of identijcation, natural concepts abstracted away from their quantum mechanical use, especially the concept of observable which is so central to quantum mechanics.

It would be instructive to have a point-by-point comparison of the developments in Hintikka and Halonen's paper with the general considerations in Mackey (1963, Ch. 2). Mackey gives a clear and rigorous general development of observables, compatible in terms of the initial assumptions, with both classical mechanics and quantum mechanics. Of course his framework is in a richer mathematical setup, but the logic to be derived from it is essentially identical with the logic derived by Hintikka and Halonen by a different route. I mention especially Mackey's work because of its development of the logic of questions and question-valued probability measures. It is especially relevant to the specific assumptions of Hintikka and Halonen that Mackey shows that observables that are questions can be a sufficient basis for all observable~.

I just want to mention in conclusion one example of my own of how the abstract logic of quantum mechanics can be interpreted and be useful in quite different domains. In Suppes (1966) I introduced a generalization of Boolean algebra to that of quantum-mechanical algebra of sets, namely, let St be a non-empty set and let F be a non-empty family of subsets of St. Then F is a quantum-mechanical algebra of sets on St if and only if F is closed under complementation, that is if A is in F, 2 is in F , and F is closed under the union of disjoint sets, that is if A and B are in F and A n B = 0 then A U B E 3 . It is obvious that the major restriction here is that the union of events is defined only for disjoint events. Such quantum-mechanical algebras will satisfy a number of restricted applications but are not sufficiently general to meet all needs. The important point of application here is rather different; namely, if we consider an ordinary probability space and consider a fixed event A, then the set of all events B that are independent of A form such a quantum-mechanical algebra. Put another way, this is just a way of showing that the binary operation of independence is closed under complementation and union of disjoint sets. This represents a use of such algebras that reaches far beyond the range of quantum mechanics itself.


My sympathy for the direction of the work of Hintikka and Halonen is also reinforced by mentioning much older work of mine on three-valued logics which introduce the concept of meaningfulness in the sense of invariance. The third value of the three-valued logic is meaningfulness. Applications to various physical theories are discussed in Suppes (1 959, 1965). It would be useful to combine the concept of meaningfulness I used in these earlier papers with the concept of identifiability used by Hintikka and Halonen.

REFERENCES

Mackey, G. W.: 1963, Mathematical Foundations of Quantum Mechanics, New York: Benjamin.

Suppes, P.: 1959, 'Measurement, Empirical Meaningfulness and Three-Valued Log- ic', in: C. W. Churchman and P. Ratoosh (Eds.), Measurement: DeJinitions and Theories, New York: Wiley, pp. 129-143.

Suppes, P.: 1965, 'Logics Appropriate to Empirical Theories', in: J. W. Addison, L. Henkin, and A. Tarski, Theory of Models, Amsterdam: North-Holland, pp. 364- 375.

Suppes, P.: 1966, 'The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics', Philosophy of Science, 33, 14-2 1.

MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

LOGIC AND PROBABILITY IN QUANTUM MECHANICS

ABSTRACT. 'To what extent did quantum mechanics contribute to changing some basic ideas in logic and probability' is a question that has been deeply investigated by Suppes since the early sixties. After many attempts developed in the framework of the logico-algebraic approach to quantum theory, the problem concerning the 'right' structure of the quantum events seems to be in a sense still open. Different forms of quantum logic arise in the orthodox von Neumann approach as well as in the unsharp approach, characterized by the acceptance of 'fuzzy' propositions.

1. INTRODUCTION

'To what extent did quantum mechanics contribute to changing some basic ideas in logic and in probability theory' is a question that has been deeply investigated by Patrick Suppes on several occasions. Some important papers on these problems appeared in the early sixties, a 'renaissance period' for the logico-algebraic approaches to the foundations of quantum theory. In our analysis we will particularly refer to 'Probability Concepts in Quantum Mechanics' (Suppes, 1961 ); 'The Role of Probability in Quantum Mechanics' (Suppes, 1963); and 'The Probabilistic Argument for a Nonclassical Logic of Quantum Mechan- ics' (Suppes, 1966).

The main thesis defended in these articles can be briefly summarized as follows:

(1) Quantum Theory gives rise to a radical conflict with classical probability theory. The basic reason is the non-existence of joint probability distributions for pairs of observables that are incompatible (i.e., not simultaneously measurable j. For instance, the joint distribution of position and momentum cannot exist.

(2) As a consequence, the most common interpretation of Heisen- berg's uncertainty relations turns out to be weak and misleading. It is not sufficient to claim that position and momentum cannot be simultaneously measured precisely. One should rather conclude that position and momentum cannot be simultaneously measured at all.

P Humphreys (ed.), Patrick Suppes: Scientific Philosopher, Vol. 3, 147- 167. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

148 MARIA LUISA DALLA CHIARA AND ROBERTO GIUNTINI

(3) This has a bearing for the question concerning the logic of quantum mechanics, as shown by the following argument.

Let us assume the premises:

(P.l) in a physical theory the working logic is the logic of events (or propositions) to which probability is assigned;

(P.2) the algebra of events should satisfy the requirement that a probability is assigned to every event;

(P.3) in quantum mechanics, probability may be assigned to an event such as position in a certain region and momentum within given limits, but the probability of the conjunction of two such events does not necessarily exist.

Conclusion: The logic of quantum mechanics is not classical.

The question arises: what is the 'right' structure of the quantum events? Suppes proposes a particular class of algebraic structures that he calls quantum mechanical algebras (a convenient weakening of the Boolean structures of events of classical probability theory).

Quantum mechanical algebras turn out to coincide with structures that are more frequently termed orthomodular orthoposets. In such structures, the events are partially ordered; the negation (complement) of any event is always defined; however the conjunction (inf) and the disjunction (sup) of two events do not necessarily exist. They exist for pairs of orthogonal events, where the first event precedes the complement of the second event.

The non-closure of the structure of the events under conjunction and disjunction is regarded as a natural consequence of the non-existence of joint distributions of incompatible observables.

To what extent are quantum mechanical algebras (or orthomodular orthoposets) faithful algebraic descriptions of the quantum events?

What are the most adequate ideas of negation, conjunction and disjunction in quantum logic?

These questions have been deeply investigated after the sixties and apparently we are still far from a definite answer.

We will discuss this problem in the framework of the logico-algebraic approach to quantum mechanics, to which the following (besides Sup- pes) have greatly contributed: Mackey, Varadarajan, Gudder, Greechie,

LOGIC AND PROBABILITY IN QUANTUM MECHANICS 149

Jauch, Piron, Beltrametti, Cassinelli, Ptiik, Pulmannovii and many others.

We will particularly refer to the unsharp approach to quantum theory first proposed by Ludwig, PrugoveEki, Davies and developed, among others, by Kraus, Mittelstaedt, Busch, Lahti, Cattaneo.

2. STATE-EVENT STRUCTURES IN QUANTUM MECHANICS

Our basic notion will be the concept of state-event structure, a system consisting of a set of states S and a set of events &. From an intuitive point of view, states may be regarded as pieces of information about objects, or physical systems (a kind of individual concept in Leibniz's sense); whereas events correspond to possible properties or propositions.

Let u, u, w, . . . represent elements of S; while a, b, c, . . . are elements of &. The minimal conditions that are required are the following

(any state associates a probability value to any event).

(2) Extensionality

In other words: events that are probabilistically indiscernible are identified. Similarly for states.

(3) & is closed under a weak complement operation such that:

(4) & contains a certain event 1 such that:

Let 0 := lL be the impossible event. S permits us to define an order relation on &:


DEFINITION 2.1.

a C b H V u [ u ( a ) <_ z ~ ( b ) ] .

DEFINITION 2.2. Orthogonality.

a l b o a C b'.

u l v H 3a E E[u(a ) = 1 and v ( a ) = 01.

One can prove:

LEMMA 2.1. The structure (&, c ,' , 1 , 0 ) is a bounded involutive regular poset. In other words:

(a) is a partial order with maximum 1 and minimum 0; (b) is an involution:

(ib) a = a"; (iib) a L b + b1 L aL;

(c) a l a and b l b + a l b (regularity).

This represents a kind of minimal structure that it seems reasonable to require for a state-event system.

Further conditions that may strengthen (S, E ) are the following:

(5) A partial sum @ is defined on E:

(a) a l b + a & b ~ E ; (b) V u [ u ( a @ b) = u ( a ) + ~ ( b ) ] .

Condition (5) may be reinforced by requiring that, when defined, a @ b is the sup of a, b:

(5*) Additivity.

(a) a l b + a $ b ~ & ; (b) V u [ u ( a & b) = u ( a ) + u(b)] ; (c) a @ b = a U b

(where: a , b a u b and Vc[a , b c c + a U b 5 c]).


(5**) 0-additivity. Let {a i } be a countable sequence of pairwise orthogonal events.

(where U is the infinitary sup).

(6) The non-contradiction principle.

(only the impossible event is orthogonal to itself).

(7) Orthomodularity.

a C b +- 3 c [ a l c and a @ c = b].

(8) Weak determinism. ( S is sufJicient and £ is unital)

(any event which is not impossible is satisfied with certainty by at least one state).

Weak determinism turns out to imply the non-contradiction principle, but not the other way around.

(9) (£, E) is a lattice, i.e., the sup a U b and the inf a n b exist for any pair of events a, b.

(10) (£, II) is a distributive lattice:

a fl (6 U c ) = ( a n b) U ( a fl c ) a U (b fl c ) = ( a LJ b) fl ( a U c) .

An event structure (£, E,' , I , 0) (which is at least a bounded involutive regular poset) will be:

(a) an (orthomodular) orthoposet, when it satisfies (orthomodularity), the non-contradiction principle and conditions (a), (c) of (53") (additivity);

(b) an (orthomodular) ortholattice, when it is an (orthomodular) orthoposet and a lattice;


(c) a Boolean algebra when it is a distributive ortholattice.

One may require that our state-event structure (which satisfies at least the minimal conditions) is closed under a necessity-operator 0, that transforms any event a into a 'finer' event Oa. This permits us to distinguish between sharp and unsharp events. The interest of this modal operator (that was first proposed in this framework by Cattaneo and Nisticb, 1989) will be made clear later, by looking at some concrete examples.

Let us first define:

DEFINITION 2.3. The positive and the negative certainty domains of an event a (D (a) and Do (a)).

(1 1) The necessity operation (0). A necessity operation on (E , 5, ', 1 , O ) is a map q : & -+ £ such that:

(a) for any state u and any event a:

u (na) = 1 iff (u(a) = 1; u (0a) = 0 iff u l D l ( a )

(the positive certainty domain of a and Oa coincide; the negative certainty domain of Oa is the maximal possible one).

(b) for any events a, b:

Oa C a; OOa = Oa;

if a C b then Da Ob; I I OOa = ma, where Ob := (O(b ))

(0 behaves like a modal Ss-operator).

DEFINITION 2.4. Sharp and unsharp events. a is sharp if a = Oa; unsharp otherwise, (Sharp events coincide with their necessitation.)


Let Es represent the subset of the sharp events of E. Es is not empty, since the certain and the impossible event are always sharp.

LEMMA 2.2. If (S, E) satisjes the minimal conditions, (ES, giL , I , 0) is an orthoposet. In other words: sharp events always satisfy the non-contradiction principle.

(12) The order is determined by the positive certainty domain:

Condition (12) implies that every event is sharp. Let us now define a very general notion of compatibility between

events. Suppose that (S, E) satisfies at least the minimal conditions and condition (5).

DEFINITION 2.5. Compatibility between events (Comp(a , b)).

Comp(a, b) 9 ga l , b l , c[aq @ bl @ c E E and a = a l @ c a n d b = b l $ c ] .

In the particular case where (E, I I , I , 1 , 0 ) is an orthomodular orthoposet and condition (5*) holds it turns out:

Cornp(a, b) 9 the substructure generated by

a and b is a Boolean algebra.

As a consequence, the sup and the inf of two compatible events always exist. And, obviously, in a classical (Boolean) event-structure, any two events will be compatible.

The abstract notion of (spectral) observable and of joint observable of two observables in a given (S, E) can be defined as follows.

DEFINITION 2.6. (Spectral) observable in (S, E) . Let 23 (Rn ) represent the set of the Bore1 subsets of Rn. An n-ary (spectral) observable in (S, E) is a map A : Z3(IWn ) --+ E which satisfies the following conditions:

(a) A(Rn ) = 1


(b) A(Rn - X ) = A ~ X ) ' (c) A(U{Xn)) = U{A(Xn)},

where {Xn) is a sequence of pairwise disjoint elements of B(Rn). When a = A(A) (where A E B(Rn )), a can be read as 'the value for

the observable A lies in A'. If u is a state and A is an n-ary observable, the composition uA determines a probability distribution:

DEFINITION 2.7. Joint probability distribution and joint observable h la Gudder. Let A and B be respectively an n-ary and an m-ary observable in ( S , E). A and B are said to admit a joint probability distribution in the state u iff there exists a probability distribution p on B ( R ~ + ~ ) such that for any E %(Rn), A E B(Rm):

whenever A( r ) fl B(A) E &. We say that A and B admit a joint observable when there exists an

observable C on 23(Rn+" ) such that Vr E B(Rn ), VA E B(Rm ):

(a) C ( r x P) = A ( r ) (b) C ( R ~ x a) = B(A).

DEFINITION 2.8. Compatibility, complementary and probabilistic complementarity between observables. Let A : B(Rn) -+ E and B : B(Rn) -+ E be two observables. A, B are called:

(i) compatible iff VA, I? E %(Rn ): Comp(A(A), B (I?)). (ii) complenzentary iff for any bounded A, I? E B(Rn) such that

A@), B(F) # 1;

(iii) probabilistically complementary iff for any bounded A, I' E B(Rn ) such that A(A), B ( r ) # 1;

A last condition that may be required is the following


(1 3) The o-convexity of S . Let { X i ) be a sequence of weights (such that X i E [O,l] and xi X i = 1 ) and let {ui} be a sequence of states. Then

3 wVa [W ( a ) = C Xiui ( a ) ] . i

Different examples of state-event structures (satisfying different subsets of our set of conditions (1)-(13)) can be found both in the abstract and in the Hilbert-space axiomatization of quantum theory.

First Example (h la Mackey)

Let (3 be a set ofprimitive observables A and let S be a set of states. Any pair (A , A ) where A E (3 and A is a Borel set is called a question. Any state associates a probability value with any question. Two questions (A , A ) and ( B , r) are equivalent iff Vu E S : u((A, A) ) = u ( (B , I?)). Any equivalence class [ (A , A)] of equivalent questions is an event. Let & be the set of all events. The probability value associated by a state u to an event [ (A , A ) ] is defined by

Extensionality for states is required. Any primitive observable A determines a spectral observable A such

that

for any Borel set A. One requires that (3 is closed under the Borel functions (for any

A and any Borel function f , (3 contains an observable B such that B = A f - I ) . Further, for any A E (3 the following conditions hold:

(i) Vu : :([(A, R ) ] ) = 1. (ii) for any countable sequence Ai of pairwise disjoint elements of

B(R) :


(iii) For any sequence {ai} of pairwise orthogonal events, £ contains an event b such that

One can prove that, in this example, the set of events has the structure of a a-complete orthomodular orthoposet. Further, the set of all events {[(A, A)]aEBcw,} determined by a single observable is a (a-complete) Boolean algebra.

Similarly for n-ary observables. If A, B are different spectral observables, generally the conjunction

A(A) n B(r) does not exist (in agreement with Suppes' requirement).

The Orthodox Hilbert-Space Exemplijication

The set S of the states is identified with the set of all statistical operators in the Hilbert space 7-l associated to the physical system. The set O of the primitive observables is identified with the set of all self-adjoint - - operators A in 1-I. Here, the spectral measure A of A associates to any A E 'S (R) a projector A( A) in 1-I. There is a one-to-one correspondence between £ = {[(A, A)] I A E 0 and A E B(R)}, {A@) I A E O and A E 'S(R)} and the set of all projectors of 1-I.

Similarly for n-ary observables. On this basis, one may conclude that events can be represented as

projectors of 7-i. However, differently from the abstract case (a la Mackey), here the conjunction A ( r ) f l B(A) of the events A(I'), B(A) ('the value for the observable A lies in r and the value for the observable B lies in A') always exists; for projectors have a lattice-structure.

Further, the following relations hold (for non-constant observables):

(a) complementarity and probabilistic complementarity and probabilistic complementarity are equivalent;

(b) complementarity implies incompatibility, but not the other way around;

(c) two complementary observable do not admit a joint probability distribution for any state.

At first sight this situation seems to be somewhat strange from an intuitive point of view. When A and B are two complementary observables


(like, for instance, position and momentum), a joint probability distribution of A and B does not exist for any state. However, any state associates a probability value with any conjunction A(A) f l B ( r ) for any Borel set A, I?. Anyway, this value will be different from 0 only in the case where either A(A) or B ( r ) represent the certain event. This seems to go beyond what is required by the common formulation of the uncertainty relations. Even for 'very large' closed intervals A and r will have:

Second Example (FuzziJication of Mackey)

In our first example (a la Mackey) every question (A, A) and every event [(A, A)] represents a kind of 'clear' property. From an intuitive point of view, one could say that, whenever ZL((A, A)) # 1,O the uncertainty involved in such a situation totally depends on the 'ambiguity' of the states and not on the 'ambiguity' of the property. However, it makes sense to consider also genuine ambiguous properties and to fuzzify our notion of event a la Mackey. In order to illustrate the difference between the two cases, from an intuitive point of view, let us refer to the following statements (expressed in the natural language):

(I) Hamlet is 1.70 m tall. (I) The Italian President Scalfaro is an honest man.

Apparently, the semantic uncertainty involved in our first example seems to depend on the incompleteness of the individual concept associated to the name 'Hamlet' (in Leibniz's sense). On the contrary, the uncertainty of our second statement is mainly determined by the fuzziness of the concept 'honest'.

In orthodox quantum mechanics, only examples of the first kind are considered. On the contrary, a characteristic feature of unsharp quantum theory is to investigate also examples of the second kind. Following an approach proposed in Cattaneo and Laudisa (forthcoming), let us first introduce the notion of (macroscopic) localization.

DEFINITION 2.9. A (macroscopic) localization is a Borel function w : B(R) x R + [O, 11 such that


(i) {x I w(A,x) = I} G a (ii) {X I w(A, x ) = 0) C R\A.

DEFINITION 2.10. A fuzzy characteristic function of A is a function wa : R -+ [O, 1) such that for a given localization w, wa(x) = ~ ( n , X ) vx E R.

The set of fuzzy events E is now identified with the set of all equivalence classes [(A, wa)] where (A, wA) is a fuzzy question; A belongs to a set of primitive observables O and WA is a fuzzy characteristic function of a Borel set A.

Obviously, any sharp question (A, wa) corresponds to a limit case of a fuzzy question (A, wa) where is the sharp characteristic function of A. From an intuitive point of view, the fuzziness of a question like (A, wA) may be regarded as depending on the accuracy of the measurements which will test our question and also on the accuracy involved in the operational definition of the physical quantity corresponding to A.

Differently from the Mackey case, here a primitive observable A does not determine a unique spectral observable A. However any pair (A, w) uniquely determines the spectral observable A : B(R) -+ E such that A(A) = [(A, wa)]. Similarly for n-ary observables.

We require that sharp events in E satisfy the conditions of the Mackey example. Further:

(1) extensionality holds even for fuzzy events; (2) a partial sum is defined for all pairs of orthogonal events (i.e. for

all pairs a , b such that Vu[u(a) 5 1 - u(b)]).

A Hilbert Space ExempliJication of the Fuzzy Approach

Again let 7-l be the Hilbert space associated to a physical system. S is identified with the set of all statistical operators in 7-l (like in the orthodox case). E is identified with the set of all eflects in 7-l, i.e. with the set of all linear bounded operators such that

Vw E S : Tr(wE) E [0, 11,

where 'Tr' is the trace functional. Let us put: w(E) = Tr(wE).


Any projector is an effect but not the other way around. Let us again identify the primitive observables with the self-adjoint operators of 7-l. Now the set of the spectral observables of (S, 8) properly includes the set of the spectral measures of the primitive observables. Further, any pair (A, wA) (where A is a primitive observable and wa is a fuzzy characteristic function of A) determines the effect A w i l (A). If (A, wa) is sharp (i.e., WA is the sharp characteristic function of A), Awz1 (A) is a projector.

Fuzzy and sharp events (effects and projectors) have many structural differences. Let (S , &') and (S, E ~ ) represent respectively the state- event structures constructed with the set &' of the projectors and with the set £E of the effects.

(£ ', c , ~ , I , 0) is a complete orthomodular ortholattice. Further, (S, E') satisfies o-additivity (condition (5 * *)), weak determinism (condition (8)) and the order of &' is determined by the positive certainty domain (condition (1 2)).

( f E , &,I , 1,O) instead is only a bounded regular involutive poset. A partial sum @ is defined for pairs of orthogonal effects: however E @ F is not generally the sup of E and F (condition (5) holds; however condition (5*) is generally violated). Weak determinism fails and the order of & E is not determined by the positive certainty domains. A typical event that does not coincide with the impossible event, and is never certain (for any state) is $11 (the semitransparent effect).

A necessity operation can be naturally defined on fE as follows:

O E = P Ker(I1-E) 7

where K e r ( F ) := ($ E 7-l I F$ = Q}, Q is the null vector and P ~ ~ ~ ( ' - ~ ) is the projector whose range is Ker (I - E).

It turns out that: E is sharp (i.e., O E = E ) iff E is a projector. In other words, the projectors represent the sharp substructure of the effects. The necessity operation permits us to define an intuitionistic- like complement " :

(the intuitionistic complement E" of E is the necessity of the fuzzy complement E~ of E).

This gives to £E the structure of a Brouwer-Zadeh poset. In other words ( E ~ , &, I, 1 , O ) is a bounded involutive regular poset. Further, the intuitionistic-like complement " satisfies the following conditions:


(i) a C a"" (weak double negation). (ii) a n a" = 0 (non-contradiction principle).

(iii) a C b + b" C a" (weak contraposition). (iv) a" a L (" is stronger than l). (v) a"' = a"" (when applied to intuitionistically negated events, '

and " coincide).

The effect system (S, E ~ ) has a property that may appear, prima facie, surprising. Differently from the projector-case, spectral observables A, B in (S, E ~ ) may admit a joint observable (in the sense of Definition 1.7) even if they are probabilistically complementary. For in the effect system, probabilistic complementarity does not generally imply complementarity. This has been shown by Davies (1976) and further investigated by Busch and Lahti (1984).

As a consequence, we obtain a kind of opposite situation with respect to the projector case. Here a conjunctive event A(A) fl B ( r ) does not generally exist, even if the joint observable of A and B does exist!

3. QUANTUM LOGICS

What about the 'natural' logics corresponding to the different state- event systems? The easiest way to associate logics to such systems can be carried out in the framework of an algebraic semantics. Let C be a sentential language with at least the connectives 1 (not), A (and), V (or). Let J? be the class of all algebraic structures satisfying a given set of conditions. Here we consider only structures U = (U, L,' , 1 , O ) that are at least bounded involutive posets.

A possible model of L (with respect to R) is defined as a pair t)32 = (U, p) where U E R and p is a partial function which transforms formulas a of L into elements p(a) of U, according to the conditions:

(i) p (1a) = p(a) '- (if p(a) is defined, p ( w ) = p(a)') (ii) p ( a ~ 4 ) - p(a) np(P) (if p(a), p(P) aredefinedand P ( ~ ) ~ P ( P )

exists then p (a A p) = p(a) fl p(P)). (iii) p(a v p) - p(a) U p(P) (similarly).

DEFINITION 3.1. R-consequence. a kfi ,L? (P is a R-consequence of a ) iff:

(i) for any (U, p) (U E R) if p(a), p(P) are defined then p(a) 5 p(,B);


(ii) for at least one (U, p): p(a), p(P) are defined.

In this way, any class .R characterizes a given logic LR. LR will be axiornatizable where there is a calculus with a decidable notion of proof, determining a derivability relation k L ~ such that

a kfi P iff cr kLa p foranya, p.

As is well known, orthodox (von Neumann-Birkhoff) quantum logic (QL) is characterized by the class of all orthomodular ortholattices, and this logic is axiomatizable. However the decidability of QL is still an open problem.

What about the other structures that we have considered in Section 2? The case of the orthomodular orthoposets (Suppes' quantum-me-

chanical algebras) is critical. As far as we know, the axiomatization of this logic is still an open problem. Strangely enough, quantum mechanical algebras that seem to be quite natural from both the intuitive and the physical point of view, turn out to be problematic at least in two respects:

(1) no axiornatized corresponding logic is known. (2) no genuine Hilbert-space model (which includes the projectors and

is not a lattice) is known. Genuine models of orthomodular orthoposets (that are not lattices) can be constructed in the framework of pre-Hilbert spaces (as shown in Cattaneo and Marino, 1986).

An axiomatizable logic that turns out to be 'slightly' stronger than the logic L~~~ (characterized by the class of all quantum mechanical algebras) is transitive partial classical logic (characterized by the class .RTPB of all transitive partial Boolean algebras).

Finally, let us consider the case of the logics suggested by the effect- structures and by the unsharp approaches in general. As we have seen, effects give rise to BZ-posets which are not lattices. Any lattice which is a BZ-poset is called a BZ-algebra. One can prove (Giuntini, 199 1 b):

LEMMA 3.1. Any BZ-poset can be embedded into a BZ-algebra.

Let L~~ be the logic characterized by the class of all BZ-algebras. The language of L~~ will contain a primitive intuitionistic-like negation -, and a necessity operator L, defined as: La = - l a . The semantic interpretation of - will be the obvious one.


LBZ turns out to be a fairly tractable logic, with nice metalogical properties: it is axiomatizable and decidable (Giuntini, 199 1 b, 1992). It represents a form of paraconsistent quantum logic, which gives rise to violations of the non-contradiction principle. Its fragment in the connectives 1, A, V is a sublogic of tukasiewicz infinite many-valued logic. However, LBZ appears to be stronger with respect to what is, at first sight, required by our effect-structure. For, in the LBZ-semantics, the meanings of conjunction and disjunction are always defined.

A possible physical interpretation of such situations can be found by constructing particular models of LBZ. Let (S, E) be the effect- structure in a Hilbert-space and let us consider the frame (S, I), where I is the orthogonal relation between states:

u l v H 3a E & : u(a) = 1 andu(a) = 0. -

A set of states X is called a simple proposition of ( S , I ) iff X = x'I

(where X' := {u E S I Vw E X(ulv)}) . Now, a proposition of (S, I) is defined as a pair ( X I , Xo) where XI and Xo are simple propositions and X I x:. The pair ( D l (a), Do (a)) of the positive and the negative certainty domains of any effect turns out to be a proposition in this sense.

From an intuitive point of view, a proposition may be regarded as the extensional meaning of a physical sentence, determining both the set of the physical states for which the sentence certainly holds and the set of the physical states for which the sentence certainly does not hold.

Propositions of (S, I) have the structure of a BZ-algebra where:

(i) ( X I , XO) L (YI, YO) XI c YI and Yo E Xo. (ii) ( ( X I , ~ 0 ) ) ~ = (Xo, XI).

(iii) ( ( X I , X O ) ) ~ = (Xo, x:).

As a consequence, these efSect-propositional models are models of the logic- LBZ. Here conjunction and disjunction receive a concrete physical interpretation in terms of the positive and the negative domains of sharp and unsharp physical events. Actually, the logic corresponding to this particular structure turns out to be somewhat stronger than L ~ ~ : it represents a form of 3-valued fuzzy quantum logic ( L ~ ' ~ ) , which is characterized by the class of all BZ-algebras satisfying also the following conditions:

(i) O(a U 6 ) = ma U ob.


(ii) Oa _C banda C O b * a L b.

Also L~~~ turns out to be axiomatizable (Cattaneo, Dalla Chiara, Giun- tini, 1993).

As a conclusion, we might end with a somewhat trivial remark: the question of what is the 'right' structure of the quantum events and what is the most adequate logic corresponding to it does not seem to admit a unique (non-dogmatic) answer. As also happens in other logical situations, we have discovered a variety of different structures which apparently reflect different aspects of the quantum universe. All this certainly agrees with Patrick Suppes' pluralistic philosophy.

Dipartimento di Filosojia, Universita di Firenze, via Bolognese, 52, 501 39 Firenze, Italy

REFERENCES

Beltrametti, E. and Cassinelli, G.: 1981, The Logic of Quantum Mechanics, Reading, MA: Addison-Wesley.

Busch, P.: 1985, 'Indeterminacy Relations and Simultaneous Measurements in Quantum Theory', International Journal of Theoretical Physics, 24, 63-92.

Busch, P. and Lahti, P.: 1984, 'On Various Joint Measurements of Position and Momen- tum Observables in Quantum Theory', Physical Review, 29, 1634-1 646.

Busch, P., Lahti, P. and Mittelstaedt, P.: 1991, The Quantum Theory of Measurement, Berlin: Springer-Verlag.

Cattaneo, G., Dalla Chiara, M. L. and Giuntini, R.: 1993, 'Fuzzy-Intuitionistic Quantum Logics', Studia Logica, 52,419-442.

Cattaneo, G. and Laudisa, F.: (forthcoming), 'Axiomatic Unsharp Quantum Theory (From Mackey to Ludwig and Piron)', Foundations of Physics.

Cattaneo, G. and Marino, G.: 1986, 'Completeness of Inner Product Spaces with Respect to Splitting Substances', Letters in Mathematical Physics, 11, 15-20.

Cattaneo, G. and Nisticb, G.: 1989, 'Brouwer-Zadeh Posets and Three-Valued tukasiewicz Posets', Fuzzy Sets and Systems, 33, 165-190.

Dalla Chiara, M. L.: 1986, 'Quantum Logic', in: D. Gabbay and F. Guenthner (Eds.), Handbook of Philosophical Logic, Vol. 111, Dordrecht: Reidel, pp. 427-469.

Davies, E. B.: 1976, Quantum Theory of Open Systems, London: Academic Press. Dishkant, H.: 1972, 'Semantics of the Minimal Logic of Quantum Mechanics', Studia

Logica, 30, 3-30. Giuntini, R. : 199 1 a, Quantum Logic and Hidden Variables, Mannheim: Bibliographis-

ches Institut.


Giuntini, R.: 1991 b, 'A Semantical Investigation on Brouwer-Zadeh Logic', Journal of Philosophical Logic, 20,411433.

Giuntini, R.: 1992, 'Brouwer-Zadeh Logic, Decidability and Bimodal Systems', Studia Logica, 51, 97-1 12.

Goldblatt, R. I.: 1974, 'Semantic Analysis of Orthologic', Journal of Philosophical Logic, 3, 19-35.

Kraus, K.: 1983, States, Effects and Operations, Berlin: Springer-Verlag. Ludwig, G.: 1983, Foundations of Quantum Mechanics, Vol. I, Berlin: Springer-Verlag. Mackey, G.: 1963, Mathematical Foundations of Quantum Mechanics, New York:

Benjamin. Ptik, P. and Pulmannovi, S.: 199 1, Orthomodular Structures as Quantum Logics,

Dordrecht: Kluwer. Suppes, P.: 1961, 'Probability Concepts in Quantum Mechanics', Philosophy of Science,

28,378-389. Suppes, P.: 1963, 'The Role of Probability in Quantum Mechanics', in: B. Baumrin

(Ed.), The Delaware Seminar, Vol. 11, New York: Wiley, pp. 319-337. Suppes, P.: 1966, 'The Probabilistic Argument for a Nonclassical Logic of Quantum

Mechanics', Philosophy of Science, 33, 14-21.


Dalla Chiara and Giuntini have given an excellent survey of the present state of investigation of the logical structure of standard nonrelativistic quantum mechanics. I just want to add a couple of remarks. One is concerned with the generalization to qualitative probability of the material they discuss in the first part of their paper, and the second concerns the logical structures associated with nonmonotonic upper probabilities, which arise in a natural way in the EPR kind of situation.

Qualitative Probability. Given the interest in the general logical structures associated with quantum mechanics, it is natural to ask how far we can go with a purely qualitative concept of probability. For example, can we go as far as needed to derive the logic we desire? I will not try to answer this question but sketch how one can proceed. In doing so I use directly the organization of ideas in Section 2 of Dalla Chiara and Giuntini's paper.

The first thing to do is to replace the states, which assign probabilities, by qualitative relations. So replacing each state u we now have a binary relation that is interpreted as a 5 b meaning a is no more probable than b. We define equivalence in the standard way: a = b iff a 5 b


and b 5 a , Such qualitative relations 5 satisfy, Pet us say, de Finetti's axioms, which are not strong enough to prove a numerical probability representation.

We can then state their initial conditions 1, 3 and 4 in the following fashion:

( V a : O d a i , l

(2) Extensionality

(3) Complementation: discussed later

(4) 3 event 1

We can also define the inclusion relation C of Definition 2.1 :

a L biff (Vd)[a 5 b].

We cannot define orthogonally in terms of qualitative probability with no separate algebraic structure on events available. So we can take as axioms the properties of their Lemma 2.1 :

3.1. a = a l l 3.2. a b - + b' L a' 3.3. a l b iff a & bL 3.4. a L a & b l b -+ a l b 3.5. 0 = lL

Continuing, then we have

(5) Additivity

(a) a l b - + a @ b ~ C (b) Vk [ a l e & b l c + [ a$ c 5 b @ c i f f a 5 b]]. (c) a @ b = a U b ,

where U is as defined in their paper.

The extension to a-additivity is obvious.


(8) Weak determinism

(9)-(10) do not involve any further use of probability. Definition 2.3 can be appropriately modified:

Dl(.) = {dl 15 a} Do(.) = {<I a 5 01

(1 1) Necessity operation. For any probability relation 5 and event a:

1 < Oa iff 1 5 a.

I stop at this point, for I only wanted to show that numerical, as opposed to a weaker concept of qualitative, probability is not needed for much of the logical theory derived from the foundations of quantum mechanics - a purely qualitative foundation is enough for much of the work.

Nonmonotonic Upper Probabilities. Recently Zanotti and I (1991) showed how a nonmonotonic upper probability could serve as a generalized joint distribution of the correlation of events that arise in the EPR type of experiment. The logical structure is naturally just Boolean algebra but the measure on these Boolean algebras is the weak one of an upper probability that is not monotonic. Its properties are these:

DEFINITION 1. Let 0 be a nonempty set, F a Boolean algebra on 0, and P* a real-valued function on F. Then 0 = (0, F, P*) is an upper probability space if and only if for every A and B in F

1. 0 5 P*(A) 5 1; 2. P*(O) = 0 and P*(0 ) = 1; 3. I f A n B = 0, then P*(AU B ) 5 P*(A) + P*(B) .

An upper probability P* is nonmonotonic if there are events A and B such that

A L B , but P * ( A ) > P * ( B ) ,


contrary to a familiar property of any probability measure. Here is a brief account of how nonmonotonic upper probabilities

are compatible with EPR correlations. Four random variables A, A', B, B' with the four correlations A B , AB', A 'B and A'B' are given theoretically. Then for various angles of spin measurement, there can be no joint probability measure compatible with the given correlations, but - and this is the point - there can be a joint nonmonotonic upper probability measure compatible with the given correlations.

My question is: how should we think about the logic of this setup? It is very different from the kind of examples discussed by Chiara and Giuntini, and perhaps represents still another of the many different ways to represent various quantum structures.

REFERENCE

Suppes, P. and Zanotti, M.: 1991, 'Existence of Hidden Variables Having Only Upper Probabilities', Foundations of Physics, 12, 1479-1499.

PART VII

LEARNING THEORY, ACTION THEORY, AND ROBOTICS

W. K. ESTES

FROM STIMULUS-SAMPLING TO ARRAY-SIMILARITY

THEORY

ABSTRACT. The development of mathematical models of cognition from 1950 to the 1980s was shaped to an important degree by efforts to account for three empirical phenomena - probability matching in simple multiple-choice learning, generalization as a function of commonality of learning and test situations, and the learning of perfect discriminations between similar stimuli. Among the models of the 1 9 5 0 ~ ~ linear models accounted for the first of these and stimulus sampling theory for the first two, but the third remained elusive for more than two decades. The third of the phenomena was brought into the fold with the emergence of array-similarity theory in the late 1970s. The critical change from stimulus sampling theory was the replacement of an additive similarity function with a multiplicative function, the product rule. Anal- yses within the general framework of the work of Estes and Suppes on foundations of linear and stimulus-sampling models shows that the stimulus-sampling and array- similarity models are isomorphic in some very special cases, exhibit close agreement in their predictions whenever stimulus structures do not allow significant nonlinearity of similarity measures, but in many cases, especially those involving discrimination or categorization of multidimensional stimuli, may diverge widely. The nonlinearities in the array-similarity model make prohibitively difficult the derivation of functions like the learning curves and descriptives statistics of data that were the bread-and-butter products of linear and stimulus-sampling models, but application and testing of both types of models can now be readily accomplished via implementation of the models on high speed computers.

Three basic facts about learning have been cornerstones in the development of mathematical learning theory: (1) probability matching; (2) transfer of learning in direct relation to the commonality of training and test situations; and (3) discrimination between similar stimuli. The first two of these were treated with considerable success by the family of learning models on which Patrick Suppes and I worked jointly in the 1950s, but the goal of handling all three within a unified theory eluded us. There would have been no difficulty if we had been willing to aug- ment our theory with ad hoc mechanisms, but we persisted in feeling that a parsimonious solution could be achieved by some revision of our core assumptions. I think now that we were right, and in this paper I

Fi Hurnphreys (ed.), Patrick Suppes: Scientific Philosop/pher; Vol. 3, 171-192. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

172 W. K. ESTES

will try to retrace the path that led to what now seems to be a satisfactory conclusion to several decades of effort. I will begin with a brief review of the earlier models in relation to the cornerstone facts, and then show how a revision of a single axiom brought our original objective within reach.

LINEAR MODELS AND PROBABILITY MATCHING

The term probability matching refers to a phenomenon frequently observed when an individual is learning to anticipate, or predict, an uncertain event. Suppose, for example, that an observer is viewing a radar screen on which images of planes belonging to either country X or country Y appear from time to time and that actually the two types of planes appear in random sequence with fixed probabilities. The characteristic result is that, over a series of trials, the observer's probability of predicting that the next image will be of a given type tends asymptoti- cally to match the true probability. (For reviews of the relevant research literature, see Estes, 1964, 1972.) The popularity of what were termed linear models of learning (Bush and Mosteller, 1955; Estes and Suppes, 1959a) during the 1950s owes a great deal to the ability of these models to predict probability matching.

I will confine attention to the simplest member of the linear model family, the version first applied to probability learning by Estes and Straughan (1954) and analyzed in detail by Estes and Suppes (1959a). In a typical experiment simulating the radar example, the learner attempts on each of a series of trials to predict which of two events, El or E2, will occur following a ready signal; although the learner is not informed of the fact, the events actually occur with fixed probabilities, which may be denoted T and 1 - T . With px,, denoting the probability that a learner who has had history x through the first n - 1 trials of the experiment will predict event El on trial n, the learning axioms of the model can be expressed in two linear difference equations:

If El is the outcome of trial n, then

and if E2 is the outcome of trial n, then

STIMULUS-SAMPLING TO ARRAY-SIMILARITY 173

where a is a constant, which may be termed a 'learning parameter', with a value in the interval 0 5 a 5 1. Clearly, over a uniform sequence of El trials, p,), would tend to unity and over a uniform sequence of E2 trials, px l , would tend to zero. Our main interest, however, is in predicting the learner's behavior over a random series. Toward that end, we compute the expectation of p,),+l by weighting both sides of Equations 1 and 2 by n and 1 - n, respectively, and summing to obtain

Finally, we weight both sides of Equation 3 by the probability of history x (the history comprising the preceding trial sequence) and sum the products over all x, yielding

where p, denotes the unconditional probability of an El prediction on trial n. In customary terminology, p,,, is referred to as the choice probability for a particular learner and p, as the expected probability in a population of identical learners (identical in having the same value of the parameter a). It can be proved (Estes and Suppes, 1959a) that over a series of trials under the specified conditions, p, must approach a fixed 'asymptotic' value, p; setting both p,+l and p, in Equation 4 equal to p and solving the resulting equation, we obtain the simple result

and conclude that the model does indeed predict probability matching.

STIMULUS SAMPLING THEORY

By appropriate application of the 'learning operators' expressed in Equa- tions l and 2, predictions can be derived for situations in which event probabilities vary over trials or depend in various ways on the learner's responses. Many of these predictions have been tested with successful results (Estes, 1972), but our interest here is in turning to a critical limitation on the model - its inability to address the problem of generalization from learning under one set of stimulating conditions to tests conducted under different conditions. The reason for this limitation is that the

174 W. K. ESTES

linear models include no means for representing stimulus properties, a lacuna that is filled in a closely related class of models subsumed under the rubric stimulus sampling theory (Atkinson and Estes, 1963; Estes, 1950,1959; Estes and Suppes, 1974).

The principal additional constituent of stimulus sampling models is a set, S , representing the stimulating situation in which learning occurs. The elements of S, N in number, represent components or aspects of the stimulating situation, and it is assumed that, owing to variation in both the physical environment and the internal state of the learner, only a random sample of these elements is active on any learning trial. Only active elements, corresponding to aspects of the situation perceived and attended to by the learner, undergo learning. The connection between the stimulus set and the learning process is a 'conditioning function', C , which partitions S into subsets Cj corresponding to the response alternatives being learned. In the simple probability learning experiment discussed above, the set S would be partitioned on any trial into subsets C1 and C2, comprising the elements currently associated with the responses, henceforth denoted Al and A2, of predicting event El or E2, respectively.

Several variants of stimulus sampling theory have been distinguished (Estes, 1959), but it will suffice for present purposes to consider only the fixed sample-size version, in which all samples drawn from S are assumed to be of the same size, ON, where 8 is a constant with a value between 0 and 1, and to have equal probabilities of occurrence. It will be apparent that the sample on any trial, like the master set S, is partitioned into subsets of elements associated with different responses. The connection between the stimulus set representation and response probability is provided by the assumption that the probability of any response, Ai, on any trial is equal to the proportion of elements in the trial sample that are associated with Ai. Learning is conceived to be essentially the same process as in the linear model except that the 'learning operators' corresponding to Equations 1 and 2 are expressed in terms of the classification of individual elements of s.' With the probability that element i of S is associated with response j denoted by f i j , these operators take the form:

If El is the outcome of trial n, then for element i,

STIMULUS-SAMPLING TO ARRAY-SIMILARITY

and if E2 is the outcome of trial n,

For the experiment in which El and E2 occur with probabilities .jlr and 1 - 71-, a derivation analogous to that of Equation 3 yields for the expected (unconditional) probability that element i is associated with response Ai on trial n + 1

and it will be apparent that probability matching occurs at the level of individual stimulus elements. It is intuitively obvious, and can readily be demonstrated (Estes, 1959) that Equation 8 implies

which is identical to the probability recursion for the linear model (Equa- tion 3) with l - 6' substituted for a as the 'learning parameter'. More generally, it can be proved that as the number of elements in S becomes large, the stimulus sampling model approaches isomorphism to the linear model (Estes and Suppes, 1959b). The equivalence holds, however, only for the modelling of learning that occurs in a single stimulating situation. I will turn now to the task of showing how the stimulus sampling model can interpret experiments that include learning and testing in two or more distinct, though possibly related, stimulating situations.

For concreteness, let us suppose that in the example, discussed above, of an individual learning to predict the incidence of different types of aircraft, the learner is given experience at different times with radar screens placed at different locations, both the details of the display panels and the probabilities of types X and Y being different at the two locations. We will represent the two stimulating situations in the model by two sets, S1 and S2, which are of equal size and have a proportion g of elements in common. We will denote the responses of predicting type X and type Y by A1 and A2, respectively, and assume that initially all elements of both sets are associated with A2, producing the state shown in the topmost panel of Figure 1. During the first series of learning trials, aircraft of type X appear on all trials, leading to the situation portrayed in the middle panel of Figure 1: all elements of S1 but only a fraction g of the elements of S2 are associated with response A1, so on tests at this point, response A1 will have a probability of

176 W. K. ESTES

unity in situation 1 and a probability equal to g in situation 2. Since no training has yet been given in situation 2, one says that the learning that occurred in situation 1 has transferred, or generalized, in part to the somewhat similar situation 2. Transfer of training in proportion to the commonality between training and test situations is a long familar empirical phenomenon, and one of the early successes of the stimulus sampling model was its demonstrated ability to predict this transfer (Atkinson and Estes, 1963; Schoeffler, 1954).

To bring out a critical problem that emerged for the model, we will suppose that in this example the environment is such that planes of type X appear on all trials in situation 1 and planes of type Y in situation 2, and that the learner's task is to acquire the ability to respond correctly on all trials in both situations. The task seems simple, but consider what happens in terms of the model. If, following the training series in situation 1 just described, the learner is given a series all in situation 2, the state of learning as represented in the model will be as shown in the lowermost panel of Figure 1. Now, on a test in situation 1, probability of correct responding will be reduced from unity to 1 - g. Clearly, if training continues, whether alternately in the two situations or in a random sequence, the classification of the elements in the intersection of sets S1 and S2 will continue to switch back and forth between C1 (associated with response A1) and C2 (associated with A2), and the learner will never develop the capability of producing uniformly correct responses in both situations. In the terminology of learning research, if g is greater than zero then only an imperfect discrimination between S1 and S2 can be learned. The reason why this prediction is critical with regard to the model is, of course, that one would expect a perfect discrimination to be acquired easily by normal human learners, an expectation supported by numerous experiments on discrimination between stimuli that have elements in common (e.g., Robbins, 1970; Uhl, 1964).

REVISIONS OF THE STIMULUS-SAMPLING MODEL

Over a period of years there were numerous efforts to solve the 'overlap problem' by adding various mechanisms to the basic structure of stimulus sampling theory, but none yielded fully satisfactory solutions (Bush and Mosteller, 1951; Estes, 1959; Restle, 1955, 1961). Because the

STIMULUS-SAMPLING TO ARRAY-SIMILARITY

Initial state

Train A1 in S t

Test in S2

Train A2 in S 2

Test in S1

Fig. 1. Portrayal of learning and transfer in the stimulus sampling model. Two situations are represented by sets S1 and S2, having a fraction g of elements in common. Initially (top row), all elements of both sets are associated with response A2. Next (middle row), training is given in S1 till all of its elements, including those common to S2, are associated with Al, the probability, pl, of A1 in S 2 increasing from 0 to g. Finally (bottom row), training is given in S 2 until all of its elements are again associated with A2, reducing the probability of A1 in S1 to 1 - g.

theory had achieved some striking success in accounting for probability learning and other phenomena, there was understandable resistance to the idea of tampering with any of the basic assumptions; but ultimately it became clear that some revisions of the original structure were essential. The revisions that now seem essential to an adequate joint account of generalization and discrimination came in two steps, the first having

178 W. K. ESTES

to do with the set-theoretical representation and the second with the algorithms for computing stimulus similarity and response probability.

It gradually became clear during the 1960s that the original idea of representing a stimulus situation by a homogeneous set of elements (identical except for their assignment to the subsets, Cj, defined by the conditioning function) did not allow convenient representation of the multidimensional stimuli with which we are concerned in all but the simplest forms of learning. A solution to this problem that was put forward almost simultaneously by investigators with otherwise quite different orientations was to represent stimuli in terms of their values on dimensions or attributes (Bower, 1967; Underwood, 1969). To illustrate with a concrete example, suppose that the stimuli in a simple discrimination, or categorization (henceforth I will use only the term categorization), experiment are drawings of plane figures that are either triangles or squares and either dark or light, with Al being the correct response to triangles and A2 the correct response to squares, regardless of brightness. The notion of a stimulus attribute, such as form or brightness, seems foreign to the stimulus sampling model, but it can be implemented by means of constraints on sampling probabilities. For this example, we will define two stimulus sets, S1 and S2, each of which is available for sampling with probability 0.5 on any learning trial. Set S1 comprises three elements, which we denote el, e3, and e4; and S2 comprises three elements e2, e3, and e4, two of which are common to S1. The constraints on sampling are imposed by the experimenter, and take the form illustrated in Table I. Elements el and e2 (corresponding to a dark triangle) occur together with probability 0.5 whenever S1 is available, which occurs only when A1 is correct; el and e3 (a light triangle) occur on the remaining S1 trials; elements e2 and e j (a dark square) constitute the sample with probability 0.5 when S2 is available, which occurs only on trials when A2 is correct; and e2 and e4 constitute the sample on the remaining S2 trials.

Expressions analogous to Equation 8 may now be derived for the probability that any element is associated with a given response category. Taking Al as the reference category, and letting fi denote the probability that e; is associated with A1, we obtain: for el,

STIMULUS-S AMPLING TO ARRAY-SIMILARITY

TABLE I

Design matrix for simple discrimination (categorization) learning experiment.

Available set Sampled elements Correct response

Note. Cell entries are probabilities of stimulus samples (element pairs) on trials when the indicated response is correct.

for e3

for e2

and for e4

To obtain Equation 10, we note that with probability 0.5, A2 is correct on trial n and S1 is not available, so f must remain unchanged; with probability 0.5, S1 is available, and the new value of f is 1. For Equation 1 1, f 3 is similarly unchanged with probability 0.5 (whenever the trial sample is either ele4 or e2e4); on half of the remaining trials, f 3 goes to 1 and on half to 0. Analogous reasoning yields Equations 12 and 13.

Our main interest is in what the model predicts about the asymptotic state of learning. Again letting f i denote the asymptotic value of fi,, and setting fi,n+l = fi,, = fi in each of Equations 10-1 3 and solving for fi, we obtain f 1 = 1, f 3 = 0.5, f 2 = 0, and f 4 = 0.5. Asymptotic response probability in the model is the sum of the appropriate f i for a given

1 80 W. K. ESTES

type of trial divided by the sample size. Thus, again letting p denote asymptotic probability of an A1 response, on trials with S1 available, we have p = (1 + 0.5)/2 = 0.75 and, on trials with S2 available, p = (0 + 0.5)/2 = 0.25, so predicted categorization performance is far from perfect.

THE ARRAY-SIMILARITY MODEL

The reformulation of the stimulus sampling model that overcomes this inherent limitation is known in the literature as the context model (Medin and Schaffer, 1978; Nosofsky, 1984) when applied to categorization and more generally as the array-similarity model (Estes, 1986).~ The most important innovation lies in the manner of computing similarity between multielement stimuli, the additive function of the stimulus-sampling model being replaced by a multiplicative function. For purposes of this paper, it will suffice to consider only finite stimulus sets having the property that the pairwise similarities of all nonidentical elements are equal. (A full exposition of the model has been presented by Estes, 1994.) The measure of similarity between any two elements is either unity (if they are identical) or s , a parameter with a value in the interval 0 < s 5 1. The similarity between any two samples of elements, sl

and s2, is computed by an algorithm known as the product rule. One sample, say s 1 , is taken as the referent and a product is formed in which a factor 1 is entered for each element of s 1 that is also in s2 and a factor s for each element of sl that is not in s2. Thus, for example, if sl

comprises the elements el, e2, and e3 and s 2 the elements el, e4, and es, the measure of similarity computed by the product rule is 1 * s * s = s2.

No systematic comparisons of the stimulus-sampling and array- similarity models have been carried out, but some salient correspon- dences and differences can be conveniently brought out by examination of a few specific cases. First, to look at stimulus generalization, I will return to the example portrayed in Figure 1, limiting consideration to the base in which sample size is unity, that is, a single element is sampled on any trial. Letting N denote the number of elements in either S1 or S2, the number common to S1 and S2 is M = gN and the number unique to either set is N - M = (1 - g)N. At the beginning of the first training series (row 1 of Figure I ) , none of the elements of either set are associated with either response A or response A2; at the end of that


series (row 2 of Figure I), all elements of S1 are associated with Al, and the probability of A1 when S1 is sampled, denoted pl in the figure, is unity. To predict the probability of Al if set S2 is sampled at this point (a test for generalization), computation from the array-similarity model proceeds in two steps. The first step is to compute the summed similarity (termed the global similarity) of the sample to all stimuli (in this case elements of S2) associated with that response and the global similarity to all elements associated with the alternative response. The second step is to obtain an expression for response (categorization) probability from the global similarities by a specialization of the choice model of Luce (1963). This expression is simply the global similarity for the reference response (A1 in this instance) divided by the sum of the global similarities.

To carry out these computations, we use the fact that M elements of S2 are associated with A1 and the remaining N - M with A2. The element sampled from S2 falls in the former subset with probability g and in the latter with probability (1 - g). When the element sampled is in the first subset, its global similarity to the full subset is 1 + ( M - l ) s , because by our assumptions, its similarity to itself is 1 and to each other element of the subset is s; and its global similarity to the other subset is ( N - M ) s . Hence, probability of response A l when this element, which I will denote ec, is in the sample is

When the element sampled is in the second subset, its global similarity to the first subset is M s and to the second subset 1 + ( N - M - l ) s , and the probability of A1 in the presence of this element, denoted eu, is

(15) P(AI ( e,) = Ms/[lMs + (1 + ( N - M - l)s}1

= M s / [ l + N - l ) ~ ] .

Finally, we weight Equations 14 and 15 by the probabilities of sampling an element from the common or unique subset of S2 and sum, obtaining for a test on which S2 is sampled at random,

W. K. ESTES

TABLE I1

Hypothetical memory array arising from the category learning experiment illustrated in Table I.

Elements Category in trial sample A1 A2

Note. Cell entries are numbers of stored representations at the end of a learning series of 4n trials.

the same result as that predicted by the stimulus-sampling model. This equivalence arises because our sampling assumptions result in the occurrence of only first powers of s in the expressions for global similarities. In the absence of this limitation, the equivalence holds only in special cases.

We turn now to an application of the array-similarity model to the categorization experiment represented in Table I. In the terminology of the array-similarity framework, it is customary to speak of memory storage rather than association. If an experiment having the design shown in Table I were conducted with each stimulus sample occurring equally often, then the outcome of a series of 4n learning trials would be as illustrated in Table 11, n representations of the el e3 and el e4 stimuli being stored in the Al column of the memory array and n representations of e2e3 and e2e4 in the A2 column. A distinctive conceptual difference between the stimulus sampling and the array-similarity models is that, in the latter, the makeup of stimulus samples is preserved in memory, whereas, in the former, that information is lost, and only information about occurrences of individual elements is preserved. If, following this learning series, a test is given with el e3, application of the product rule yields n(l + s) and n(s + s2) for its global similarities to the Al


and A2 columns of the memory array, respectively, and the predicted response probability is

Now, if the similarity parameter, s, happened to be equal to 113, the response probability would be 0.75, just as in the stimulus sampling model, but in general the predictions of the models differ. In particular, if s is equal to 0, then the response probability is unity, and the array- similarity model is seen to predict the result that both intuition and related research indicate should be expected for normal human learners.

The replacement of an additive with a multiplicative rule for computing similarity has enabled us to overcome a major limitation of the stimulus sampling model. The price paid for this advance is that the expression for predicting asymptotic response probability is no longer parameter free, but requires estimation of the free parameter, s. Howev- er, unlike the illustrative example of Tables I and 11, most experiments conducted to test and apply categorization models are rich enough in data so that this requirement poses only technical problems that are routinely handled by standard methods. (See, for example, Estes et al., 1989; Nosofsky, 1992). The critical question now is whether the array similarity model can handle phenomena, such as probability matching, that were a main source of support for the stimulus-sampling model.

The experiment whose design is illustrated in Table I is converted into a simple probability-learning experiment if we require all of the cell entries (stimulus probabilities) in each column to be equal. For an experiment in which the learner's task is to predict on each trial whether event El or E2 will occur (i.e., whether Category Al or Category A2 is correct) and these events have probabilities n and 1 - n, respectively, all four entries under A1 in a design table analogous to Table I would be 7114 and all four entries under A2 would be (1 - 7r)/4. If, say, 400 learning trials were allowed, then all four cell entries under A1 in a representation of the resulting memory array analogous to Table I1 would be 75, and all four entries under A2 would be 25. Then, on a trial with any one of the stimuli, say ele3, the global similarity to the Al array would be 75 + 150s + 75s2 and the global similarity to the A2 array would be 25 + 50s + 25s2. Entering these similarities in an expression of the same form as Equation 17 yields

W. K. ESTES

and of course the same result holds for each of the other three stimuli. Thus, the array-similarity model yields the same parameter-free probability-matching prediction as the stimulus sampling model. This result can be shown to hold for all of the experimental designs for which probability matching has been predicted by the stimulus-sampling or linear models. The essential constraints are that the probabilities of the events that the learner is trying to predict are constant over trials (and independent of the learner's responses) and that the sampling probabilities of all stimulus elements are independent of the trial type (i.e., are the same whether A1 or A2 is the correct response). It might be rnen- tioned that the restriction to two response alternatives is inessential; the probability-matching prediction holds for any finite number of response alternatives in the array-similarity model, as in the stimulus-sampling and linear models.

SOME COMPARISONS BETWEEN MODELS

To give a better perspective on the relationship between the stimulus- sampling and array-similarity models than is afforded by the examples presented above, I will review the concepts and assumptions of the stimulus-sampling model somewhat more formally and then discuss the changes in basic concepts and axioms that produce the array-similarity model. In the review, I will follow the exposition of the stimulus- sampling model by Estes and Suppes (1974) but dispense with formal notation except where it is essential for clarity.

The basic concepts are a stimulus set, S, consisting of N elements; a conditioning function on S; a set of alternative responses, Ai; a set of trial-outcome event^,^ E j ; a sample space, X; and a probability measure on the sample space. A trial of an experiment comprises the pentuple (C, T , s k , i , j ) where C denotes the partition of S into subsets associated with the various responses; T denotes the stimuli made available for sampling; sk is the stimulus sample on the trial, and i and j denote the response (Ai) and outcome event ( E j ) occurring on the trial. The principal axioms, stated informally, are as follows.


Restrictions on Event Probabilities

The probabilities of stimuli made available for sampling and the probabilities of outcome events may be conditional only on preceding observable events (stimulus presentations, outcome events, and responses).

Sampling

All stimulus samples of equal size are equiprobable, and sampling probabilities depend only on the stimulus presentation, not on past events or the conditioning partition of S .

Learning

At the end of a trial on which outcome-event Ej occurs, all stimulus elements in the trial sample are associated with response Aj (i.e., are in the set Cj of the conditioning partition), regardless of their previous status.

Response Probability

The probability of a response Ai in the presence of a stimulus sample is equal to the proportion of elements in the sample associated with the response (i.e., in Ci).

Only a few of the basic concepts are changed in going from the stimulus- sampling to the array-similarity model. The concept of a stimulus set, S , is unchanged, but the conditioning partition is defined, not on S , but on the set of samples that can be drawn from S . In this exposition, we are confining attention to the fixed-sample size version of stimulus- sampling theory. Thus, for any set S and fixed sample size u (where u = ON), we can define a set U whose elements, (U) in number, represent all of the different samples of size u that can be drawn from S . The conditioning function is defined on U rather than on S: The subset Ci of U includes all of the samples (i.e., elements of U) that are associated with response A;. The learning axioms are unchanged from stimulus-sampling theory except that they are stated in terms of the conditioning function on U : at the end of a trial on which outcome- event Ej occurs, the element of U sampled on the trial is associated

186 W. K. ESTES

with response Aj (i.e., is in the set C j of the conditioning partition), regardless of its previous status. The axioms for response probability in the array-similarity model are new: Given a test on which some element of U is sampled, first the global similarity of this element to each subset Ck of U is computed by application of the product rule; then the probability of response A, is given by the global similarity of the sampled element to Ci divided by the sum of its global similarities to all of the Ck, k = 1,2, . . . , r , where r is the number of alternative responses.

Owing to the different axioms for response probability, predictions of the two models often differ widely, the differences usually depending on the magnitude of the similarity parameter of the array model, but in some important special cases predictions are identical. I will discuss just one of these cases, the specialization of stimulus-sampling theory known as the pattern model (Estes, 1 959).4 This case arises when sample size is constrained to be equal to 1, each element of S being sampled with probability 1 / N on any trial. Applied to the simple probability learning experiment, the learning recursion for the pattern model, given by Equation 9 for the general case, takes the form

where pn is again the probability of response A1 on trial n, .ir the probability of outcome-event E l , and c the learning parameter, i.e., the probability that the outcome event of a trial is effective in producing learning (the current conditioning partition remaining unchanged with probability 1 - c).

The corresponding case of the array-similarity model arises analo- gously when sample size is unity. The set U is then identical to the set S , and its elements are sampled with probability 1/N. Pairwise similarities of the elements are either 1 or s, the former for a comparison of an element with itself and the latter for all other pairs. In this case, the learning axioms are the same as those of the stimulus-sampling pattern model, and lead to the same learning recursion (Equation 18). To demonstrate this equivalence, we need only show that in this case the expressions for response probability in the two models agree. From the assumptions of the stimulus sampling model, discussed above, it follows that if, on any trial, N1 of the N elements of S are associated with response Al (i.e., are in C1), then the probability of Al is Nl /N. Applying the special case of the array-similarity model to the same sit-

STIMULUS-S AMPLING TO ARRAY-SIMILARITY 187

uation, we note that the memory arrays for A1 and A2 contain N1 and N - N1 elements of U, respectively. An element of the A1 array is sampled with probability Nl I N , and its similarities to the Al and A2 arrays are 1 + (NI - 1 ) s and ( N - Nl ) s, respectively, yielding an Al probability of [I + (Nl - l )s ] / [1 + (N - l)s]. An element of the A2 array is sampled with probability (1 - Nl/N) , and its similarities to the Al and A2 arrays are N l s and 1 + ( N - NI - I)s, respectively, yielding an Al probability of Nls / [ l + ( N - l )s] . Weighting each of these conditional probabilities by the associated sampling probability and adding, we obtain for the unconditional probability of response A1

which establishes the equivalence. For larger sample sizes, linear recur- sions in response probability like Equation 18 are not generally available, and usually the only practical recourse is to generate learning functions trial-by-trial by means of a computer program.

It is interesting to note the critical role of technological advance in the evolution of models. In the 1950s, both exploring the empirical implications of models and applying the models to experiments had to be accomplished mainly by deriving explicit functional expressions from the models, corresponding to quantities that could be computed from experimental data. The popularity and influence of linear and stimulus sampling models owed a great deal to the ease with which these derivations could be accomplished. Replacing the linear similarity function of the stimulus sampling model with a multiplicative function in that earlier period would not have been regarded as a practical tactic, for few interesting predictions regarding learning functions and the like could have been derived from the resulting models. In the 1990s all has changed, and it is easier to explore the properties of array-similarity models and apply them to experiments by means of computers than it was to accomplish those tasks for the earlier models via explicit derivations. Analyzing and testing the newer models raises new problems, but there seems to be no likelihood that the trend will ever be reversed.

188 W. K. ESTES

ACKNOWLEDGMENT

Preparation of this chapter was supported by Grant BNS 90-09001 from the National Science Foundation.

Department of Psychology, Harvard University, 33 Kirkland St., Cambridge, MA 021 38, U.S.A.

NOTES

The learning process of the model is actually deterministic: At the end of a trial whose outcome is E j , all elements in the trial sample are assigned to the subset Cj of S , which includes all elements associated with Aj. In this exposition, however, we are interested in predictions about learning that can be made in advance of a trial, when the trial sample is unknown, so we use the fact that the probability of sampling any element of S is equal to 8 and employ a probabilistic formulation.

A different approach to this problem, developed by Gluck (1992), is to combine the formalism of the stimulus-sampling model with the learning mechanism of an adaptive network model. It remains to be determined whether the two approaches can be brought together in a single theory.

I am dispensing here with a distinction made by Estes and Suppes (1974) between an observable trial outcome, Oj , and an unobservable 'reinforcing' event, E j , that is assumed to occur when outcome O j produces a learning effect on a given trial.

The pattern model, like the linear model, owes its viability not so much to its capability of describing a few simplified experiments as to its frequent appearance as a module in more complex models. For a timely case in point, see Suppes (1 992).

REFERENCES

Atkinson, R. C. and Estes, W. K.: 1963, 'Stimulus Sampling Theory', in: R. D. Luce, R. R. Bush, and E. Galanter (Eds.), Handbook ofMathematica1 Psychology, Vol. 2, Wiley, New York, pp. 121-268.

Bower, G. H.: 1967, 'A Multicomponent Theory of the Memory Trace', in: K. W. Spence and J. T. Spence (Eds.), The Psychology of Learning and Motivation: Advances in Research and Theory, Academic Press, New York, pp. 230-327.

Bush, R. R. and Mosteller, F.: 1951, 'A Mathematical Model for Simple Learning', Psychological Review, 58, 3 13-323.

Bush, R. R. and Mosteller, F.: 1955, Stochastic Models for Learning, Wiley, New York. Estes, W. K.: 1950, 'Toward a Statistical Theory of Learning', Psychological Review,

57,94-107.


Estes, W. K.: 1959, 'Component and Pattern Models with Markovian Interpretations', in: R. R. Bush and W. K. Estes (Eds.), Studies in Mathematical Learning Theory, Stanford Univ. Press, Stanford, CA, pp. 9-52.

Estes, W. K.: 1964, 'Probability Learning', in: A. W. Meiton (Ed.), Categories of Human Learning, Academic Press, New York, pp. 89-128.

Estes, W. K.: 1972, 'Research and Theory on the Learning of Probabilities', Journal of the American Statistical Association, 67, 8 1-1 02.

Estes, W. K.: 1986, 'Array models for Category Learning', Cognitive Psychology, 18, 500-549.

Estes, W. K.: 1994, Classijcation and Cognition, Oxford Univ. Press, OxfordlNew York.

Estes, W. K., Campbell, J. A., Hatsopoulis, N., and Hurwitz, J. B.: 1989, 'Base-Rate effects in Category Learning: A Comparison of Parallel Network and Memory Storage-Retrieval Models', Journal of Experimental Psychology: Learning, Mem- ory, and Cognition, 15,556-571.

Estes, W. K. and Straughan, J. H.: 1954, 'Analysis of a Verbal Conditioning Situation in Terms of Statistical Learning Theory', Journal of Experimental Psychology, 47, 225-234.

Estes, W. K. and Suppes, P.: 1959a, 'Foundations of Linear Models', in: R. R. Bush and W. K. Estes (Eds.), Studies in Mathematical Learning Theory, Stanford Univ. Press, Stanford, CA, pp. 137-179.

Estes, W. K. and Suppes, P.: 1959b, Foundations of Statistical Learning Theory, 11. The Stimulus Sampling Model, Technical Report No. 26, Institute for Mathematical Studies in the Social Sciences, Stanford University, Stanford, CA.

Estes, W. K. and Suppes, P.: 1974, 'Foundations of Stimulus Sampling Theory', in: D. H. Krantz, R. C. Atkinson, R. D. Luce, and P. Suppes (Eds.), Contemporary Devel- opments in Mathematical Psychology, Vol. 1, Learning, Memory, and Thinking, Freeman, San Francisco, pp. 163-1 83.

Gluck, M. A.: 1992, 'Stimulus Sampling and Distributed Representations in Adaptive Network Theories of Learning', in: A. F. Healy, S. M. Kosslyn, and R. M. Shiffrin (Eds.), From Learning Theory to Connectionist Theory: Essays in Honor of William K. Estes, Vol. 1, Erlbaum, Hillsdale, NJ, pp. 169-199.

Luce, R. D.: 1963, 'Detection and Recognition', in: R. D. Luce, R. R. Bush, and E. Galanter (Eds.), Handbook of Mathematical Psychology, Vol. 1, Wiley, New York, pp. 103-189.

Medin, D. L. and Schaffer, M. M.: 1978, 'Context Theory of Classification Learning', Psychological Review, 85,207-238.

Nosofsky, R. M.: 1984, 'Choice, Similarity, and the Context Theory of Classification', Journal of Experimental Psychology: Learning, Memory, and Cognition, 10, 104- 114.

Nosofsky, R. M.: 1992, 'Exemplars, Prototypes, and Similarity Rules', in: A. F. Healy, S. M. Kosslyn, and R. M. Shiffrin (Eds.), From Learning Theory to Corznectionist Theory: Essays in Honor of William K. Estes, Vol. 1, Erlbaum, Hillsdale, NJ, pp. 149-167.

Restle, F.: 1955, 'A Theory of Discrimination Learning', Psychological Review, 62, 11-19.

Restle, F.: 1961, Psychology of Judgment and Choice, Wiley, New York.

190 W. K. ESTES

Robbins, D.: 1970, 'Stimulus Selection in Human Discrimination Learning and Trans- fer', Journal of Experimental Psychology, 84, 282-290.

Schoeffler, M. S.: 1954, 'Probability of Response to Compounds of Discriminated Stimuli', Journal of Experimental Psychology, 48,323-329.

Suppes, P.: 1992, 'Estes' Statistical Learning Theory: Past, Present, and Future', in: A. F. Healy, S. M. Kosslyn, and R. M. Shiffrin (Eds.), From Learning Theory to Connectionist Theory: Essays in Honor of William K. Estes, Vol. 1, Erlbaum, Hillsdale, NJ, pp. 1-20.

Uhl, C. N.: 1964, 'Effect of Overlapping Cues upon Discrimination Learning', Journal of Experimental Psychology, 67,9 1-97.

Underwood, B. J.: 1969, 'Attributes of Memory', Psychological Review, 76, 559-573.


One of my great experiences in the first ten years of academic life was working with Bill Estes from 1955 to 1960. Of course our interaction has continued over the years, but it was particularly intense during that five-year period, which began when we were both fellows at the Center for Advanced Study in the Behavioral Sciences in 1955-56. Before that time I had had an interest in learning theory of at least a desultory sort, had tried to study Hull, Tolman, Guthrie, and Thorndike, but without a really serious intellectual engagement. When Bill Estes began explaining to me his work in statistical learning theory in the fall of 1955 I took to it immediately and realized very soon that here were theoretical ideas I could understand and perhaps contribute to. As I remarked in Suppes (1992) I regard Estes' first paper in statistical learning theory (Estes, 1950) as one of the seminal papers in psychology in the twentieth century. It marks a genuine turning point and heralds in the first of many significant papers by Estes and others, the development of stochastic theories of learning in the second half of this century.

My own work with Bill Estes ended with our intense scrutiny of the foundations of stimulus-sampling theory, published in 1974, many years after most of the work was done. It is evident from Estes' paper that he has continued to tackle the problems of extension and modification in a fundamental way. He gives in his present paper an excellent introductory and critical analysis of his array-similarity theory which is a natural extension of the earlier work. Work on similarity in various directions by many psychologists shows how important the concept is to many parts of psychological theory. Estes' work is particularly salient for the way he has brought it at a fundamental level into general


learning theory. It is worth noting that the concept of similarity is one also that marks a convergence of interest between philosophers and psychologists. Without attempting a detailed exposition, I cannot resist recalling that the three qualities that give rise to associations in Hume's theory (173911888) of the understanding are resemblance (i.e., similarity), contiguity in time or place, and cause and effect. Hume uses the term resemblance instead of similarity but the role it plays in his theory is just what, by and large, we think of the modern theory of similarity playing. In many ways it is surprising how long it has taken psychologists to really begin to develop a thorough and sophisticated theory of similarity, but there is no doubt that it is here to stay at the center of many different theoretical developments and will be a concept that receives intense scrutiny in the decades ahead. Estes emphasizes the subjective nature of similarity and this is already very evident in Hume's thinking as well. I quote from an early passage (p. 11) in the Treatise of Human Nature

'Tis plain that in the course of our thinking, and in the constant revolution of our ideas, our imagination runs easily from one idea to any other that resembles it, and that this quality alone is to the fancy a sufficient bond and association.

What will be fascinating to watch is how the continued intense development of connectionism as an approach to learning, similar to but different from Estes' in certain respects, also incorporates and uses in a deep way concepts of similarity. For example, the real question is whether similarity will come out as a construct from still more primitive ideas or whether it will need to be built in as an explicit concept. Certainly the tendency of connectionism and the development of neural networks will be to favor the former rather than the latter solution.

This suggests that a successful neural theory will lead to a construction of Estes' similarity parameter and it would not need to be estimated from data. I am quite happy to express my skepticism about this, at least in the present development of neural networks. They are certainly not prepared to incorporate the many varied ways in which similarity plays a role in human experience. In fact, I have tried to make the case against ever having a satisfactory reduction of psychology to neural concepts in Suppes (1991). In my judgment, there is no foreseeable time in the future when the kind of parameters that are estimated, like Estes' similarity parameter, will be computed from more primitive neural constructs. In what is in many ways a much simpler situation, the relation between chemistry and physics, it is impossible to compute

192 W. K. ESTES

directly from fundamental principles most of the stable chemical properties of matter and, in my view, so it shall be for many of the salient psychological properties of human experience in relation to their neural and physiological substrata.

REFERENCES

Hurne, D.: 1739/1888, Treatise of Human Nature, Oxford, Clarendon Press. Estes, W. K.: 1950 'Towards a Statistical Theory of Learning', Psychological Review,

57,94-107. Suppes, P.: 1992, 'Estes' Statistical Learning Theory: Past, Present and Future', in:

A. F. Healy, S. M. Kosslyn, and R. M. Shiffrin (Eds.), From Learning Theory to Connectionist Theory, Vol. 1, Hillsdale, NJ, Lawrence Erlbaurn, pp. 1-20.

Suppes, P.: 1991, 'Can Psychological Software Be Reduced to Physiological Hard- ware?', in: E. Agazzi (Ed.), The Problem of Reductionism in Science, Dordrecht, Kluwer Academic Publishers, pp. 183-198.

RAIMO TUOMELA AND GABRIEL SANDU

ACTION AS SEEING TO IT THAT SOMETHING IS THE CASE

ABSTRACT. In this paper the notion of intentional activity in its broadest sense is characterized in precise terms. This is done by means of the notion of seeing to it that something, say R , is the case. Contrary to what has been generally assumed in the logic of action, it is claimed that one cannot unintentionally see to it that R is the case, seeing to it being necessarily intentional. Indeed, it is claimed that the notion of effective intention will in turn clarify the notion of seeing to it (Stit). Many properties of intentions can be applied to Stit, by analogy, and as a result precise conditions for Stit are given at the end of the paper. The paper is also concerned with the more basic notion of an agent A performing an action having as its necessary 'result' a state of affairs R (DOA R ) and with the notion of an agent A being an indirect agent of R, viz., his performing an action with R as its result by using a means Q (DOA (Q, R)). A logical semantics for DO and Stit is given in terms of 'agent-trees'. This semantics draws on game-theoretical notions such as games and strategies and, especially, analyzes Stit in terms of the notion of an agent's having a winning strategy (over a state) and his using it.

1. INTRODUCTION

In this paper the notion of intentional activity in its broadest sense will be characterized in precise terms. This will be done by means of the notion of seeing to it that something, say R, is the case. Contrary to what has been generally assumed in the logic of action, we claim that one cannot unintentionally see to it that R is the case, seeing to it being necessarily intentional. Indeed, it will be claimed that the notion of effective intention will in turn clarify the notion of seeing to it (Stit). Many properties of intentions can be applied to Stit, by analogy, and as a result precise conditions for Stit can be given. Before doing that, however, we shall be concerned with the more basic notion of an agent A performing an action having as its necessary 'result' a state of affairs R (DOA R) and with the notion of an agent A being an indirect agent of R, viz., his performing an action with R as its result by using a means Q (DOA (Q, R)). A logical semantics for DO and Stit will be given in terms of 'agent-trees'. This semantics has many resemblances to that of Belnap and ~ e r l o f f . ~ However, our framework is essentially different

P Hurnphreys (ed.), Patrick Suppes: Scientific Philosopher, W)l. 3, 193-221. @ 1994 Kluwer Acadetnic Publishers. Printed in the Netherlands.

194 RAIMO TUOMELA AND GABRIEL SANDU

from theirs, in that we draw more on game-theoretical notions such as games and strategies..

2. PERFORMING AN ACTION

Roughly speaking, our idea of agency will be represented in the following way: we shall compare two possible situations or 'moments', mo and ml , which occur at two distinct instants, that is, the time of mo is earlier than the time of ml . In the model we shall be using, nothing happens without a cause, and when causes are non-human, they will be ascribed to a particular agent called 'Nature'. So if we notice that a state of affairs 1 R obtains or holds at the moment mo (briefly: 1 R holds at mo), but R holds at ml , then we shall infer that the change is due to some agent's action at mo. Thus R is the result brought about by some agent's action at mo, and we shall say that mo is a choice point or decision point for that agent. It then becomes clear that for arbitrary R, if R holds at ml and we want to find out who performed the action with R as its result, or - as we will say - 'did' R, we have to look backwards for a moment mo at which the negation of R holds and see to whom mo was assigned as a decision point.

It is worth emphasizing that according to our model the world is assumed to function in the 'normal way'.3 However, nature can act unexpectedly and, as said, we account for this by allowing that nature acts as a kind of agent (viz., simulates to some extent the effect of real agents). In cases where the world can behave in different alternative ways from the point of view of the action situation in question we assume that Nature has a decision point with two or more alternatives.

One way of making these ideas more precise is to use the idea of branching time represented in the form of a tree.4 A tree is a non-empty partially ordered set T = {T, <), where T is a set of discrete moments and, for each x E T, the set {x E T : x < m) is linearly ordered by < (which is intended to be the 'earlier' relation). A least element of T, if any, is called its root. Intuitively, each branch of T will represent a history made up of moments (possible situations) mo, ml , etc. With each such tree T we associate a set Ttime of instants of time which represents the time of T, linearly ordered by <. Each instant of time i, belonging to Ttime determines a sequence of 'parallel' moments in the tree as in Figure I .

ACTION AS STIT

T time

Fig. 1.

In Figure 1 the instant i, is associated with the set of parallel moments {mo, ml , m2, m3), and conversely, each moment m in T corresponds to an instant of time i, E Tiime. Consider now a moment mo in a tree T situated on a history (branch) h and an instant i, E Ttime such that i,, < i,. Clearly, the moment mo and the instant i, determines a sub-tree T z of T whose root is mo and whose terminal points (leaves) are the moments determined by i, in T.

Consider now a tree T in which the nodes are assigned as decision points to agents belonging to a set, X, of agents, denoted by A, B, Nature, etc., such that each node is assigned to exactly one agent. The idea is that the branches starting from a node m represent different ways in which a history can notionally go on in the future as a result of something that the agent to whom m is assigned as a decision point does at m.

A possible situation or moment m in a tree T consists of states of affairs which hold at m, or, alternatively, consists of the set of state- describing sentences true at m. We stipulate two rather obvious things about the moments m of T. (i) If m is a choice point for some agent A, then at least one of the set of sentences associated with the moments next to m in T is different from the set of sentences true at m; (ii) the set of sentences true at m is closed under the rules of inference of propositional logic.

These requirements should be intuitive, (i) indicating that m is a real choice point, in the sense that the agent A may either let the world stay as it is, or he may bring about a change in it.

We shall call any tree which fulfills all the conditions described above an agent-tree and denote the class of all agent-trees by Tag.

Figure 2 provides an example of an agent-tree. We notice that 1 R holds at mo but not at ml . We therefore infer that

this change is due to something A did at mo. That is, at ml it is true


T time

Fig. 2.

that A is the direct agent of R. The same is true at m2, since, B left R 'untouched' at ml .

Let us fix an agent-tree T that will remain unchanged in the sequel. We are now ready for our first definition, in which 'DOAR7 reads 'A is the direct agent of R':

DEFINITION 1. 'DOAR' is true at the moment rn in T if and only if the following conditions hold:

(i) R holds at m; (ii) the set C = {ml : rnl < m and 1 R holds at m l } is non-empty

and the last (in the order of <) rnl E C is a decision point of A.

Our analysandum is the performance of an action, a doing, as we might say. Because there is the linguistic problem that one cannot say that an agent performs a state or does a state, we have chosen to say that our agent is the agent of a state. Furthermore, an agent being a direct agent of a state R means that he has the last decision point in the situation at hand, and thus not even Nature can change the situation after our agent has made his choice. When 'DOAR' is true we can say that A does or performs an action having (the exemplification of) R as its conceptually inbuilt 'result' - we will use the clumsy sayings 'A does R ' and 'A performs R' interchangeably here. We are here dealing with standard action or performance of an action with a result

ACTION AS STIT 197

of the kind R , as contrasted with seeing to it that R, allowing R to be an agentive state if needed. (However, our account does not deal with the movements involved in the action having the result R.) Thus, 'A opened the window' would be translated by 'DOA R ' with R as the state of the window's being open. Note that, as A has the last decision point in this situation, it is not possible here, for instance, that he brings about R by asking or ordering somebody else to bring it about. Although we are are not directly dealing with the specific contentual features of A's action, viz., the movements by means of which he effects the result R, our present account seems to be as close as one can come in formal terms to capturing direct agency - direct performance of an action - in the case of a specified state. Such direct agency includes the possibility of using tools as long as the tools are not the agent's other actions. In contrast, how the general metaphysical problem of agency is solved is here left open.

DOA R is not required to be intentional with respect to R. However, if intentionality is needed, it can be made explicit in the tree T in the following way. With each instant of time i, E Ttimc, and each agent A, a set INTi, of sets IntTu is associated. We think of each IntyO in I N T ' ~ as containing the sentences denoting the sets of states of affairs intended by A at mo to hold at 2,. (We will not in our model explicitly deal with the knowledge and beliefs that intentionality presupposes.)

We can say somewhat loosely that A intends a state of affairs R if and only if he has the future-directed or present-directed intention to 'do R' or if he has the 'intention-in-action' (or if he endeavors or wills - please use your favorite term) to bring it about. The latter is the case when he has the intention of bringing about something Q and regards bringing about R as a means to (or, occasionally, a part of) Q. He can then be said to 'mean' or intend R by his action. It can be argued that all future-directed and present-directed intentions are 'act-related'.5 One necessarily intends, by one's actions, that something be the case. If an action of a specific kind, say P, is in question we have a case of the agent's intending to do or bring about R by P.

We can now define a notion of intentional action, 'DO; R ' meaning A's intentionally performing or doing R:

DEFINITION 2. 'DO2R' is true at the moment m in T if and only if the following conditions hold:

198 RAlMO TUOMELA AND GABRIEL SANDU

(i) R holds at m; (ii) the set C = {ml : ml < m and 1 R holds at ml ) is non-empty

and the last (in the order of '<') ml E C is a decision point of A (cf. (i) and (ii) of Definition 1);

(iii) R E IntT1 E INT'~".

It can easily be seen that, at each moment m in T: DOIR - DOaR. That is, if an agent acts intentionally, he also acts simpliciter.

3. GENERATED ACTIONS

As we know from common sense and also from action theory, a great many actions are performed by performing something else, viz., many actions are non-basic or generated actions. Here is an example:

By checkmating Jack, Jim gave him a heart attack.

In this example the state of affairs of Jack's being checkmated (brought about by Jim) is usually understood as being the cause of the state of affairs of Jack having a heart attack, and the former state of affairs is said to causally generate the latter.6

Generated actions can be introduced on our trees in a straightforward way. We define a binary operator DOA(P, R ) ('by doing P , A did R') which is intended to represent generated actions and thus indirect agency.

DEFINITION 3. 'DOA(P, R)' is true at the moment m in T if and only if the following conditions hold:

(i) P , R hold at m; (ii) the last node mo (in the order of <) before m at which 1P holds

is a decision point of A; (iii) the last node ml before m at which 1 R holds is a decision point

of either Nature or A; (iv) ml > mo and for any decision point m* such that mo < m* <

m, m* is assigned to either Nature or A.

Definition 3 could handle most cases of causally generated actions even without mentioning the agent A in its clauses (iii)-(iv). However,

ACTION AS STIT 199

there are cases in which mention of the agent is needed. For example, our agent A may lift his left arm in several ways: he may lift it in the normal way or he may use, for instance, his right arm to lift it. In the former case we have a directly performed action. In the latter case we are dealing with a generated or indirect action. We can take the action with the result P to be the agent's moving his right arm in an appropriate way. Assuming that the world cooperated in a normal way, our agent had the last decision point before m. Our clause (iii) accepts this as a case of indirect lifting of A's left arm, because it allows A to have the last decision point.

Let us now state some of the consequences of our definitions. The reader may check himself that for any moment m in T the following hold:

In (6) and (7) T is a tautologous state. As to the proofs, for instance, (1 1) is proved in the following way: assume that DOA(P, R) holds at m, i.e., clauses (i)-(iv) of Definition 3 hold. But then, by (iv) the last node before m at which -DOA(P, R) is true is assigned to Nature. Therefore the consequent of (1 1) holds at m.

Let us briefly comment on (1)-(11). (1) is the obvious success axiom. (2) is our counterpart to the Davidsonian insight that in the case of generated actions, what the agent does is a 'basic' action, the rest being 'up to Nature'. It is not unreasonable to think of natural causation as closed with respect to conjunction. Thus (3) and (4) seem warranted in the case of the causal notion of doing, and we see no arguments against accepting them. (5 ) must be understood to concern simultaneous action. (6) and (7) seem to require no more discussion. (8) as well as (3) and (4) do not seem compelling but not harmful either.


(9) and (1 0) show that our dyadic operator DO(P, R) satisfies two of the requirements imposed by Goldman on generated actions: irreflexivity and asymmetry. (1 1) is simply an iteration principle.

As before, DO(P, R ) is not required to be intentional with respect to either P or R. However, intentionality can be introduced in a natural way. For this purpose, the sets IntA have to be enlarged so that they contain not only propositional symbols like P or R but also symbols of the form (R by P) . The meaning of (R by P ) E 1ntY E INT'- was, in effect, commented on earlier. (Note, however, that we simplify matters by assuming in effect that P occurs at i,, although of course normally it occurs before; cf. A's ventilating the room by opening the window.)

DEFINITION 4. 'DO>(P, R)' is true at m in T if and only if the following holds at m:

(0 DOA (P, R ) ; (ii) P, R E I N T ~ E lNTim (cf. (ii) of Def. 3);

(iii) R E IntT1 E 1NTirn (cf. (iii) of Def. 3); (iv) (R by P ) E IntTO E lNTim.

As in the case of the operators DO and DO*, intentional action is action simpliciter, that is, the following holds for any m in T: DO> (P, R ) --+ DOa (P, R).

It should be pointed out that (3), (4), (8) and (11) do not hold if we replace DOA (P, R ) by DO; (P, R), unless we impose some closure conditions on the set IntA. These closure conditions will guarantee that (ii)-(iv) of Definition 4 are satisfied. The formulation of these conditions is straightforward after a careful inspection of (3), (4), (8) and (1 1). For instance the analogue of (3) will hold at every m in T if the following conditions are satisfied for any moment mo:

(Cl) If P E Int?' E INT~- and R E IntY E 1NT2-, then ( P & R) E I n t y E lNTim.

(C2) If (R by P ) E I n t y E lNTim and (Q by P ) E I n t y E INT'- then ((R & Q) by P ) E I n t y E lNTim.

Although (Cl) and (C2) might hold when P and R are simple enough, there seem to be counterexamples in the general case: A may fail to 'put together' the various things he intends when matters get complicated. Similar conditions are necessary for the validation of (4), (8) and (1 1) but we shall not pursue this matter further here.

ACTION AS STIT 201

Both DOA (P, R) and DO; (P, R) are intended to capture only an aspect of the complex meaning of action generation, i.e., the causal- generative aspect. However, causal generation does not exhaust the whole meaning of action-generation.7 Witness the following:

By moving the queen, Jim checkmated Jack.

The usual meaning of this sentence is that the act of moving the queen is also an instance of the act of checkmating. Cases of this sort can be accounted for in the following way: the sentences which are true at a given moment m in T are understood in relation to the participants' mutually shared cultural ingredients, especially linguistic rules and conventions. Thus the rules of chess might apply at moment m to the sentence 'Jim's queen is located at E-6' and allow a redescription of it in that particular game context as 'Jim checkmates Jack'. Jim made a choice and it had a consequence redescribable in these two ways. Our theory can accordingly be complemented with an account of the possible ways to correctly redescribe the states generated by the participants of the game. Thus we can require that at each such moment there is some set of norms, conventions, rules, etc. which redescribe the state-describing sentences in a suitable way.8

4. SEEING TO IT THAT SOMETHING IS THE CASE

The notion of (intentional) action that has occupied us in the previous section is not the broadest conceivable notion of intentional activity. We shall in this section characterize a broader, achievement-related notion that can be expressed in terms of the agent's seeing to it that something is the case. The intuitive difference between the notion of performing or doing in our earlier sense and the notion of seeing to it can be seen by looking at Figure 2: on the basis of Definition 1, it is true at both m2 and ml that DOAR. Also DO;R might hold both at m2 and at ml . However, intuitively speaking, we would not like to say that A saw to it that R is the case at m2. This is because A lacked the opportunity to control the potential consequences of his actions up to m2. For instance, at ml , B could have done something as a result of which 1 R would have obtained. The fact that R comes to obtain and the history reaches m2 instead of m3 is not so much due to the effort of A but rather to the


Fig. 3.

decision of B at ml to omit doing 1R. A different situation is pictured in Figure 3.

The difference between Figure 2 and Figure 3 consists in the fact that, although in both cases it is true at m2 that DOA R (and supposing that R E Intyo E I N T ~ ~ , DOiR also holds at m2), in the first case A has no guarantee at mo that DOAR will turn out to be the case at i,, while he has such a guarantee in the second case. It seems to us that only in the second case can we speak of A's seeing to it that R is the case at m2, and that the essential thing about an agent A's seeing to it that R is the idea of A's 'making sure' that R obtains, or the idea of A's exercising intentional control and protection over the state of affairs R according to an intentional plan with respect to R. This plan need not be an explicit one and may be conceived as a set of skills that we shall codify by the notion of strategy in a game. We will now consider an example: suppose that at the instant i,,,, A intends that R be the case at the instant i,, i,, < i,; that is, R E IntTO E INT'-. Let us make a distinction between A and the environment, viz., the rest of the world. There are 6 possible histories (hl-h6) starting from the instant i,,, before A's action, represented by the tree in Figure 4 (logically speaking, there are only four possible histories, but we prefer to distinguish between Nature and the other agents in X): History 1. R (e.g., the door is closed) obtains at imo and will continue to obtain at i,;

ACTION AS STIT

Fig. 4.

History 2. R obtains at i,, but will cease to obtain because of the interference of B (e.g., B comes and opens the door at an instant before im); History 3. R obtains at i,, but will cease to obtain because of the interference of Nature (e.g., a strong wind opens the door); History 4. 1 R obtains at i, and will continue to obtain; History 5. 1 R obtains at i, but B interferes (B does R) and R will obtain at i,; History 6. 1 R obtains at imo but Nature interferes (Nature does R) and R will obtain at 2,.

Given A's intention that R be the case at i,, then he will do everything necessary to make sure that R will obtain at i,, starting with i , . Depending on which of the above histories will be realized, the agent A will have to act in a certain way, in order for him to see to it that R is the case at i, (cf. Figure 5):

Case 1. If history 1 turns out to be the case, then A does not need to do anything;


Fig. 5.

Case 2. If history 2 turns out to be the case, then A will interfere at an instant 2,. < 2, and do R(h7); Case 3. If history 3 turns out to be the case, then A will have to act as in Case 2 (h8); Case 4. If history 4 turns out to be the case, then A will interfere and do R(h9); Case 5. If history 5 turns out to be the case, then A does not need to do anything; Case 6 is similar to Case 5.

It has been claimed by logicians of action that the locution 'sees to it that . . . ' can be accepted as a general umbrella term for all a ~ t i o n . ~ For instance, Belnap and Perloff claim that a sentence P is agentive for an agent A if and only if P can be paraphrased as ' A sees to it that P ' , or, briefly, 'StitAP' (Belnap and Perloff, 1990, p. 179). However, while we agree with much of what these authors claim about the Stit-operator, we think that this central thesis of agency is not quite correct and that,

ACTION AS STIT 205

in any case, we need a better conceptual analysis of the notion of seeing to it that something is the case than is available in the literature.

The notion of seeing to it should be taken to concern states of affairs, however abstract and complex. The activity of somebody's seeing to it that something is the case is, at least in our view, necessarily intentional, although Belnap and Perloff do not point it out and seem not to think so. That is, in all cases, if you see to it that R , then you intentionally see to it that R. Accordingly, if I see to it that the door is closed, then this action is necessarily intentional. If I inadvertently close and open the door, that may be an instance of doing or performing (DO), but has nothing to do with my seeing to it that the door is closed. Let us consider again the example of closing the door from Figure 5. Seeing to it that the door is closed expresses intentional control over the state of the door's being closed and requires success (viz., the agent has not seen to it that the door is closed unless it is closed). This in turn involves the idea of the agent having and c a v i n g out a strategy through which he, if necessary, not only brings about R but also prevents all the possible interferences by other agents who might want to bring about 1 R . In our example, this meant that the agent A would close the door, if it is open (case 4), or he would omit to open the door, if it is closed (case I), or, in case some other agent would open the door, the agent in question would interfere and close the door (cases 2 and 3). Let us once again emphasize that in all the cases above the agent can be roughly said to exercise an intentional control over the situation. Notice also that our operator DO is necessarily change-related, while this is not true of Stit.

Given the above discussion, it can be seen that our theory does not make true Belnap's and Perloff's 1990 thesis of agency. There are at least two central cases (both against the 'only if' part of the thesis) showing this. First, an agent can perform something R and thus be a direct or indirect agent of R; yet he can fail to see to it that R. That is, there are cases in which DOA R and even DO; R hold but StitA R fails to hold; this will be shown below. Secondly - and this does not depend on the truth of our first point - there are, however, unintentional and non-intentional performances of R, and they cannot be cases of seeing to it that R. Why? If it is true that A sees to it that R ('R' being a state description), then R attributes intentional activity to A in the general sense of his exercising intentional control over R. If A sees to it that R he must be taken to intend that R, and A intends that R only if he is committed by his actions to the obtaining of R. Thus he is committed to


do whatever he can that the obtaining of R in his view requires him to do. More precisely, if A intends that R there will be actions of his that are in his view required of him which are conducive to the obtaining of R and which he must be capable, in his view, of performing with some non-negligible probability.

We consider that the notion of seeing to it that a state, say R, obtains, concerns intentional activity (achievement-related activity) in the most general conceivable sense. Broadly speaking, we think that the notion of Stit is analyzable in terms of effective intention involving commitment and the actions carrying out that intention according to an intention- containing plan.

We shall now list some logical properties that we take to apply to the notion of seeing to it. We shall use Stit for an operator on state descriptions meant to capture the notion of seeing to it that a state of affairs is the case. We claim that, on conceptual grounds, the following properties apply to StitA(-), where the argument place is for a state description:

Of these requirements, (12) is simply the success property applicable to all achievements. Property (13) excludes contradictory intentional actions. Requirement (14) is also natural, since it is not possible for an agent to see to it that a tautological state (T) obtains, because it obtains independently of an agent's interference. Property (15) is based on the following property, the analogue of which holds for intentions (viz., if one intends Q then he intends Q to come about as intended):

(C3) If Q E INTT E I N T ~ ~ then also StitAQ E INT? E INT '~ .

Some further remarks are still due. First, the basic possibilities related to intentional activity according to the present account are: (1) StitAR, (2) %itA+, and (3) lStitAR & lSt i tAIR. These alternatives are disjoint and jointly exhaustive.

Secondly, let us note that the 'intensionality' property applicable to many modal operators, that is, if R logically entails S then M(R) logically entails M(S), M being a modal operator, does not apply to the case where M = stit.'' This is because seeing to it that . . . is intentional and thus the agent might not realize that he is seeing to it that S (cf.

ACTION AS STIT 207

Davidson's example of a typist who intentionally makes six carbon copies without really knowing it) when he is seeing to it that R, even if R and S are logically equivalent state-describing sentences. However, we may consider the following principle (for an arbitrary m in T):

(C4) If 'R ++ S' is valid (i.e., holds at every moment in every tree T in Tag) and, in addition, for arbitrary A, (R E I n t y E INT'- if and only if S E IntTo E INT '~ ) holds for any instant 2, E Tim in every T E Tag, then %titA R ++ Stitas' is valid.

In some cases of conjunctive states, R and S , we have

although we will not try to specify for which conjunction (G) holds. The issue of how to combine the Davidsonian event-approach with

the present kind of approach in terms of the Stit operator is of course important, but the matter cannot be pursued further here.

We have above been dealing mostly with action in the sense of performance or doing, viz., DO and DO*, and with Stit. In passing we have also spoken about the bringing about of a state. Bringing about is a notion closely related to doing, although it does not coincide. As the notion of bringing about is not central to our paper, suffice it here to make the following suggestion as to how to characterize it: an agent A brings about a state R (viz., DOA R) at a possible situation if and only if (a) A is the direct agent of R or (b) A is an indirect agent of R (viz., DOa (P, R) for some P) , or there is a state R* such that A is a direct or an indirect agent of R* and R is a causal or conceptual consequence of R*. Omissions should be included among the actions in (a) and (b). (We will not investigate here the logical properties of causal and conceptual consequence.)

5. SEEING TO IT THAT SOMETHING IS THE CASE: A GAME-THEORETICAL APPROACH

Let us now characterize for an arbitrary m in T what it means to say that it is true at m that an agent A saw to it that the state of affairs Q obtains. Of course, our aim is that our definition validate the truth of the logical principles (12)-(15) and of the principle (C4) mentioned above. We shall define the notion of Stit indirectly, using the notion of A's being able at mo (mo E T ) to see to it that the state of affairs Q obtains at


the instant i,(i, E Time). For this purpose we shall consider a game G(A, mo, m, Q ) played by A against the set of all the other agents in the set X. The game is played on the sub-tree T;, of T whose root is mo and whose terminal points (leaves) are the parallel moments associated with i, in T . Intuitively, A will try to show that he saw to it that Q holds at i,, by following a branch (history) which leads to leaves in Tzo at which Q holds.

DEFINITION 5. Let Tg) be a tree whose root is mo and whose leaves are the parallel moments associated with the instant i,. We say that Q is unavoidable in TGI if Q is true in all the leaves of the tree. Otherwise, Q is said to be avoidable.

A game G(A, mo, m, Q ) is a set of plays or rounds. A play of the game G(A, mo, m, Q) is a sequence of choices prompted by decision points of the agents in X situated in the sub-tree TG, of T . The choices will be made according to the following rules: let the agent from X to whom mo is assigned as a decision point choose an immediate next decision point ml (mo < ml) in the sub-tree T;. The play will proceed on the sub-tree TG. When the play reaches the last node m, before i,, let the player who is assigned m, choose a leaf m in the tree T;,. If

(i) Q holds zt m; (ii) there is a node m* such that m* < m, m* is assigned to A and Q

is avoidable in T$

then A wins the play. He wins the game G(A, mo, m, Q) if he wins all the plays of the game, i.e., if he wins against any possible choices of the other agents in X.

An example. Let us look at the tree T;) shown in Figure 6. A play of G(A, mo, m, Q ) is, for instance, the sequence of choices (ml , m3, m6) such that first B chooses ml, then C chooses m3 and finally A chooses mb. Clearly A wins the play. Another play would be (m2, m4, m7), which A also wins. The third play is (m2, m ~ , ms) with A as the winner. These are all the plays of the game G(A, mo, m, Q) and thus A wins the whole game.

A strategy for a player A in a game G(A, mo, m, Q) is, informally, a method which gives A his choice for each move in the game. This

ACTION AS STIT

Fig. 6.

method is usually codified by a set of functions. Each function corresponds to a move by A and is defined on the set of possible choices made by the opponents of A previous to the move in question. Since, in the sequel, we will not need the formal definition of a strategy, we will not give one. The following example indicates what we have in mind. For instance, the strategy of player A in the game G(A, mo, m, Q) played on the tree in Figure 6 is the singleton consisting of any function f : N --+ {m6, m7, m8}, where N = {ml , . . . , ms}. We say that a player A uses his strategy in a game if all his choices in any arbitrary play of the game are values of the appropriate functions which constitute his strategy.

A strategy for a player in a game G(A, mo, m, Q) is a winning one if, by using it the player wins every play of the game (i.e., the whole game). In the above example, if f is such that

then { f ) is a winning strategy for A in the game in question.


We can now give the main definition for the notion of A's ability to see to it that a state Q obtains. Let us thus use 'CanStitAQ' to mean that A can see to it that Q. We define:

DEFINITION 6. 'CanStitAQ' is true at the instant i, E Ttime if and only if A has a winning strategy in the game G(A, mo, m, Q) played on the sub-tree TG, of T.

Notice that, if it is true at mo that A can see to it that Q holds at the instant i,, Q is not supposed to hold at every moment (leaf) determined by i, in the tree T;. Q has to hold only at those leaves which are the last chosen ones in arbitrary plays of the game G(A, mo, m, Q). We shall call every such leaf a terminal point in the game G(A, mo, rn, Q). Clearly the notion of terminal point is relative to that of winning strategy, since a player may have more than one winning strategy in the same agent-tree. Now we can define what A's seeing to it that Q, in symbols 'StitAQ9, amounts to:

DEFINITION 7. 'StitAQ' is true at m in T if and only if there is a node rno < m such that:

(i) A has a winning strategy in the game G(A, mo, m, Q) played on T;, .

(ii) A uses his winning strategy and wins a play of the game G(A, mo, rn, Q) in which m is a terminal point;

(iii) Q E Intyl E INT;.

The reader may check for himself that all the logical principles (12)- (15) are true in the tree T for any moment m E T and history h. For instance, (1 5 ) @titA Q - StitA(StitAQ)) can be proved in the following way: assume StitAQ holds at m, i,e., there is a moment mo < m such that conditions (i)-(iii) from Definition 7 hold. We claim that the following conditions hold:

(1) A has a winning strategy in the game G(A, rno, m, S t i t ~ Q ) played on TZO;

(2) m is a terminal point in the game G(A, mo, m, S t i t ~ Q ) that A wins by using his winning strategy.

ACTION AS STIT

T time

Fig. 7

Clearly, (I)-@) together with (C3) will prove that StitA(StitAQ) holds at m.

As for the proof of (I), let A play G(A, mo, m, StitAQ) in the same way as he played G(A, mo, m, Q ) and let him use in the former the winning strategy he had in the latter. (2) follows trivially from (ii) of Definition 7.

In a similar way it can be shown that the principle (C4) holds in our model.

It may be interesting to point out that there are situations in which it is true for two distinct agents A and B that they see to it individually that a state of affairs R holds at m. The tree shown in Figure 7 is a case in point.

Let us assume that R E I n t y E I N T ~ ~ and also R E Int?;;' E I N T ~ ~ . The reader may check that both StitAR and StitsR hold at m2. The former follows from the fact that A has a winning strategy in G(A, mo, m, R ) and rnz is a terminal point, and the latter from B's having a winning strategy in G(B, ml, m, R) with m2 a terminal point. In the first game, A's winning strategy is FA = {ml ? r n ~ }. The fact that m2 is a terminal point follows from the fact that A has to win against


any choice of his opponents, and m2 is one of them. In the second game, B's winning strategy is FB = {m2).

The truth of StitA(StitBR) in our theory is in contradiction to the theory of Belnap and Perloff. In their theory, lStitA(StitBR) is valid. The contradiction, however, is only apparent, given the fact that the class TAg of agent-trees is different in the two approaches. In Belnap's and Perloff's case a decision point in a tree is ascribed to more than one agent.

Let us consider an example which shows the possibility of such nest- ed Stits. Here is a case where A has the (institutional or informal) power to make B bring about R:

A sees to it that B sees to it that the bill will be paid.

In this case A makes B pay the bill (by ordering him to do so, for instance) and, as a reasoned consequence of this, B pays the bill. While B's action is A's means to the end R (= the bill is paid), B's action is still sufficiently free to be regarded as an intentional action. Here A makes B act: he exercises intentional agentive control over B and thus brings about B's paying the bill. Both A and B are partially responsible for the payment of the bill (and would have been partially responsible for a failure of the payment - had such a failure taken place due to B's negligence). The state of the bill's having been paid can be said to have both A and B as its agents.

One undesired consequence of the present framework is the validity of the sentence

However, the validity of (*) can easily be blocked by restricting StitA StitBQ to hold only if A's first decision point in the relevant game is earlier than B's. Then the following sentence, illustrating the 'Qui facit per alium' principle is provably valid:

In analogy with the above we can define related notions. For instance, we can introduce a binary Stit for indirect (as contrasted with direct) seeing to it that something is the case in analogy with our binary DO*:

ACTION AS STIT 213

DEFINITION 8. 'Stit* (R, Q) ' (viz. 'By seeing to it that R A sees to it that Q') is true at m in T if and only if there is a moment ml < m such that the following conditions are fulfilled:

(i) 'StitA R ' is true at ml ; (ii) 'Q' is true at m and Q E IntTz E INT'",

(iii) the decision points between ml and m are assigned to none.

The notion of Stit can be used to characterize 'negative' actions as well. "

6. COMMENTS ON THE THEORY OF BELNAP AND PERLOFF

We shall now show that our theory of Stit makes more Stit-sentences true than the theory of Belnap and Perloff. Actually, the tree in Figure 7 illustrates a case in which the two theories yield different consequences. We saw that StitAR holds at ma. As a matter of fact we can construct the scenario which led to m2 starting from mo: A intends at mo that R hold at 2,. However, A need not bring about R himself but may instead delegate or give this task to B who will eventually do the job. This is in fact a reasonable thing for A to do, given B's intention at ml that R hold at im. Were B not to bring about R at ml but would do something different instead the result of which would be the history h2 (or h3), then A would interfere at the moment m4 and correct the situation by bringing about R himself. So it is right to say that it is A who sees to it that R holds at m2.

We shall not discuss all the details of the theory of Belnap and Perloff. We shall only reproduce here a few definitions that make comparison possible. With each moment m of the tree a choice set is assigned to each agent which is a partition of all the histories passing through m. A member of the choice set, which is thus a set of histories, is called a possible choice for the agent in question. A central restriction is that a choice set for an agent at m must keep together histories that are undivided at m; that is, no agent can make a choice that includes one of the two undivided histories but excludes the other. If histories from later moments mo and ml are in the same possible choice for A at an earlier moment m*, and if mo and ml inhabit the same instant, then Belnap and Perloff call mo and ml choice equivalent for A at m*. Here is the definition we are interested in:


DEFINITION (Belnap and Perloff). StitAR holds at m if and only if there is a decision point mo < m for A such that the following conditions are fulfilled:

Positive condition: R is true at all moments that are choice-equivalent to m for A at mo. (Thus, the prior choice of A at mo 'guarantees' that R holds at the instant i, determined by m).

Negative condition: R is not true at some moment m' parallel to m, situated in a history which passes through mo.

The point mo is called by Belnap and Perloff 'witness'. The reader may notice that the negative condition (ii) is identical to our notion of avoidability. Let us turn to the positive condition and see how it applies to our concrete example (cf. Figure 7).

Any possible choice for A at mo must keep together the histories h2 and h3 which are undivided for A at mo. Thus, the only 'reasonable' choice set for A at mo is {{hl, h2, h3), {h4, hs}}. Thus m2, m3 and ms are choice equivalent for A at mo. However, R does not hold at ms and thus the positive condition defined above is violated. Hence, StitAR does not hold at m2.

The above example shows that Belnap and Perloff's positive condition is too strong. What is wrong with it is not the fact that StitAR holds at m2 only if A can 'guarantee' at an earlier moment mo that R will hold at i,, but the way in which the existence of such a guarantee is implemented: it is required that R hold at all moments which are choice equivalent to m2 at mo. Belnap and Perloff do not take into account the fact that such a guarantee exists even if A has as possible choices histories which intersect the instant i , at moments at which 1 R holds, provided he can 'neutralize' such histories by his later choices. It is here that the notion of strategy proves to be useful and the sort of guarantee Belnap and Perloff are looking for was made explicit in our system through the notion 'can see to it that'.

The essential formal difference between our approach and that of Belnap and Perloff may be summarized as follows. Consider this:

LEMMA. I f StitAR holds a t a moment in T, then there is exactly one witness for StitAR.

ACTION AS STIT

Fig. 8.

This lemma holds in the theory of Belnap and Perloff (as proved by hella as), l2 but does not hold in ours. The fact that this lemma holds in the former is a direct consequence of the positive condition, which, as we pointed out above, is rather strong. Consider also Chellas's comments: "Uniqueness of witnesses may occasion disquiet, for it means that a choice cannot be subsequently overturned, that even the agent is powerless to choose differently later on."" It should be pointed out that Chellas thinks the above lemma is a consequence of the negative condition of Belnap and Perloff that he finds too strong. As we saw, things look different from our perspective and it is the positive condition that we found too strong.

In our theory, a counter-example to the above lemma is illustrated by Figure 8

Readers may check for themselves that A has a winning strategy in both G(A, mo, m, R) and G(A, ml, m, R) (assuming, of course, that R E IntyO E I N T ~ and R E Int;' E INT~- .)

Another consequence of our theory is that the following holds for any moment m and agent A:

'StitAQ9 is true at m --+ A made a prior choice at mo determining Q(m0 < m).


Thus there is a conceptually inbuilt temporal assumption, which entails that the Stit operator cannot be meaningfully applied to sentences which do not satisfy this condition. This is an extra reason (in addition to Chellas's criticism of past-Stit) for giving up Belnap's and Perloff's agentiveness condition, viz., that for all agentive sentences $, StitQ is meaningful.

7. CONCLUSION

Our investigations have thus shown that there are basically three different kinds of (intentional) action, all related to the coming about or maintenance of a state: (1) performances (doings); (2) bringings about; and (3) Stits. We have, in particular, noticed that there are bringings about which are not performances. Furthermore, there are performances and bringings about which are not Stits; and there are Stits which are neither performances nor bringings about (cf. the lighthouse keeper who keeps the light on without having to bring about a change in the world). l 4

ACKNOWLEDGEMENT

We are grateful to Professor Nuel D. Belnap for comments on the penultimate version of this paper.

Department of Philosophy, R 0. Box 24 (Unionink. 40B), 00014 University of Helsinki, Finland

NOTES

' Every action is assumed to have a conceptually necessary 'result' in the sense of von Wright (1971), p. 67. Thus the action of opening the window has the state of the window's being open as its result (and the state or the event of the room's becoming cool as its 'consequence').

See Belnap and Michael Perloff (1990) and Belnap (1991). This seems to be an idealization, and it is a major idealization if we are to believe

Suppes (1984). See Belnap and Perloff (1990) and also Aqvist (1974).

ACTION AS STIT 217

See Wilson (1989) and also Tuomela (1994). ' Such actions have been investigated in the literature. Thus, e.g., Goldman (1970) and Tuomela (1977) from the actions-as-events camp and Chisholm (1971), Porn (1978), as well as Holmstrom-Hintikka (1991) from the agent-causality or actions-as-modalities camp have investigated generated actions.

See Goldman (1970) and also Tuomela (1984). ' Besides being non-generated actions, basic actions are often taken to be actions in the agent's 'repertoire' (cf. Tuomela, 1977). That is, they are actions which the agent can intentionally perform (at least in normal circumstances). Of such actions it holds true that the agent can see to it that they become performed. The class of such basic actions forms a proper subclass of the actions which the agent can intentionally and directly perform in the sense of this paper (see next section for an example to show this). "f. Belnap and Perloff (1990), Kanger (1972), Holmstrom-Hintikka (1991), and others. "' For an opposite view, see, for instance, Kanger (1972). " For instance, we may define an intentional omission or refraining to bring about or maintain a state R , which can be agentive, viz., of the kind S t i t ~ R , as follows: A (intentionally) omitted bringing about or sustaining R at the moment m , if and only if S t i tAIR holds at m. Another 'negative' activity that can be defined with the help of the notion of Stit is letting something happen. We suggest, without further discussion, that A's intentionally letting R happen be taken to amount to st it^ sti it^ l R ) , possibly together with a statement to the effect that R will be brought about by another cause, unless A interferes, this latter fact being believed by A to be the case. l 2 For the proof of this lemma, see Chellas (forthcoming). 13 Chellas (forthcoming), p. 23. l 4 Joint actions can be analyzed in terms of operations corresponding to DO, DO*, and Stit. This we have done in Tuomela and Sandu (forthcoming).

REFERENCES

Aqvist, L.: 1974, 'A New Approach to the Logical Theory of Actions and Causality', in: S. D. Stenlund (Ed.), Logical Theory and Semantic Analysis, Essays Dedicated to Stig Kanger on his Fiftieth Birthday, Reidel, Dordrecht, pp. 73-91.

Belnap, N. D.: 1991, 'Backwards and Forwards in the Modal Logic of Agency', Philosophy and Phenomenological Research, LI, 777-807.

Belnap, N. D. and Perloff, M.: 1990, 'Seeing to it that: A Canonical Form for Agen- tives', in: H. E. Kyburg, Jr., R. P. Loui, and G. N. Carlson (Eds.), Knowledge, Rep- resentatiort and Defeasible Reasoning, Kluwer Academic Publishers, Dordrecht, pp. 167-190.

Chisholm, R.: 1971, 'On the Logic of Intentional Action', in: R. Binkley, R. Bronaugh, and A. Marras, (Eds.), Agent, Action and Reason, University of Toronto Press, Toronto and Buffalo, pp. 38-69.

Chellas, B. F.: forthcoming, 'Time and Modality in the Logic of Agency', Studia Log ica.


Goldman, A.: 1970, A Theory of Hutnan Action, Princeton University Press, Princeton. Holmstrom-Hintikka, G.: 1991, 'Action, Purpose and Will A Formal Theory,' Acta

Philosophica Fennica, 50. Kanger, S.: 1972, 'Law and Logic', Tl~eoria, 38, 105-1 32, Porn, I.: 1978, Action Theory and Social Science: Some Formal Models, Reidel,

Dordrecht. Suppes, P.: 1984, Probabilistic Metaphysics, Blackwell, Oxford. Tuomela, R.: 1977, Human Action and Its Explanation: A Study on the Philosophical

Foundations of Psychology, Reidel, Dordrecht. Tuomela, R.: 1984, A Theory of Social Action, Reidel, Dordrecht. Tuomela, R.: 1994, The Importance of Us: A Philosophical Study of Basic Social

Notions, Stanford University Press. Tuomela, R. and Sandu, G.: forthcoming, 'Joint Action and Group Action Made

Precise'. von Wright, G. H. : 197 1, Explanation and Understanding, Cornell University Press,

Ithaca. Wilson, G. M.: 1989, The Intentionality ofHuman Action (revised and enlarged edition),

Stanford University Press, Stanford.


I enjoyed the typically clear and careful presentation of Tuomela and Sandu's ideas about intention and action. They remind me of the days many years ago when Raimo Tuomela was a graduate student at Stan- ford and we argued extensively about many questions concerning the theory of action and decision-making. Because Raimo was spending a sabbatical at Stanford in the Fall of 1992, we had an opportunity to informally discuss his paper and it was useful, as would be expected, in clarifying my own thinking about it. I am sorry not to have had the opportunity to talk to his coauthor, Gabriel Sandu, as well. I do have several comments that I think are pertinent to the paper.

Normative Versus Descriptive Theory. Tuomela and Sandu are not explicit as to whether their theory is one of a normative or a descriptive character. It seems to me that in the absence of any detailed psychological description of intentions, it has to be treated as a normative theory. This is also reinforced by property (1 3) which excludes contradictory intentional actions. It is subject matter for novelists and therapists to discuss and deal with contradictory intentions. In other words, I would claim it is part of the descriptive facts of life about intention that we

ACTION AS STIT 219

often hold what we would formally regard as contradictory intentions, just as we hold partial beliefs reflected in subjective probabilities that are not coherent, i.e., could not be represented by any probability function. Intention is in these matters not surprising. It goes along with desires, beliefs, values, etc., all of which in real life do not satisfy plausible normative properties. But if the theory is normative we are still left with the problem of describing in appropriate folk-psychologlcal terms what we regard as a realistic descriptive theory. As far as I know, and I do not claim to be as thoroughly conversant with all of the literature as I might, this is not a problem whose solution has been developed in any detail in the literature as yet.

Expected Intentions. All Raimo and Gabriel's formulations, almost without exception, of intentions and seeing-to-it-that properties are in terms of states of affairs that are either known or will happen with certainty. This seems to divorce the theory rather strongly from complex states of affairs in the real world. It is well known that we simply cannot have what we value or what we desire, but must deal with expected desire or expected value. The same seems to be true of intentions. We cannot simply have an intention, but must really operate with expected intentions just as we operate with expected values or desires. This means that the theory, either normatively or descriptively, must be much more complicated, as is clear, for example, in the economic applications of the theory of value as we pass from ordinal values with certainty to maximizing expected value with uncertainty. Thus I may say in an ordinary way, 'I intend to go to San Francisco today', but then someone interjects, 'But what will you do if it turns out that the forecast is right and there is a very heavy rainstorm. Will you still go?' and I could easily reply, 'No. In that case, I intend to stay home and finish a paper that is a month overdue.' It is quite true that we do not ordinarily state the various uncertainties surrounding intentions, but they are very much there, often barely below the surface, and easily filled in under the simplest sort of questioning.

Varieties of Intention. I mentioned already the absence of any detailed psychological considerations. I discussed this point with Raimo and he correctly pointed out that there is really not an extensive literature in experimental or scientific psychology, as we shall call it in order to


distinguish it from folk psychology, on intentions. But there is some and there are lots of related theories; for example, the very extensive literature on selective attention which certainly has close relations to intention, and also on experimental studies of desires, drives and values. What is important is that there are several different levels or kinds of intention, well recognized by novelists and therapists as I already mentioned.

The typical and most important case is intentions that are not con- scious but that are used by observers to explain actions. A favorite of mine, that I have used in a couple of papers, is the young man who goes up as often as possible to show his attractive sixth-grade teacher his work. Often what he gives as an excuse for showing it to her is rather flimsy. What observers recognize is that his real intention is to have the opportunity to talk to her and to be close to her, not to show his work. This goes the other way as well. There are several empirical studies showing that teachers in elementary schools in the United States tend to give more attention and respond more frequently to the boys than the girls - I am here speaking of the typical case where the teacher is a woman.

We do not have to be Freudians to recognize the slippery and sub- tle character of the concept of intention when applied to the emotional life of any person, child or adult. Raimo and Gabriel are not alone among philosophical theorists of action in not dealing with this aspect of intentions. Of course there are many routine uses of intention, as in 'I intend to go to the store to buy a bottle of milk', but many of the really interesting cases do not have anything like such prima facie simplicity. Even in these cases it is plausible to argue that the process of forming the intention, as opposed to the result of having it, is unconscious and mostly inacessible to the concepts of folk psychology.

Weakness of Logic Alone. As I have already remarked, when we pass in the theory of desire from ordinal desires, typical of the classical theory of demand in economics, to expected theory of value, the complications are very large indeed. The same is true of intentions. Once we leave questions of simple logic, the much more general and powerful theory of intention that we really should want both normatively and descriptively cannot live by logic alone. This is evident already in the case of decision making. What we can say from the standpoint of logic or Boolean algebra about decisions is relatively restricted. What can be

ACTION AS STIT 22 1

said in that framework is of course clear, but it is not nearly adequate. I feel the same about a purely logical approach to intention at the level of the assumptions that Tuomela and Sandu state. Interesting though the assumptions may be, we cannot hope to formulate in their terms alone an adequate theory. This does not mean that it is not an interesting task to do what they have done. I commend the authors especially for their clarity. I am just pushing the point that it should be recognized in the theory of action, that anything like the kind of results produced by them cannot be regarded as the last word in the theory of intention. Much more is needed to have a theory rich enough to deal even with the central facts, let alone the periphery.

On this point I also want to note that Raimo himself is moving in this direction with his consideration of strategies and games in the latter part of the paper and in other publications of his, for example, his interesting discussion of collective action and the free-rider problem. In Tuomela (1992) it is evident enough that he too is not satisfied just with questions satisfying the logic of intention, and I do not mean to suggest that he is so satisfied.

REFERENCE

Tuomela, R.: 1992, 'On the Structural Aspects of Collective Action and Free-riding', Theory and Decision, 32, 165-202.

COLLEEN CRANGLE

COMMAND SATISFACTION AND THE ACQUISITION OF

HABITS

ABSTRACT. This article develops the idea that the meaning of a command can be analyzed by stating conditions on a satisfactory response to the command. It argues that in evaluating command satisfaction, consideration must be given not only to resu!t judgments but also to judgments about process. It identifies two kinds of process judgments - one in terms of action properties, the other in terms of action execution - and two kinds of result judgments - one at the level of events, the other at the level of specific outcomes. It discusses when judgments of each kind are called for and examines the circumstances under which process considerations intrude into judgments that are primarily in terms of result. Finally, it identifies a role for habits in command semantics, drawing on work done to design robots that can understand ordinary English commands.

Over the past few years, in work Patrick Suppes and I have done on robots that learn new tasks through verbal instruction, the notion of a robot's needing to acquire habits has surfaced from time to time. It quickly became apparent to us that robots need repetition and practice in the performance of a new task, just as people do. What has been more surprising, however, is the extent to which repetition and practice, and the concomitant acquisition of habits, are required for robots to understand ordinary English commands requesting action. My purpose in this article is to examine the extent to which a consideration of habits must enter into the semantics of commands, and in so doing also to identify a place for habits in the design of intelligent robots. First, I must extend the analysis of command satisfaction begun in our earlier work (Suppes and Crangle, 1988).

SATISFACTION CONDITIONS FOR COMMANDS

Consider the following simple command:

(1) Put the cup on the table.

P Humphreys (ed . ) , Patrick Suppes: Scientific Philosopher; Vol. 3, 223-241. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

224 COLLEEN CRANGLE

If in response to this command the agent - robot or human - tips the cup, spills its contents, and then places the cup upside down on the table, in ordinary circumstances we would consider the request to have been poorly or only partially satisfied. The intention behind the command is most likely for the cup to be placed right way up, its contents undisturbed. Consider another example, the directive

(2) Go to your room

addressed by a parent to a child. The intention behind this command typically is for the child to go directly to her room, without dawdling, without taking a detour to the playroom, and without stopping to pat the dog. The further intention is typically that she stay in her room for some period of time.

It is characteristic of a great many commands requesting action that they leave unexpressed most of the details required to perform the action. At the same time, however, the command carries with it a host of conditions on what counts as a satisfactory response. These conditions encompass not only considerations of result - the cup is on the table or it is not, the child is in the room or she is not - but also process considerations - how the cup was moved to the table, how and when the child ended up in the room.

It has long been acknowledged that one way to analyze the meaning of an utterance is to state the conditions under which the utterance is satisfied. For a statement, giving satisfaction conditions amounts to providing conditions under which the statement is true. For a question, conditions on a correct answer are sought. For a command, the conditions of interest are conditions on a satisfactory response to the command. The basic semantic notion is that of satisfaction in a model, and while it is relatively easy to make the intuitions behind this notion explicit in the case of elementary formal languages, considerable difficulties arise when the complexities of a natural language such as English are introduced. Still the basic intuition prevails, for commands in part because we generally make the reasonable assumption that the agent's response to a command proceeds from his understanding of the command, not from mere willfulness.

One major source of difficulty lies in determining what kinds of satisfaction conditions should apply. As noted above, the satisfaction of a command can be judged purely in terms of result: the cup is on the table or it is not, the child is in the room or she is not. But if we desire

THE ACQUISITION OF HABITS 225

anything more than this somewhat crude semantic evaluation, process considerations inevitably intrude. In our earlier work we introduced the distinction between result semantics and process semantics but offered little analysis of the distinction and did not examine in any detail the question of when result conditions or process conditions are called for. These issues will be discussed in some detail in the next two sections.

First, some straightforward remarks about the two kinds of conditions are needed. While result conditions concern the outcome of a response to a command, process conditions pertain to the action taken to achieve an outcome. Result conditions are most naturally thought of as pairs, with one member of the pair representing a satisfactory outcome, the other an unsatisfactory outcome, as in (the cup is on the table, the cup is not on the table). Note that a satisfactory outcome for a response to a command may itself be the taking of an action, but these action outcomes should not be confused with process conditions. Consider, for example, the following command addressed to a hospital attendant or a mobile robotic courier:

(3) Continue to deliver meals to room 27.

The action outcome pair (meals are being delivered to room 27, meals are not being delivered to room 27) is a result condition. A process condition would refer to the way in which delivery was continued - perhaps there was a break in service, for instance. I turn now to examine these process conditions in greater depth.

PROCESS SEMANTICS

Language is replete with mechanisms for characterizing actions and thereby making process-oriented distinctions. Familiar aspectual distinctions such as progressive and non-progressive, perfective and non- perfective, durative and punctual all serve as indicators of process. In addition, a wide variety of adverbials qualify described actions. Howev- er, it is not at all a straightforward matter to specify appropriate process conditions for evaluating the satisfaction of a command.

Part of the difficulty lies in the fact that there is a level of detail beyond which language cannot go. A command such as

(4) Walk carefully

226 COLLEEN CRANGLE

is simple and concise, but conditions laying out the requirements of careful locomotion in a particular context may be beyond the descriptive powers of language. In fact, as Patrick Suppes has remarked in various discussions, there are limits on the extent to which the details of an action can be laid out, no matter what descriptive means and depth of detail are relied on. Suppes points out that even for the simple action of throwing a pair of dice, the detailed analysis of the motion of the dice is a matter of great mathematical difficulty, with a full description of the process by which a result such as double sixes is reached in fact being impossible.

Furthermore, in order to fully specify conditions on the action an agent takes in response to a command, we need some way to specify details of the agent's cognitive, perceptual, and motor functioning. Consider commands such as

(5) Lift the chair gently

and

(6) Look diligently for the empty bottle on the shelf

It is hard to see how satisfaction of such commands can be evaluated without including aspects of the agent's functioning. At this point, a retreat to the simpler framework provided by robots often becomes necessary. Later in this article I briefly describe some of our own work using robots.

Adverbials of inanner such as 'gently' and 'diligently' certainly demand process evaluations in terms of the agent's functioning. But contrast these adverbials, which stipulate details of action execution, with adverbials that simply qualify the event that is to take place by stipulating a property that the action must have. Here are examples.

(7) Cook the spaghetti for 20 minutes.

(8) Ring the bell loudly.

(9) Answer at once.

In each case a property of the requested action is given: there must be a 20-minute spaghetti-cooking activity, a loud bell ringing, an immediate answer. Process evaluations are required for (7), (8) and (9), but it is not clear that these have to include details of the agent's operation.


A further distinction must therefore be introduced, that between action property and action execution. A judgment about process satisfaction may be in terms of action properties or in terms of action execution. Note that even simple commands unadorned by adverbs may demand appraisal in terms of action execution, as evidenced by the following set:

(10) Pound the nail into the plank.

(1 1) Drive the nail into the plank.

(12) Hammer the nail into the plank.

(1 3) Tap the nail into the plank.

Verbs are rich in process distinctions, a fact that has long been a challenge to lexical semantics. Nonetheless, adverbials remain central to questions about command satisfaction. It has been notoriously difficult to explain adverbs of action. Familiar problems such as the difference between

(14) Clumsily he trod on the lawn

and

(1 5 ) He trod on the lawn clumsily

have received considerable attention - see, for example, Vendler (1 984) - but there is as yet no systematic account of the way action descriptions are modified by adverbials. Furthermore, in philosophical analyses of events and actions such as those provided by Davidson (1970) and Bratman (1987), while the notion of intentionality has received much attention there has been less of a concern with properties of actions and details of action execution.

The two kinds of process evaluations I have identified are just a beginning in understanding how the plethora of process-oriented distinctions found in language affect judgments about a command's satisfaction. Later in this article I will return to process judgments when I discuss circumstances under which process considerations intrude into judgments that are primarily in terms of result. Table I presents the two kinds of process judgments together with sample commands requiring each kind of evaluation.

228 COLLEEN CRANGLE

TABLE I

Kinds of process judgments and sample commands.

RESULT SEMANTICS

JUDGMENT IN TERMS OF ACTION PROPERTIES

JUDGMENT IN TERMS O F ACTION EXECUTION

Despite the importance of process considerations in the evaluation of commands, result semantics has a role to play in the analysis of commands, one that cannot be eliminated. Consider, first of all, commands requesting actions that have well-defined end points, as in

Cook the spaghetti for 20 minutes. Ring the bell loudly. Answer at once.

Lift the chair gently Pound the nail into the plank. Hammer the nail into the plank

(1 6) Write the letter.

This command will elicit a result judgment: has the end point been reached, that is, does a terminating condition such as the letter's having been signed and sealed hold? The agent may have been writing and may have since stopped writing, but unless the terminating condition holds the command will not be judged to have been satisfied. Process considerations may also intrude into satisfaction judgments for these commands, but result judgments are sure.

This characteristic - an action's having a well-defined end point - is a familiar one in the extensive literature on tense and aspect. See, for example, Comrie (1 976), Vendler (1 967), and Freed (1 979). Vendler 's original classification of activities, accomplishments and achievements has spawned much discussion over the years, and the distinctions that have been recognized as a result are significant to judgments about command satisfaction. For instance, consider a punctual event that is the culmination of some process (an achievement in Vendler's terms). Examples are winning the prize, reaching the summit of the mountain, finding the red book on the shelf - respectively the culmination of competing in an event, climbing a mountain, searching for a book.


A command requesting the performance of a punctual end-point event cannot in itself elicit process conditions for there is no process to evaluate. What may be evaluated, however, is the process leading up to the punctual event: how the race was run, for instance, how the mountain was climbed, how the book was searched for. Primarily, though, these commands elicit result judgments.

Result semantics plays a role in yet other command evaluations. In many cases of aspectual complementation found in 'stop', 'start', 'continue', 'resume' and 'repeat' commands, result semantics is exactly what is needed. Take the following 'stop' command:

(17) Stop carrying the box.

Its satisfaction is judged primarily in terms of result: the box continues to be carried or it does not. But simple result semantics is not enough. Consider the command

(1 8) Put the book on the shelf.

A satisfactory response at the result level would have the book on the shelf, the intended result achieved. However, the event of the book's being shelved may be realized by many different specific actions that may produce many different specific outcomes. The book may be grasped clumsily and its cover torn, on placement its spine may face in or out on the shelf, it may be placed upside down or right way up. A second distinction introduced in our earlier work was that between events and specijic outcomes. Closer analysis of the 'stop' command in (17) reveals that what is needed is not a simple result semantics that offers pairs such as (the box is being carried, the box is not being carried) but an analysis that is at the level of specific outcomes. A command to stop doing something expresses more than the simple intention that the agent interrupt that activity. It carries with it the usually unexpressed intention that the agent do something else instead (Crangle, 1989). For the command in (17), for instance, the agent should not simply freeze; typically the object being carried should be put down carefully or handed over to someone else. There are even circumstances in which it should be dropped without delay. What is needed for these 'stop' commands is an analysis that offers pairs of action outcomes such as (the box is being carried, the box is put down), (the box is being carried, the box is handed over), or (the box is carried, the box is dropped). Similarly, for the command

230 COLLEEN CRANGLE

(1 9) Stop running

a result evaluation is needed, one that offers specific outcome pairs such as (the agent is running, the agent is walking) or (the agent is running, the agent is standing still).

Judgments at the level of events have their own place. They are typically elicited by commands requesting repeated or habitual behavior. Consider, for example, the injunctions found in a book of etiquette.

(20) Be considerate.

(21) Escort your visitor to the front door.

(22) Remove soup plates to the sideboard one at a time.

(23) Place the vegetable spoons at the corner of the table, about ten inches from it, their bowls pointing in opposite directions.

(24) Watch the theater entrance for your escort unobtrusively, not with fixed and desperate gaze.

Here we encounter exhortations to engage in certain kinds of habitual behavior, some expressing broad generalities as in (20), others address- ing specific conventions as in (22) and (23). We tend to evaluate satisfaction of such admonishments not so much in terms of any specific outcome as over the long haul, making a judgment, for instance, as to whether a few failures to accompany a visitor to the door mean the injunction is not being obeyed.

It is interesting to note that although only some of the above commands make explicit reference to process - 'one at a time', 'with fixed and desperate gaze' - process considerations would probably enter into most satisfaction judgments for these commands. A possible exception is (23); under ordinary circumstances a straightforward evaluation of the location and orientation of the spoons is all that would be required. In terms of the distinction introduced earlier, any process evaluations made would typically be in terms of action properties not action execution. It would be assumed that details of action execution have been thoroughly taken care of by the time a call for habitual behavior is issued.

THE ACQUISITION OF HABITS

THE INTRUSION OF PROCESS JUDGMENTS

It is instructive to examine just when it is that process considerations intrude into satisfaction judgments that are primarily in terms of result and at the level of events. Consider the following command addressed to a human or robotic janitor:

(25) Pick up litter.

This command directs the agent to engage repeatedly in the activity of finding and retrieving individual items of refuse. We judge satisfaction in terms of the results achieved - litter is left behind or it is not - and we do so less in terms of specific outcomes - there is no litter Wednesday 2:35 p.m., there are three scattered pieces Friday 5:40 p.m., and so on - than in general terms, that is, at the level of events. Furthermore, we typically do not concern ourselves with how each item of refuse is removed, for the request assumes that all details of action execution at this lower level will be taken care of adequately. The satisfaction of (25) is therefore typically judged in terms of results at the event level.

Process considerations intrude, however, when something can go wrong and does. Suppose certain pieces of litter start getting left behind - discarded wooden ice-cream sticks, for instance, and crumpled candy wrappers. Suppose it turns out that all small objects are being ignored, perhaps because the agent regards only big items as trash, perhaps because the agent has poor vision, or perhaps because the agent lacks the fine motor control needed to retrieve small items. In each case, a flaw in the process of picking up litter produces an unsatisfactory response to the command. Any subsequent evaluation of the agent's response to the command would have to include an evaluation of process to check that the flaw has been eradicated. This evaluation would be in terms of action execution, that is, it would refer to details of the agent's functioning. Although for human agents there is limited access to the processes by which cognitive, perceptual, and motor functioning are governed, there are various methods for making the appropriate process-oriented appraisals: interrogating the person, subjecting her to tests of visual acuity and manual dexterity, for instance. In the case of robots, investigation of process, though technically complex, is entirely possible and indeed necessary if a satisfactory response to a command is to be achieved.

COLLEEN CRANGLE

TABLE I1

Kinds of result judgments and sample commands.

There is another set of circumstances in which process judgments intrude into evaluations that are primarily result oriented. Consider the command

JUDGMENTS INTRUDE. . .

(26) Hit the ball

JUDGMENT AT THE LEVEL OF EVENTS

JUDGMENT AT THE LEVEL OF SPECIFIC OUTCOMES

addressed to an agent learning how to play tennis. Result judgments are naturally important here and they are primarily at the level of specific outcomes - exactly where the ball lands on each stroke is crucial to evaluating responses to the command. Process considerations inevitably intrude, however, when a new task is being taught. Initially, for tennis instruction, evaluations are made about how the player holds the racquet, uses her wrist and elbow, positions her feet, and so on. These process evaluations fall away as she masters her stroke, but return if she has to adapt her skill to changed circumstances, a different racquet design, for instance.

In Table 11, I show the two kinds of result judgments I have discussed along with sample commands and an indication of when process judgments intrude. I turn in the next section to examine interactions between the different kinds of satisfaction conditions.

Be considerate. Pick up litter.

Stop carrying the box. Hit the ball.

WHENEVER FLAWS SURFACE AND DURING VERBAL INSTRUCTION

THE ACQUISITION OF HABITS

HABITS, LEARNING, AND SATISFACTION CONDITIONS

I began this article with the observation that a great many commands requesting action leave unexpressed many details of action execution and specific outcome. A request as simple as

(27) Put the wrench away

says nothing about how the wrench is to be held and nothing about its precise placement. What has not yet been emphasized, but is crucial to an understanding of commands requesting action, is that the speaker will typically have in mind not some precise spot but a target location - somewhere around the middle of the top shelf, for instance, or anywhere on the countertop. The speaker will often also be quite neutral as to the object's orientation while it is being moved and when it is deposited on the shelf. Even when details of action execution and outcome are specifically intended, because of the difficulty and tedium of expressing them precisely, something like a negotiated settlement is generally arrived at. The agent being addressed offers a best response to the command and the person giving the order determines if that response is good enough. If it is not, she repeats the order or gives corrective instruction. During this interchange, process considerations and judgments about specific outcomes predominate in evaluations of command satisfaction.

There is, however, a point at which settlements no longer have to be negotiated. Consider the command

(28) Go to the door.

It carries with it many unexpressed intentions concerning how close to the door the agent should come, and whether she should stop at a point that does not impede the door's opening or perhaps block entry to the door. People typically respond to this request by drawing on habits acquired over time through experience with many comings and goings. But a very young child will not automatically stand in the right place or face the right way in response to this command, for the appropriate habits of arrival and departure have not yet been ingrained.

Habits have received relatively little attention from philosophers and cognitive scientists. It is only in the study of animal behavior that habits have been accorded any kind of prominence. The ordinary understanding of a habit is as behavior ingrained by frequent repetition of an act. The behavior is often thought to be almost involuntary, particularly in the case of bad habits. A habit is also sometimes thought

234 COLLEEN CRANGLE

of as a tendency or disposition to act in a certain way brought about by frequent repetition. Although I believe it would be fruitful to analyze habits in probabilistic terms, as suggested by the notions of disposition and tendency, in the context of this discussion it is instructive rather to examine several lines of development in the design of intelligent robots.

A recent debate in robotics argues the merits of two different approaches to intelligent robot design. The first approach adopts the perspective characteristic of much mainline research in artificial intelligence over the past few decades. See, for instance, Nilsson (1980) and Latombe (1 991). It assumes that an intelligent agent will need to reason about its world and the tasks it undertakes using representations or models of that world. These robots typically include 'task planners' in their design. To do its job, a task planner needs geometrical, physical, and kinematical descriptions of all objects in the robot's environment, including the robot itself. A task planner takes a task-level specification, that is, one that specifies an action largely by its effect on objects, and produces a manipulator-level program, one that directs the detailed movement of the robot. Task-level robot programming languages are thus largely result oriented rather than process oriented. In its most idealized form, this approach assumes that with enough information about the world and itself, the robot can, from scratch, satisfactorily perform any task within its capability.

In contrast, the second approach believes in the efficacy of trial and error, that is, in the importance of process (Brooks, 1991). Each robot is designed as a network of relatively independent 'primitive' actions or behaviors. For example, 'avoid hitting things' is one primitive behavior, 'stand up' is another (for legged robots), 'explore' (i.e., wander around) yet another. These behaviors operate largely in parallel, and during development of the robot they are added incrementally. This approach abjures centralized or hierarchical control in its robots and makes only limited use of representations of the robot's world. It seeks to rely rather on techniques that through trial and error will organize the primitive behaviors in such a way that they will, in concert, exhibit the desired higher-level behavior. For example, one well publicized experimental robot finds and retrieves empty soda cans, a goal that was built into it through the choice and design of its primitive behaviors.

The debate over these two approaches is often thought to be primarily about centralized versus decentralized control in intelligent agents and about the role of representations in intelligent behavior. But at a more


general level it is a debate about whether or not robots need habits. In the second approach, the robot is essentially constructed as a collection of built-in habits, habits of standing, avoiding, and wandering about, for instance. Through repetition and practice a new habit emerges, that of finding and collecting empty soda cans, for instance. The robot's behavior is unreflective and, by all ordlnary notions of consciousness, unconscious. In contrast, the first approach builds robots that engage in continual reflection, reasoning about their world, their capabilities, and the tasks they are to undertake. There is no obvious place in this framework for repetition, practice, correction or refinement. There is no obvious place in this framework for habits.

The second approach to robot design - the trial-and-error habit- forming approach - has made a significant contribution to the study of intelligent agency. The development of habits is central to the proper functioning of intelligent agents, not least of all to their ability to respond appropriately to commands requesting action. However, habits alone are not enough. Healthy skepticism should be held toward one of the key assumptions of this second approach, namely that increasingly complex robot behavior can be achieved by incrementally adding new behaviors, that is, new built-in habits, without imposing any form of centralized or hierarchical control. Some organizing scheme is surely needed if these robots are to exhibit high-level skills such as adjudicating between competing goals.

In our own work on intelligent robots, our approach has been to explicitly build into the robot mechanisms for acquiring habits through interaction with a human operator (Crangle et al., 1987; Suppes and Crangle, 1990). These habits are developed over time in response to verbal commands requesting action. The first time a command such as 'Go to the door' is issued to a robot in a particular set of circumstances, the robot's response typically only approximates the desired response. A process of negotiated settlement then ensues. The operator issues qualitative feedback to correct or confirm the action taken, either a congratulatory command such as 'That's fine' which indicates that the response was acceptable, or a corrective command such as 'Further to the left' which indicates that the response was unacceptable. The corrective feedback is non-determinate in that it does not let the robot know exactly what its response should have been. It merely indicates what the robot can do to improve its response on subsequent attempts. The corrective feedback is either in terms of result - 'Much further to

COLLEEN CRANGLE

TABLE 111

A dialogue for developing a habit of correct responses to a command.

I ORIGINAL COMMAND ISSUED: Put the wrench on the shelf.

RESULT FEEDBACK: Not that far. CONGRATULATORY FEEDBACK: That's fine.

ORIGINAL COMMAND REPEATED: Put the wrench on the shelf.

RESULT FEEDBACK: A little further to the right.

MORE RESULT FEEDBACK: Further. PROCESS FEEDBACK: Be more careful. RESULT FEEDBACK AGAIN: A little to the left now.

CONGRATULATORY FEEDBACK: Good.

the right', 'A little bit further' - or in terms of process - 'Be more careful', 'Faster next time'. Over time, the robot's repetition of the action together with the refinement achieved through verbal feedback inculcates in the robot a habit of correct responses to the command.

In Table 111, I show the initial portion of an instruction dialogue used to induce in the robot a habit of correct responses to the following command:

(29) Put the wrench on the shelf.

Before a habit of correct responses has been ingrained in the robot, judgments about action execution and specific outcome predominate in evaluations of the command's satisfaction. The proposition I would defend in general is that as an action becomes ingrained as habit in an agent, satisfaction judgments for commands requesting that action change from judgments of action execution and specific outcome to result judgments at the level of events and process judgments in terms of action properties. A consequence of this proposition is that satisfaction conditions for a command do not adhere in the language itself but are determined by the circumstances of the command's use, including the agent's capabilities and past responses to the command. That is as it should be. Consider, in particular, imperatives that may be used to


Fig. 1.

L

express either a request for habitual behavior or a request for specific action. For example, while it is hard to construe

Result

(30) Take regular exercise

Judgments in

as referring to anything other than habitual behavior,

judgments at terms of the level of action events properties

Result Judgments in judgments at terms of the level of action specific execution outcomes

(31) Mow the lawn

may be understood as a request for specific action or a call to practice a certain kind of habit. Past experience is surely crucial in determining the appropriate satisfaction judgments for this command.

In the figure above I show the four principal satisfaction judgments covered in this article and I indicate the changes brought about by the agent's acquiring a habit of correct responses to the command.

I have sought in this article to lay out a framework for evaluating command satisfaction, and in addition to advocate a more prominent place for habits in the semantics of commands. A full account of command satisfaction will undoubtedly turn out to be more complex than demonstrated here, but to be complete it will, I believe, have to encompass the different kinds of satisfaction judgments that are made and acknowledge the role played by habits.

Intelligent Interface Technology, 848 Cambridge Avenue, Menlo Park, CA 94025, U.S.A.

238 COLLEEN CRANGLE

REFERENCES

Bratman, M.: 1987, Intention, Plans, and Practical Reason, Harvard University Press, Cambridge, Massachusetts.

Brooks, R. A.: 1991, 'Intelligence Without Representation', Artificial Intelligence, 47, 139-1 60.

Comrie, B.: 1976, Aspect: an Introduction to the Study of Verbal Aspect and Related Problems, Cambridge University Press, Cambridge, Massachusetts.

Crangle, C.: 1989, 'On Saying "Stop" to a Robot', Language and Communication, 9(1), 23-33.

Crangle, C., Suppes, P., and Michalowski, S.: 1987, 'Types of Verbal Interaction with Instructable Robots', in: G. Rodriguez (Ed.), Proceedings of the Workshop on Space Telerobotics, JPL Publication 87-13, Vol. IT, NASA Jet Propulsion Laboratory, Pasadena, California, pp. 393-402. Reprinted in P. Suppes: 1991, Language for Humans and Robots, Blackwell Publishers, Cambridge, Massachusetts, pp. 299- 316.

Davidson, D.: 1970, 'The Individuation of Events', in: Nicholas Rescher (Ed.), Essays in Honor of Carl G. Hempel, D. Reidel, Dordrecht.

Freed, A. F.: 1979, The Semantics of English Aspectual Complementation, D. Reidel, Dordrecht.

Latombe, J.: 199 1, Robot Motion Planning, Kluwer Academic Publishers, Boston, Massachusetts.

Nilsson, N. J.: 1980, Principles of Artificial Intelligence, Tioga Publishing Company, Palo Alto, California.

Suppes, P. and Crangle, C.: 1988, 'Context-Fixing Semantics for the Language of Action', in: J. Dancy, J. M. E. Moravcsik, and C. C. W. Taylor (Eds.), Human Agency: Language, Duty, and Value, Stanford University Press, Stanford, Califor- nia, pp. 47-76.

Suppes, P. and Crangle, C.: 1990, 'Robots that Learn: A Test of Intelligence', Revue Internationale de Plzilosophie, 44(172), 5-23.

Vendler, Z.: 1967, Linguistics in Philosophy, Cornell University Press, Ithaca. Vendler, Z.: 1984, 'Adverbs of Action', in: David Testen, Veena Mishra, and Joseph

Drogo (Eds.), Lexical Semantics, Chicago Linguistic Society, Chicago, IL, pp. 297- 307.


Colleen Crangle's extension of ideas that we have worked on together in the past shows how there is no natural bottom to the depth of semantic interpretation of commands. It is the single most striking feature of ordinary language that any sharp model-theoretic cutoff in the interpretation of the semantics is bound to be unsatisfactory in some respects. This is scarcely surprising, for it is similar to the situation that


obtains in the use of mathematical models in almost every domain of science. Increasingly, we have come to learn that, whether it is a matter of physics, economics, psychology or biology, an exact and complete model can seldom if ever be given for complicated phenomena. It is not unreasonable to think that the simple utterances so familiar as commands in ordinary language actually apply to simple situations, but this is only a characteristic illusion in the understanding of what is implicit in the use of language. Colleen's examples and arguments strengthen the case for the importance of context in giving a semantics for commands in natural language. There is much to be understood about past experience that is assumed, which comes under the general heading of habits and expectations, and there is also much to be understood about current external conditions when the command is given.

As a reflection of what I have just said, I would mention one minor difference with Colleen. I would not myself be inclined to draw a sharp distinction between events and specific outcomes. It is as if we can really conceive of specific outcomes as definite atoms of experience. This again represents for me an artificial simplification, much used in science and valuable for that reason, but not intrinsic to natural language in its use or in its semantic analysis. For example, we certainly do want to speak of outcomes in a definite theoretical and abstract fashion when we introduce sample spaces in probability theory or when we have an abstract, highly specific theory to consider. But recall the situation in probability theory just as an example. In almost all advanced work in statistics, one considers not a specific probability space, but a family of random variables and relies upon the family of random variables satisfying Kolmogorov's theorem so that there exists a common underlying probability space, but this space is not unique. Moreover, in all of the conceptual and computational work specific features of an underlying space scarcely arise. To put the formulation in an even more radical way, a subjectivist such as de Finetti is even skeptical of using a probability space at all, and his random quantities, which he does not call 'random variables' for good reason, are introduced without being defined on any probability space. I would expect de Finetti to have shared my views on this even though we use somewhat different language. If we mention what Colleen would term a 'specific outcome' in natural language then the generative capacity of the language always permits us to qualify in different ways this specific outcome. Let me consider one simple example.

240 COLLEEN CRANGLE

Specific Outcomes. When someone says 'Hit the ball' usually a range of specific outcomes are expected but unstated. For example, the command might be implicitly qualified depending upon the sport in different ways. In tennis it might mean 'Hit the ball over the net', in baseball it might mean 'Hit the ball at least 50 feet', and in croquet it might mean something still different. The features that are expected arise from current circumstances and past habits and have no natural fixed finite enumeration. For this reason I would much prefer to talk about atomless Boolean algebras in terms of formal models and to speak in terms of one event being more specific than another, specific in the way in which it is more specific to say that in rolling two dice 'at least one 6 came up' as opposed to saying 'at least one even number came up'.

Skepticism about Representations. A more important point is that I very much share Colleen's skepticism about intelligent agents, especially robots we create or humans that we educate, being able simply to think rationally about their world and use continually explicit representations or models to guide their behavior. The strong belief in the psychological reality of such representations is certainly behind misguided efforts of some philosophers of language to believe that all thinking and reasoning are done in terms of discrete symbolic symbols, a mistaken psychological fantasy if ever there was one.

Rather, what we need is what Colleen emphasizes: learning and the formation of habits which do not have an explicit representation and which, above all, either in the case of robots or humans, are not acces- sible to consciousness. Just as I cannot describe or explicitly represent in my own mind consciously how I hit a tennis ball, so a robot that goes through a process of learning will not naturally have such representations. What I can say about hitting a tennis ball, or what even the best experts can say is, from the standpoint of the physical trajectory of the racket and the ball, crude and qualitative in character. Now it might be thought that we can artificially build into the robot a capacity to describe in mathematical terms what it is doing while it acquires specific habits of motion. But this would be mistaken for an obvious reason. We might be able to build into the robot the ingredients to find the derivation of various differential equations governing various pieces of the trajectory, but we would not at all be able to embed in the robot the ability to solve these differential equations and thereby to describe the trajectories. In

THE ACQUISITION O F HABITS 241

most casss nonlinear phenomena are flagrantly present which makes detailed solutions with realistic boundary conditions unfeasible. The lesson to be learned from all the problems of chaos theory, incompleteness of physical theories, etc., is that explicit representations of the motions of bodies is a mistaken enterprise either in terms of thinking about human mental processes or about the construction of robots.

In being so negative about representations, I do not want to give the impression that I think we can therefore introduce habits as a panacea for our ways of thinking about the behavior of intelligent agents. There is still much that we need to learn about the theory of habits, and above all about how they can be learned by intelligent agents of our own construction. All the same, the emphasis on habits rather than representations is, in my view, the right one.

PART VIII

GENERAL PHILOSOPHY OF SCIENCE

MARIA CARLA GALAVOTI'I

SOME OBSERVATIONS ON PATRICK SUPPES ' PHILOSOPHY

OF SCIENCE

ABSTRACT. The paper outlines the main traits of Patrick Suppes' 'probabilistic empiricism'. It combines the conviction that probability should be assigned a central role within epistemology with a serious consideration of experimentation. First Suppes' empiricism and its close connection with pragmatism is described. Then, his views on probability are discussed at length, together with his theory of probabilistic causality. Lastly, Suppes' notion of rationality is recalled. The main features of Suppes' position in the whole are identified with its pluralism and its local character. It is argued that in place of a general philosophical view Suppes works out a method allowing a representation of phenomena in which both theories and data, abstract mathematical construals and particular experimental techniques, are all given attention and find their place.

If one were to review the literature on philosophy of science in recent years, one would be faced with all sorts of claims to the effect that empiricism is dead, or at least in mortal agony. Many have claimed that the empiricist way of looking at things has proved incapable of capturing the complexity characterizing human knowledge in general, and the scientific enterprise in particular. At the same time, many have urged that empiricism has nothing to say, and should just give way to some form of relativism, or even to methodological anarchism. Contextually, metaphysical tendencies have been resumed in order to solve some of the numerous problems left open by the empiricist way of looking at things.

These attitudes have in most cases resulted from dissatisfaction with logical positivism. A criticism of the strictures imposed on epistemology by logical positivism, however, does not necessarily imply abandoning empiricism altogether. In addition to the voices pleading the cause of anarchism and/or a return to metaphysics, the revision of logical positivism has opened alternative roads to empiricism, inspired by a genuinely constructive attitude.

One of them has been indicated by Patrick Suppes, who calls it 'probabilistic empiricism'. In what follows I will try to elucidate the

P. Humphreys (ed.), Patrick Suppes: Scientific Philosophel; Vol. 3, 245-270. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

246 MARIA CARLA GALAVOTTI

main traits of this position, which is in many ways innovative. As suggested by the label attached by Suppes to his own point of view, this combines an empiricist attitude to philosophy with a probabilistic component. At its basis we find the conviction that the notion of probability should be assigned a central role within epistemology. This is clearly stated by Suppes, who says that

it is probabilistic rather than merely logical concepts that provide a rich enough framework to justify both our ordinary ways of thinking about the world and our scientific methods of investigation (1984a, p. 2. See also 1980, p. 171).

Probabilistic empiricism should then replace logical empiricism, and accordingly probability, and not logic, should be the point of departure of our investigations into philosophy of science and epistemology.

It is worth stressing how this fundamental conviction involves a shift in emphasis from the linguistic aspects of the language of science to its content. In other words, attention to the syntactical structure of scientific discourse gives way to consideration of the complex procedures, like measurement and model building, which allow phenomena to be investigated and organized within scientific theories. This marks a major difference between the form of empiricism advocated by logical empiricism and that put forward by Suppes. As a matter of fact, a similar attitude has recently become quite widespread, and logical empiricism has often been criticized for having attached too much importance to the linguistic aspects of science. Let us then turn to some of the more distinctive features of Suppes' position. To start with, it seems appropriate to characterize his 'empiricism' and to illustrate its close connection with pragmatism. Suppes' views on probability will then be discussed at length, together with the main philosophical implications of his approach.

EMPIRICISM

In his self-profile written in 1978, Suppes calls himself "the only genuinely empirical philosopher I know" (Suppes, 1979, p. 45), and by this he is stressing the influence of scientific work on his own philosophy. No doubt it receives a peculiar flavour from his twofold activity, as a philosopher and as an experimental scientist. This double militancy is at the basis of the importance Suppes ascribes to experimentation. It is certainly experimentation with all the problems connected with it that

ON PATRICK SUPPES' PHILOSOPHY OF SCIENCE 247

occupy the central place within Sugpes9 empiricism. Here the traditional distinction between the two contexts of 'discovery9 and 'justification' introduced by Reichenbach, and widely accepted by philosophers of science, is superseded by an approach in which experimentation and related problems pervade the whole 'reconstruction' of scientific investigation. The attention paid to such problems influences above all Suppes' view of theories, while being at the root of the importance he ascribes to measurement, as well as probability.

Suppes' view of theories is probably the most well-known aspect of his philosophy of science, and has been analysed in detail,' mainly focusing on the formalization of theories and on his dictum to the effect that 'to formalize a theory is to define a set-theoretical predicate'.2 Here I will briefly recall the centrality in that perspective of the notion of model, together with the pluralism this brings with it.

For Suppes models are the entities in terms of which a theory is to be defined. Models provide the connection between a theory and the phenomena under investigation, through the notion of structure. This task is fulfilled by showing that "the structure of a set of phenomena under certain empirical operations is the same as the structure of some set of numbers under arithmetical operations and relations", where the idea of 'sameness of structure' is to be taken in terms of a suitable notion of isomorphism (Suppes, 1967, p. 59). Isomorphism plays a crucial role in representing a theory in terms of its models. For Suppes

the best and strongest characterization of the models of a theory is expressed in terms of a significant representation theorem,

where this is a proof to the effect that

a certain class of models of a theory distinguished for some intuitively clear conceptual reason is shown to exemplify within isomorphism every model of the theory (Suppes, 1988a, p. 259).

Another crucial notion in this context is that of invariance, which gives a criterion of meaningfulness according to which

an empirical hypothesis, or any statement in fact, which uses numerical quantities is empirically meaningful only if its truth value is invariant under the appropriate transformations of the numerical quantities involved (Suppes, 1988a, p. 265).

The representation of a theory in terms of its models here takes the place of the 'received view' of theories delivered by logical empiricism, according to which a theory consists of a logical construction plus a set

248 MARIA CARLA GALAVOIT1

of rules assigning an empirical meaning to its terms, or at least to some of them. In a general way - Suppes says - the best insight into the structure of a complex theory is by seeking representation theorems for its models, for the syntactic structure of a complex theory ordinarily offers little insight into the nature of the theory (Suppes, 1988a, p. 254).

In other words, knowledge of the syntactical structure of a theory is not enough; in order to understand what a theory is about we should instead look for its models.

PRAGMATISM

This move away from the 'received view' of theories goes in the direction of a pragmatist philosophy of the kind upheld by authors like C. S. Peirce, W. James, J. Dewey and E. Nagel, who was Suppes' teacher at Columbia from 1947 to 1950. According to a pragmatist approach, scientific activity is perpetual problem solving and theories typically qualify as local construals. Like our own lives and endeavors - Suppes says - scientific theories are local and are designed to meet a given set of problems (Suppes, 1981 b, pp. 14-15).

Locality goes hand in hand with a peculiar pluralism, characterizing both the notion of model and that of structure, neither of which is amenable to a univocal definition. The point is extensively dealt with by Suppes in an article published in 1962 bearing the title 'Models of Data'. Since this does not seem to be among his best known papers, it is worth recalling its central ideas. The main thesis is summarized by the claim that "the relation between empirical theories and relevant data calls for a hierarchy of models of different logical type" (Suppes, 1962, p. 253). When analysing the linkage between theory and experimental data one therefore has to distinguish models of the theory from models of the performed experiments, and models of the data obtained. To a hierarchy of models there corresponds a hierarchy of the problems one typically encounters at the different levels of analysis, in the course of the complex procedure aimed at comparing theories with experiments.

This conclusion is suggested by a close inspection of the statistical methods employed at each level of such comparison, which include measurement, experimental design, estimation of parameters, tests of goodness of fit, identification of exogenous and endogenous variables,


and the like. A detailed analysis of such methods not only points to the need for a plurality of different models, it also clarifies the direction of the dependence between models. The dependence is such that in a hierarchy of models characterized by an increasing level of abstraction one does not move from top to bottom, but rather from bottom to top. In other words, given a model of the data, exhibiting a certain statistical structure of the phenomenon under investigation, one looks for a theoretical model that fits it.

What I have attempted to argue - says Suppes - is that a whole hierarchy of models stands between the model of the basic theory and the complete experimental experience. Moreover, for each level of the hierarchy there is a theory in its own right. Theory at one level is given empirical meaning by making formal connections with theory at a lower level (Suppes, 1962, p. 260).

As a direct consequence of considering theories not in any abstract way, but rather in connection with experimentation, we then have, in addition to a hierarchy of models, also a hierarchy of theories.

If someone asks 'what is a scientific theory?' - Suppes says - it seems to me there is no simple response to be given . . . What is important is to recognize that the existence of a hierarchy of theories arising from the methodology of experimentation for testing the fundamental theory is an essential ingredient of any sophisticated scientific discipline (Suppes, 1967, pp. 63-64).

This is a lesson to be learned from a careful analysis of the role played by statistical methods in experimentation and theory making. This sort of analysis is precisely what, according to Suppes, is missing from the traditional approach taken by philosophers of science,

who write about the representation of scientific theories as logical calculi [and] go on to say that a theory is given empirical meaning by providing interpretations of coordinating definitions for some of the primitive or defined terms of the calculus (Ibidem).

As Suppes points out, this lesson emerges most explicitly from some branches of the social sciences, like learning theory, to which he mostly refers. It is in fact a common feature of disciplines which have not reached a highly theoretical development to make extensive use of sophisticated methods for evaluating evidence and testing hypotheses. It is noteworthy that recent econometric research points exactly in the same direction indicated by Suppes, and propounds a pluralistic view of model building that bears strong resemblance to the approach outlined in 'Models of ~ a t a ' . ~

One can see that within Suppes' philosophy of science data receive at least the same importance ascribed to theories. The structure of data


delivered by observation is in itself the object of interest and investigation, and calls again for pluralism. Detailed analysis of such structures indicates that there is no univocal answer to the question 'what are data?' since the 'data' represent an abstraction from the complex practical activity of producing them. Steps of abstraction can be identified, but at no one point is there a clear and distinct reason to exclaim, 'Here are the data!' (Suppes, 1988b, p. 30).

Depending on the desired level of abstraction different pieces of information will then count as 'data'. In addition, a multitude of context- dependent elements will have a bearing on it.

The refusal to search for a unique characterization of important concepts is a major feature of Suppes' philosophy. He has repeatedly stressed that the complexity of phenomena and the variety of practical situations in which phenomena are investigated are such that important notions in science as well as in philosophy cannot be cooped up in some definition given once and for all. Plurality then becomes for Suppes one of the tenets of the 'new metaphysics' by means of which he fights the chimeras of a traditional view of rationality also shared by logical empir i~ism.~ The ideal of the unity of science should then be abandoned in favour of the recognition that the sciences are characteristically pluralistic, rather than unified, in language, subject matter, and method (Suppes, 1984a, p. 10).

This abandonment of the ideal of the unity of science goes hand in hand with the rejection of the neopositivistic ideal of reductionism. Suppes has convincingly and repeatedly argued that the diversity of language, subject matter and methods both in different disciplines and in different branches of the same discipline is such that reduction is either impossible, or - when possible - uninteresting and barren.5

Suppes' pragmatically oriented view of theories opposes another chimera of rationality, namely the idea that "scientific knowledge can in principle be made complete" (Suppes, 1984a, p. 2) and that the shift from old to new theories brings with it a convergence to some finite value. On the contrary, one of the main tenets of Suppes' 'new metaphysics' amounts to the claim that the collection of past, present, and future scientific theories is not converging to some bounded fixed result that will in the limit give us complete knowledge of the universe (Suppes, 1984a, p. 10).

ON PATRICK SUPPES' PHILOSOPHY OF SCIENCE

PROBABILISTIC APPROACH

Suppes' 'new metaphysics', of which pluralism and incompleteness are essential ingredients, is illustrated in the volume Probabilistic Meta- physics which appeared in print in 1984. The main purpose of this work is to establish a probabilistic approach to philosophy of science, intended to supersede the 'neotraditional metaphysics' centred on determinism, namely the idea that the future is determined by the past and that every event has a sufficient determinant cause. As a first step in that direction one finds a claim to the effect that

certainty of knowledge - either in the sense of psychological immediacy, in the sense of logical truth, or in the sense of complete precision of measurements - is unachievable (Suppes, 1984a, p. 10).

Among the arguments put forward by Suppes to support this claim, one of the most convincing comes from imprecision of measurement, arising once again from his experimentalist attitude. Measurement itself has been extensively studied by Suppes, who has contributed important work to it. Without going into details, it is worth recalling that imprecision of measurement does not come only in connection with human or instrumental errors, but arises in a more substantial way from certain developments of our century's physics, like Heisenberg's uncertainty principle.6 Uncertainty then pervades not only the level of experimentation, but is to be encountered at the level of physical theories as well. In view of this, Suppes holds that the ideal of certainty should be renounced together with the other 'chimeras of rationalism' discussed above, completeness of knowledge and unity of science.

Recognition of uncertainty "at the most fundamental level of theoretical and methodological analysis" (Suppes, 1984a, p. 99) leads directly to probability, because probabilistic methods provide 'a natural way' of working out the form of empiricism advocated by Suppes. In other words, probability is the tool that allows him to build the pars con- struens of his 'new metaphysics', once the way has been cleared from the 'chimeras of rationalism'. Its fundamental tenets are precisely that the basic laws of natural phenomena are essentially probabilistic, and that causality, as well as the theories of meaning and rationality, are probabilistic, not deterministic in character. Among other things, this involves the conviction that "our conception of matter must contain an intrinsic probabilistic element" (Suppes, 1984a, p. lo), and leaves the door open to the admission of randomness in nature.

MARIA CARLA GALAVOTTI

BAY ESIANISM

As far as the interpretation of probability is concerned, Suppes has always called himself a Bayesian and a subjectivist. However, this conception of probability strays in many ways from that of an 'orthodax' Bayesian like Bruno de Finetti, to whom he often refers in his writings. With no intention of making a comparison between the two, let me try to focus on what I regard as the most original traits of Suppes' position. A central feature is the strict connection between probability and measurement, inspired once again by Suppes' experimentalist attitude. By connecting probability with measurement he has come to the conviction that exact values should be substituted by probability intervals. A number of results on upper and lower probabilities due to Suppes and zanotti7 testify to the fruitfulness of this approach.

Another important feature of Suppes' perspective is the conviction that probability and utility are notions to be dealt with separately. In this connection he agrees with de Finetti, with whom he shares a Bayesian approach to scientific inference. It is worth mentioning that the adoption of a Bayesian framework leads Suppes to an original treatment of problems like the paradoxes of confirmation8 and the problem of total e ~ i d e n c e . ~ To the latter, he offers a solution essentially based on the idea that under Bayesian conditionalization there is no additional problem of total evidence, once coherence is satisfied.

Disagreement with de Finetti comes in connection with the fact that for Suppes the subjective theory of probability offers necessary but not sufficient conditions for a theory of rationality. For one thing, the subjective theory of probability offers no way of evaluating different probability assessments due to experts, nor for dealing with uncertain evidence. The main disagreement, however, regards de Finetti's rejection of the ideas of 'unknown' and 'objective' probability. Moreover, Suppes ascribes great importance to the notion of randomness, regarded as meaningless by de Finetti. While interpreting de Finetti's attitude towards the above mentioned topics as a result of his positivistic and reductionistic attitude, Suppes devotes a certain effort to working them out. Let me briefly recall his views in this connection.

In interpreting probability, Suppes makes the point that the question of the meaning of probability statements is not different from that of the meaning of statements about physical properties or magnitudes, like mass and weight. In all such cases


there is not really an interesting and strong distinction between subjective and objective, or between belief and knowledge (Suppes, 1983, p. 399).

The important thing appears instead to be completeness of information, and the relevant distinction to be made is that between complete knowledge in principle or in practice and incomplete knowledge with the possibility of learning more. When talking about the meaning of a probability statement, one has in the first place to ask whether it is based on complete information, in the sense that there is no additional information we can conditionalize on that will bring about a change in the probability value. This is an important feature to be considered, especially when completeness of information comes from physical theory.

Another element that according to Suppes should be taken into account is the point in time at which the utterance of the probability statement is located. As he observes, "this is especially true for properties of events as opposed to properties of objects" (Suppes, 1983, p. 400), because insofar as events are concerned, we have more information after their occurrence than before. Suppes concludes that "in talking about completeness of information it is important to stress at what point in time the matter is being discussed" (Ibidem).

Suppes' claims in this connection raise some perplexity. One might object that the problem referred to by his remarks does not have much to do with completeness of information, being related to the tension between two different contexts in which probability statements may occur, corresponding to their predictive use, on the one hand, and their descriptive and explanatory use on the other. As widely stressed by recent literature on probabilistic explanation, such contexts should not be seen as overlapping, because they exhibit a fundamental asymmetry. This in turn reflects another important difference to be pointed out, between probability statements about single events and probability statements about properties of objects. It is a matter of fact that when dealing with single events characterized by probabilistic behaviour we can only (if ever) reconstruct the history behind their occurrence after they have taken place. This, however, does not have much to do with the meaning of probability statements, but with explanation of single events. In this context, in fact, we do not just have a statement giving the probability of an event prior to its happenings, but we have in addition the information that the event itself has actually happened, no matter how small a probability it was assigned beforehand. The preceding


considerations also have some bearing on probabilistic causality, to be discussed later.

OBJECTIVE PROBABILITY

Where completeness of information really matters is in connection with the problem of 'objective probability'. It is no doubt an important feature of Suppes' point of view to recognize that probability statements acquire a special meaning when based on completeness of information in the light of physical theories. In this case one tends to attach an objective meaning to them. While de Finetti vigorously denied that there can be any sense in talking about 'objective probability', Suppes admits such a notion. In a paper presented at a conference, on which I had the chance to comment,1° he advocates a 'propensity' view of objective probability. Coherently with the general attitude characterizing Suppes' philosophy of science, he gives representation theorems for such phenomena as radioactive decay, response strengths and coin tossing. In all such cases, plus a version of the three-body problem that exhibits what he calls a 'propensity for randomness', in order to obtain a representation of the phenomena at hand one has to give structural axioms, that are built on information which is not purely probabilistic in character. Representation of decay, for example, is obtained by means of a 'waiting-time axiom' which is a structural axiom that would never be encountered in the standard theory of subjective probability as a fundamental axiom. It is an axiom special to certain physical phenomena (Suppes, 1987a, p. 345).

Suppes' conclusion is that such an axiom represents "a qualitative expression of a propensity". Still discussing the same case, he remarks that the probabilities we obtain from the representation theorem are not unique but are only unique up to fixing the decay parameter. Again, this is not a subjective concept but very much an objective one (Suppes, 1987a, p. 346).

And once again he concludes that identifying and locating the number of physical parameters to be determined is a way of emphasizing that propensities have entered and that a purely probabilistic theory with a unique measure has not been given (Ibidem).

One can certainly agree that in the cases discussed by Suppes 'objective' considerations of some sort enter into the evaluation of probabilities.


The question, however, is whether an appeal to the notion of propensity is a fruitful move in this connection.

In my comments on Suppes' papers I raised some doubts, mainly due to the fact that his treatment of the matter seemed to overlook the wide debate on propensities going on in the literature, and to underestimate the implications, mainly with regard to indeterminism, that the notion of propensity brings with it. Instead of going back to such questions, I now wish to say that I am still dissatisfied with Suppes' treatment of the matter, but on slightly different grounds. I emphasize that we agree that a purely subjective notion of probability cannot account for all evaluations of probability, and some notion of 'objective chance' is required. His theorems are very useful in order to clarify at what point, in the representation of phenomena exhibiting probabilistic behaviour, considerations which are not purely subjective are to be encountered. However, I doubt that the notion of propensity, especially if taken in the vague meaning that Suppes attaches to it, can serve the purpose of providing a sound basis on which objective chance could be founded. In other words, Suppes' appeal to propensity does not seem to involve much more than the introduction of an extra term that adds nothing to the structural axioms.

In general, objective considerations in the evaluation of probabilities are essentially dictated by scientific theories. At least in this sense, then, the notion of 'objective chance' is linked to the view of scientific theories which is adopted. As Suppes pointed out - and I agree with him - it is a lacuna of de Finetti's position to overlook the role played by considerations which are not purely subjective in the evaluation of probabilities. The main reason why de Finetti rejected the notion of objective chance altogether comes from his general attitude to philosophy, an attitude Suppes calls 'positivistic', and I have called 'anti-realist'." In other words, de Finetti's main concern was to keep probability free from metaphysics. However, in the light of a relativistic and pragmatist view of scientific theories like that upheld by Suppes, admitting that certain elements, in situations characterized by completeness of information, can guide probability evaluations, does not mean opening the door to metaphysical ontology. 'Objective chance' can simply keep the pragmatist character pertaining to theories, and the only thing that matters is a clear distinction between 'objective' and subjective elements entering in the evaluation of probabilities. Incidentally, one might recall that the other founder of subjectivism, F. P. Ramsey,


admits of a notion of objective chance supported by a pragmatically oriented view of scientific theories which bears some analogy with Suppes' own position.

INDETERMINISM

Suppes puts forward a most interesting view of indeterminism, based on the notion of stability. Its starting point is instability in mechanics, which amounts to the consideration that wide divergence in the behaviour of two systems identical except for initial conditions is observed even when the initial conditions are extremely close (Suppes, 1991b, p. 9).

Unstable systems reflect a fundamental feature of indeterminism, namely unpredictability, for their future behaviour is not predictable on the basis of their present behaviour. According to this kind of analysis the notion of randomness, which represents the extreme case of unpredictability, is not to be seen as incompatible with determinism. On the basis of results taken from the theory of unstable dynamical systems, it can therefore be shown that there is no opposition between completely deterministic systems and random systems - and that moreover - the same phenomena can be both deterministic and random (Suppes, 1988c, p. 400).

In view of this, randomness can be seen as a "limiting case of unstable determinism" (Suppes, 1991 b, p. 17). The bridge between randomness and determinism is provided by instability, in the sense that random sequences can be generated by deterministic but very unstable systems of classical mechanics. Moreover, randomness is characterized by complexity, as random sequences "are the limiting case of increasingly complex deterministic sequences" (Ibidem). Suppes' conclusion is that talk in terms of stable or unstable systems should supersede the opposition between determinism and indeterminism, and that complexity, as referred to results, should be preferred to randomness, as referred to procedures.

If the preceding remarks seem to suggest that the choice between determinism and indeterminism is essentially a matter of taste, there is an additional aspect of the question to which Suppes draws attention. This amounts to the fact that, for those who accept the standard formulation of quantum mechanics, indeterminism looks much more plausible than instability, for


unstable deterministic mechanical systems cannot be construed to be consistent with standard quantum mechanics. The conclusion of this line of argument is that standard quantum mechanics is the most outstanding example of an intrinsically indeterministic theory (Suppes, 199 1 b, p. 16).

The question of the choice between indeterminism and instability, however, remains open, insofar as there are alternative theories to the standard formulation of quantum mechanics. In view of all this, Suppes maintains that a responsible philosophy of science should leave the door open to indeterminism. This is indeed the starting point of Sup- pes' 'new metaphysics'.

CAUSALITY

A further important feature of Suppes' perspective amounts to the conviction that there is no opposition between indeterminism, or the admission of the existence of randomness in nature, and causality. In his well-known monograph of 1970 A Probabilistic Theory of Causality Suppes advocates a probabilistic theory, that has become the object of much attention and debate. Its peculiar feature, that distinguishes it from other theories of probabilistic causality, like those put forward by I. J. Good, H. Reichenbach and W. C. Salmon, is to be identified with its general formulation, which is intended to make it applicable to the various contexts where causal speech occurs. l2 Suppes' notion of causality can be formulated both in terms of events and of random variables, and is compatible with different interpretations of probability. Remarkably, no 'ultimate genuine causes' are contemplated within this theory. On the contrary, the notion of cause, genuine or spurious, is strictly linked to the specification of the set of concepts on which the set of events that can serve as causes in a given context is to be defined. This is a point stressed by Suppes not only in his monograph of 1970, but also in more recent writings. For example, in his Self-projile he writes: I do think that the insistence on relativizing the analysis of cause to a particular conceptual framework is a point on which to make a stand (Suppes, 1979, p. 24).

His notion of probabilistic causality is then characterized as intrinsically relativistic. Conceived in this way, it testifies to Suppes' relativistic and pluralistic attitude towards philosophy and epistemology.

Much of the debate on Suppes' theory of causality revolves around the problem of accounting for what can be called 'surprising events',

258 MARIA CARLA GALAVOTI'I

relative to which causes behave as counteracting, or negatively relevant to the effect. The problem is strictly linked to that of the existence of two sorts of causal talk, one in terms of kinds of events, the other in terms of single events. In this respect, probabilistic causal talk simply reflects the distinction between different kinds of probability statements, that was recalled earlier. As previously observed, to such a distinction there corresponds a tension between explanation and prediction, which is even more evident in connection with causal analysis. In fact, according to a widespread opinion causality is strictly related to explanation, and an adequate theory of probabilistic causality should be able to account for single events as well as classes of them, even when such events are in fact unpredictable. While recognizing the relevance of this distinction, in his 1970 monograph Suppes holds that "the natural setting for extended scientific analysis of causal relations is provided by classes of events rather than individual events" (Suppes, 1970, p. 80). This conviction reflects an attitude that tends to give a privileged place to prediction over explanation, and does not take causal talk as necessarily connected with explanatory talk. This accounts for the fact that Suppes has never attempted to trace a systematic distinction between the two kinds of causal analysis. Also in this connection - as with regard to genuine causes - Suppes clarifies the kind of causal analysis adopted by reference to a detailed specification of the context.

The quest for a detailed causal analysis in terms of single events is strictly linked to the quest for a specification of ultimate genuine causes. Both are rooted in the conviction that causality is an intrinsically explanatory notion. This has inspired a number of attempts at providing the probabilistic notion of causality with some concept of homogeneity, so devised as to make the specification of causes depend on maximal specification of factors which are relevant to the effect. Proposals in this direction have been made by N. Cartwright, E. Eells, W. Spohn and many others, and are the object for much debate. The position taken by Suppes is utterly skeptical: "the search for homogeneity - he says - seems as quixotic and metaphysically mistaken as the search for ultimate causes" (Suppes, 1984a, p. 56). Also in this connection his answer points in the direction of a contextualization of the notion of cause, because it is only within a specific context that it can be decided at what point one can stop scrutinizing the data and take as exhaustive the information available at a certain time.


In the light of the endless discussions raised by the problems mentioned above, Suppes' relativism and pluralism with respect to the characterization of probabilistic causality, together with his general formulation of the notion of cause, reflect a very sensible position. Clearly, this should be taken more as a point of departure, than one of arrival. In other words, the distinction among different contexts where causal talk occurs should open the way to a detailed analysis of the specific features characterizing causality within such contexts. In his 1970 monograph Suppes indicates three main conceptual frameworks where causal statements are to be found. They are the following:

One conceptual framework is that provided by a particular scientific theory; the second is of the sort that arises in connection with a particular experiment or class of experiments; and the third is the most general framework expressing our beliefs with respect to all information available to us.I3

This tripartition calls for a more articulated analysis, aiming at a detailed specification of the various kinds of experiments performed, as well as the statistical techniques adopted in order to detect causality and the probability functions being used. Obviously, the degree of theoretical abstraction and sophistication attained by particular sciences making use of causal talk is crucial with respect to a characterization of the conceptual framework in which causality occurs.

As a final remark on causality, it might be observed that a view like Suppes', according to which there is no strict linkage between causality and explanation, could be fruitfully supplemented by a notion like manipulability. A suggestion to this effect comes from econometrics, where probabilistic causality has been dealt with in a highly original fashion. It combines a functionalist approach with a manipulative notion of causality, in the framework of a view of model building which bears strong analogies with Suppes', as pointed out before. Within such a perspective, causality is not seen as a property of explanatory accounts, being rather a sort of 'qualified predictability', pertaining to those models that can serve as a basis for practical intervention on some variables, which are precisely those taken as causal.14 The distinctive feature of a causal model lies in the manipulative character of its variables, as opposed to purely predictive models which forecast the future trend of some variable under study. A generalization of a view of this sort calls for a detailed analysis of the notion of experiment and its implications, with respect to various disciplines making use of causal talk. In this connection the distinction enters between experimental and

260 MARIA CARLA GALAVOITTI

non-experimental sciences, together with specification of the techniques adopted by specific disciplines. This looks like a promising direction in which Suppes' theory of probabilistic causality could be expanded.

RATIONALITY

The view of rationality emerging from Suppes' philosophy is accordingly local and pragmatic in character.'' In the first place he distinguishes between two aspects of rationality, to which there correspond two different approaches. The first is a 'dynamical' approach, dating back to Aristotle, according to which rational action is defined as performed in accordance with good reasons, or reasons tending to the achievement of some purpose. The second approach, called 'kinematic', applies to action choice. It is ruled by the overall principle of maximizing expected utility, characterizing the utilitarian tradition of Bentham and Bayes. This has been given great impulse by our century's Bayesianism.

The 'dynamical' approach is developed by Suppes in terms of justified procedures, based on the idea that good reasons should be provided in support of the procedures to be adopted in order to attain a given purpose. Since the very concept of 'procedure' is always referred to expected results, the model based on 'justified procedures' is to be seen as complementary with respect to the model of expected utility. Once again, instead of a contraposition we find in Suppes' thought a pluralistic attitude, aimed at combining different approaches within a composite perspective.

A similar spirit pervades Suppes' attitude towards the model of expected utility. His main point in this connection is that the 'classical ' Bayesian model should be implemented in such a way as to gain applicability in many practical situations, both in everyday life and scientific practice. Suppes reaffirms the importance of allowing for estimates in terms of 'interval' probability values, and takes intuition, intention and individual judgment as crucial elements to be included in a more elastic model of Bayesian rationality. Since such things as intuition and judgment also play a central role in view of the choice of the procedures directed to a given end; they provide within Suppes' perspective a bridge between the two components of rationality. In the resulting view quantitative and qualitative elements, intuitive judgment


and deliberation, individual beliefs and objective evaluations are all admitted and complement each other.

This kind of approach has the advantage of allowing for a probabilistic treatment of notions that are not to be dealt with in the framework of the 'classical' Bayesian model, like those of obligation and free wi11.16 As a counterpart, Suppes asks for a renunciation of a general theory of rationality, and for the acceptance in its place of a local view, resulting from theoretical as well as practical elements, within a mixture that is not to be fixed once and for all.

The use of modern quantitative methods of decision making - Suppes says - is necessarily limited but powerful when properly applied. The role of judgment and practical wisdom in applying these methods will continue to be of central importance. The tension between calculation, qualitative justified procedures, and judgment will not disappear (Suppes, 1984a, p. 22 1).

CONCLUDING REMARKS

The kind of philosophy Suppes offers us is not easy to locate within the framework of contemporary epistemology. Instead of the many 'isms' - determinism, realism, individualism, and the like - which are the object of so much debate, we find in Suppes' work an investigation into science from inside. Such investigation, conducted in a pragmatic and empirical spirit, does not lead to a general philosophical view of science and reality. For Suppes philosophy reflects the local character of science, its problems are dealt with in the framework of a specific context, and so are the proposed solutions.

This approach is not bound to give us a comprehensive theory of the kind logical positivists tried to work out. What it gives us is instead a method for dissecting notions and problems in a way that makes it possible to distinguish between their various components, and allows identification of structure at different levels of abstraction. This opens the way to a representation of phenomena in which both theories and data, abstract mathematical construals and particular experimental techniques, are all given attention and find their place. The pluralistic and local character of Suppes' approach might look like a sign of weakness to those who do not wish to abandon the neopositivistic ideals of science and rationality. However, the force of Suppes' position lies in its constructive attitude towards epistemological problems. This certainly

262 MARIA CARLA GALAV07TI

answers that urge to understand human knowledge in general, and science in particular, that logical empiricist philosophers felt so strongly.

Dipartimento di Filosojia, via Zamboni, 38, 401 26 Bologna, Italy

NOTES

' See for example Suppe (Ed.) (1977), and Moulines and Sneed (1979). See his 'Self-profile' (Suppes, 1979, pp. 46-47), for some remarks on this 'slogan' as

Suppes calls it. See for example Spanos (1986). See Suppes (1981a) and (1984a). See Suppes (1979). See Suppes (1984a, pp. 85ff.) ' See Suppes and Zanotti (1977) and (1989).

See Suppes (1966a). see Suppes (1966b).

"' See Suppes (1987a, 1987b) and Galavotti (1987). " See Galavotti (1989). l 2 For a review of the debate on probabilistic causality, focusing on some of the topics discussed in the following pages see Galavotti (1991). l 3 Suppes (1970, p. 13). This point is also discussed at length in Suppes (1984b). l 4 For a discussion of the epistemological relevance of the notion of causality developed by econometricians see Galavotti and Gambetta (1990) and Galavotti (1990). l 5 Suppes' views on rationality are outlined in (198 la, 198 lc, 1984a). l6 See Suppes (1973).

REFERENCES

Bogdan, R. J. (Ed.): 1979, Patrick Suppes, Dordrecht: Reidel. Galavotti, M. C.: 1987, 'Comments on Patrick Suppes "Propensity Representations of

Probability"', Erkenntnis, 26, 359-368. Galavotti, M. C.: 1989, 'Anti-Realism in the Philosophy of Probability: Bruno de

Finetti's Subjectivism', Erkenntnis, 31, 239-261. Galavotti, M. C.: 1990, 'Explanation and Causality: Some Suggestions from Econo-

metrics', Topoi, 9, 161-1 69. Galavotti, M. C.: 1991, 'Probability and Causality', in: Atti del Congresso 'Nuovi

problemi della logica e della Jilosojia della scienza', Vol. I, Bologna: CLUEB, pp. 69-82.


Galavotti, M. C. and Gambetta, G.: 1990, 'Causality and Exogeneity in Econometric Models', in: R. Cooke and D. Costantini (Eds.), Statistics in Science, Dordrecht: Kluwer, pp. 27-40.

Hintikka, J. and Suppes, P. (Eds.): 1966, Aspects of Inductive Logic, Amsterdam: North-Holland.

Moulines, C.-U. and Sneed, J.: 1979, 'Suppes' Philosophy of Physics', in: R. Bogdan (Ed.), pp. 59-9 1.

Spanos, A.: 1986, Statistical Foundations of Econometric Modelling, Cambridge: Cambridge University Press.

Suppe, F. (Ed.): 1979, The Structure of Scienti$c Theories, 2nd edition, Urbana: Uni- versity of Illinois Press.

Suppes, P.: 1962, 'Models of Data', in: E. Nagel, P. Suppes, and A. Tarski (Eds.), Logic, Methodology and Philosophy of Science, Stanford: Stanford University Press, pp. 252-261.

Suppes, P.: 1966a, 'A Bayesian Approach to the Paradoxes of Confirmation', in: J. Hintikka and P. Suppes (Eds.), pp. 198-207.

Suppes, P.: 1966b, 'Probabilistic Inference and the Concept of Total Evidence', in: J. Hintikka and P. Suppes (Eds.), pp. 49-65.

Suppes, P.: 1967, 'What Is a Scientific Theory?', in: S. Morgenbesser (Ed.), Philosophy of Science Today, New York: Basic Books, pp. 55-67.

Suppes, P.: 1970, Probabilistic Theory of Causality, Amsterdam: North-Holland. Suppes, P.: 1973, 'The Concept of Obligation in the Context of Decision Theory', in:

P. Suppes, L. Henkin, G. C. Moisil, and A. Joja (Eds.), Logic, Methodology and Philosophy of Science IV, Amsterdam: North-Holland, pp. 5 15-529.

Suppes, P.: 1979, 'Self-profile', in: R. Bogdan (Ed.), pp. 3-56. Suppes, P.: 1980, 'Probabilistic Empiricism and Rationality', in: R. Hilpinen (Ed.),

Rationality in Science, Dordrecht: Reidel, pp. 17 1-1 90. Suppes, P.: 1981a, La logique du probable, Paris: Flammarion. Suppes, P. : 198 1 b, 'The Plurality of Science', in: P. D. Asquith and I. Hacking (Eds.),

PSA 1978, Vol. 11, East Lansing: Philosophy of Science Association, pp. 3-16. Suppes, P.: 198 Ic, 'The Limits of Rationality', Grazerphilosophischen Studien, 12/13,

85-101. Suppes, P.: 1983, 'The Meaning of Probability Statements', Erkenntnis, 19, 397-403. Suppes, P.: 1984a, Probabilistic Metaphysics, Oxford: Blackwell. Suppes, P.: 1984b, 'Conflicting Intuitions about Causality', in: P. Trench, T. Yuehling,

and H. Wettstein (Eds.), Midwest Studies in Philosophy, Causation and Causal Theories, Vol. IX, pp. 151-168.

Suppes, P.: 1987a, 'Propensity Representations of Probability', Erkenntnis, 26, 335- 358.

Suppes, P.: 1987b, 'Some Further Remarks on Propensity: Reply to Maria Carla Galavotti', Erkenntnis, 26, 369-376.

Suppes, P.: 1988a, 'Representation Theory and the Analysis of Structure', Philosophia Naturalis, 25, 254-268.

Suppes, P.: 1988b, 'Empirical Structures', in: E. Scheibe (Ed.), The Role of Experience in Science, Berlin-New York: Walter de Gruyter, pp. 23-33.

Suppes, P.: 1988c, 'Comment: Causality, Complexity and Determinism', Statistical Science, 3, 398-400.


Suppes, P.: 199 1 a, 'Indeterminism or Instability, Does It Matter?', in: G. G. Brittan, Jr. (Ed.), Causality, Method and Modality, Dordrecht: Kluwer, pp. 5-22.

Suppes, P.: 1991b, 'Can Psychological Software Be Reduced to Physiological Hard- ware?', in: E. Agazzi (Ed.), The Problem of Reduction in Science, Dordrecht: Kluwer, pp. 183-198.

Suppes, P. and Zanotti, M.: 1977, 'On Using Random Relations to Generate Upper and Lower Probabilities', Synthese, 36, 427440.

Suppes, P. and Zanotti, M.: 1989, 'Conditions on Upper and Lower Probabilities to Imply Probabilities', Erkenntnis, 31, 323-345.


Maria Carla Galavotti gives a detailed and sympathetic overview of my philosophy of science. Most of what she says I find myself in agreement with. On a few points there is a divergence in our views but probably more important are the questions she raises about positions I have not developed in the detailed way that is needed. The two large topics raised by Carla that I want to explore more carefully are, first, the relationship between causality and experiments, and second, my nonfoundational problem-solving approach to science and the philosophy of science. I follow the discussion of these subjects with some more particular comments.

Causality and Experiments. Carla remarks that already in my 1970 monograph A Probabilistic Theory of Causality I indicated that there were three main conceptual frameworks where causal statements were to be found. These frameworks (i) were provided by a scientific theory, (ii) arose in connection with particular experiments, and (iii) were concerned with our expression of beliefs. She rightly remarks that the topic is not developed as thoroughly as it needs to be, for much more needs to be said about the various kinds of experiments and how they relate to causality. I emphasize of course it is possible to do experiments that do not directly bear on causality. Kinematical experiments in physics are common and in other subjects as well, but still the great bulk of experimentation is aimed at causal questions. This means that the theory of experimentation is itself an important part of the theory of causality. The following classification is not meant to be definitive or sufficiently detailed. I also amplify it in remarks on other papers in these volumes, but it is 1 think at least a beginning.


Experiments with Control Groups. In much of empirical science the image of experimentation, at least in the social sciences and in medicine, is that of the experiment in which there are at least two groups, one the experimental group and the other a control group. In a typical medical experiment, the experimental group will be given a certain treatment and the control group either no treatment or a placebo. Often in such experimentation there is little if any underlying scientific theory, except for the theory provided by the statistical theory of experimental design and the statistical theory of evaluation of the experimental results. This remark is not meant to denigrate the importance of such experiments. Major findings using this methodology can be found in many disciplines. It is only in this century that the theory of experiments with control groups has been put on a sound statistical basis. That statistical theory plays a very important role of offering a generalized theory of experimentation when there is little else in the way of scientific theory to guide how the experiments are conducted. The pioneering work of Ronald Fisher on the design of experiments of this kind is one of the great landmarks of twentieth century intellectual thought.

Experiments Testing a Theoretical Model. Most of the great experimental triumphs in science - experiments that are historical events to be discussed for many years afterwards - are of this type. Here the experimental setup is very different from that of the first type. There is ordinarily no control group from a statistical or methodological standpoint. There is a completely different objective from that of the Fisherian control-group experiment. The object is not to reject the null hypothesis, but to have data that enable a scientist to accept the null hypothesis, where the null hypothesis now is that the theory and the experiment yield the same predictions. Wonderful experimental examples of this kind are easy to enumerate. A celebrated but less prominent one was Poisson's realization that Fresnel's theory of diffraction implied that if a small disk is held in a light emanating from essentially a point source, the center of the disks's shadow will be bright, in fact just about as bright as if no disk had been placed in the light's path. Poisson thought this was a clear refutation of Fresnel's theory. An experiment was performed confirming Poisson's prediction, contrary to his expectation.

There is another point of interesting comparison with the first class. In the case of testing a theoretical model, from a general methodological


standpoint it is felt to be most desirable to have a competing theory or competing theoretical model with which to compare the prediction of the same experimental facts. In this case rather than an experimental group and control group, there is, so to speak, a main theoretical model and an alternative theoretical model. The aim of refined statistical analysis is to compare the goodness of fit of these two models. It is a common piece of heuristic statistical advice that it is better to have almost any alternative hypothesis or model rather than none. The insight we get from comparison of two theoretical models is almost without exception superior to simply looking at the comparison of a single theoretical model and experimental data. In my own scientific experience one of the episodes I enjoyed the most was the concentration in the 1960s on the comparison of all-or-none versus incremental learning. In this case there were two very sharply defined theoretical ideas: whether the learning occurs incrementally, as suggested by the description, or in a single step, as described by the phrase 'all or none'. The existence of these two competing theoretical models spurred not only experimentation, but also much more detailed examination of experimental data than would often have been the case. My own contribution to this controversy, in the context of concept learning, is set forth in detail in Suppes and Ginsberg (1963).

Measurement Experiments. A third class of experiments is concerned with the accurate measurement of some important constant or physical scale. Great examples are to be found in the long history of the measurement of the velocity of light. There is of course an overlap with the second class of experiments because in many tests of theoretical models a certain number of free parameters must be estimated from the data, but in these cases the real objective is the test of the theoretical model, not the estimation of the particular parameters whose exact values may not be considered of fundamental importance. It is quite different in the case of measurement experiments aimed at the value of an empirical constant that itself plays independently a role in theory. Volume I of Foundations of Measurement (pp. 539-544) contains a six-page table listing various physical quantities and their dimensions which one way or another we want to measure with detailed experiments when possible. The experiments devoted to the physical quantities listed in this table fill hundreds of pages in experimental journals over the past century.


The relation of the first two classes of experiments to causal ideas is clear enough. For the case of the first class, the rejection of the null hypothesis leads to the conclusion that the treatment had a causal effect. In the second class the notion of causality must be abstracted from the theoretical model but in most cases what is intuitively meant to be causal in the theory is fairly evident. In the third case, matters are quite different. The experiments often are themselves not causal in character at all, but involve detailed measurement procedures which may themselves entail causal ideas but, more importantly, the exact measurement of various empirical quantities is the key to the detailed testing of causal models, and so measurement experiments bear upon causality in a significant even if indirect way.

Observational Experiments. Contrary to some thinking about experiments, I want to classify as experiments those investigations which are observational in nature, in the sense that the initial and boundary conditions of the system observed are disturbed as little as possible by the experimenter and are not created artificially for the purpose of the experiment. Characteristic examples would be meteorological experiments to measure thermal and turbulent conditions in clouds, astronomical experiments to observe particular features of the light from distant stars, or radio astronomy experiments aimed at similar observations about radio waves from distant objects.

It is held by some stout-hearted statisticians that we can really not make causal inferences when we cannot manipulate the experimental conditions and thus we cannot really make good causal inferences in the case of observational experiments. I believe in a more robust concept of causality. It is essential to our ordinary and deeply entrenched methods for dealing with the everyday world to be able to make such causal inferences as the following: 'The storm outside is now causing the flood in my basement'. Certainly it can be the case that we have to take stronger precautions about the correctness of our causal inferences in the case of observational experiments. It is also evident that there is a strong overlap between measurement experiments and observational experiments, and for that matter, observational experiments and experiments testing theoretical models, but in the second class of experiment I had especially in mind when controlled conditions of experimentation are deliberately created.


Science as Pluralistic Problem Solving. Galavotti catches very well in the last part of her paper, as well as in the earlier remarks on pragmatism, the skepticism with which I greet overarching philosophical theories of science. Contrary to something I might have held to at the beginning of my career, physics or psychology, for example, are not to be organized as one grand set of axiomatic systems whose interrelations are carefully dovetailed and extremely well developed. This may be possible for mathematics but it is certainly, I now think, a mistaken ideal for physics. I have come to see that physics is a science of clever problem solving, and this is true of other areas of science as well. Good physicists are past masters at knowing just what particular physical assumptions are to be made in analyzing a particular problem, something most mathematicians are very bad at in spite of the great confidence some of them show at their mastery of fundamental physical theory.

There are of course individual foundational or philosophical questions of great conceptual interest. What does not exist is an overarching philosophical foundation for the enterprise of science in the many different forms it now takes. There is, as Carla emphasizes, in my view no unity of subject matter, method or language in science, and consequently no philosophical view that will encompass in any nonsuperficial way the great plurality of scientific activity.

Now for some particular comments on Galavotti's analysis of my philosophy of science.

Probability of Single Events. In her discussion of Bayesianism, in my judgment she makes too much of the special circumstances surrounding the occurrence of single events. She says that "when dealing with single events characterized by probabilistic behavior we can only (if ever) reconstruct the history behind their occurrence after they have taken place." But it seems to me there are obvious counterexamples to this claim. One of the best would be the modern intense observation of the formation and movement of hurricanes. Long before a hurricane hits the shore to create a disastrous single event, we know a great deal about the history of the hurricane. In no sense are meteorologists committed only to analysis after the event. They continually make predictions about the future behavior of the hurricane, including the prediction of the potentially disastrous single event of its moving onshore in a populated area.


So I am less inclined than Galavotti to stress the difference between probability statements about single events and probability statements about properties of objects.

Propensities. This disagreement carries over to our running dialogue and intellectual differences about propensities. This is not the place to enter into that debate once again in full detail, but I think that Galavotti is correct that much of what I said in the past is still not sufficiently detailed about the nature of propensities. More needs to be said. One way of saying more is to take the view of probabilities as fitting within deterministic systems. In this case, we can look at properties of the deterministic system that are not in themselves probabilistic properties, but that can be related, even causally if one desires, to the generation of probabilities. Example of such concepts would be the case of the generation of randomness in the motion of the three bodies. This is discussed in my paper on propensities, which Galavotti criticized and refers to (Suppes, 1987a); in regions of unstable behavior randomness is generated. It is a propensity of instability in this case to generate random behavior. In other cases, for example the classical coin tossing cases also discussed by me in the same paper, the propensity of the system that generates probability is slight variations in initial conditions together with the symmetry of the physical object, for example, the coin being tossed. Neither the variation in initial conditions nor the symmetry of the object is inherently probabilistic in character, but their joint presence can cause the generation of probabilistic sequences. Almost certainly Galavotti will not be satisfied with these further statements. It is a topic I hope we will be able to pursue in more detail on another occasion.

Determinism or Indeterminism. Concerning the discussion of instability at the end of her section on objective probability, she quotes a statement of mine from Suppes 1991b on quantum mechanics as "the most outstanding example of an intrinsically indeterministic theory." In the spirit of Suppes 1991a and especially in view of a recent article of mine emphasizing the transcendental character of determinism (1 993) I would now want to say that quantum mechanics is equally congenial to determinism or indeterminism, and the choice of an overarching view of how to think about the world is transcendental in character. This

270 MARIA CARLA GALAVOIT1

point is not really a difference between Galavotti and me, but a point of clarification as she puts it that "Suppes maintains that a responsible philosophy of science should leave the door open to indeterminism." I certainly agree with this and now go on to say that necessarily that door is open because of the transcendental character of determinism.

It is a good thing that Carla and I continue to have rather strong dis- agreements about propensity, otherwise our philosophical conversations might become too irenic and congenial.

REFERENCES

Krantz, D., Luce, D., Suppes, P., and Tversky, A.: 1971, Foundations of Measurement, Volume I, New York: Academic Press .

Suppes, P.: 1987a, 'Propensity Representations of Probability', Erkenntnis, 26, 335- 358.

Suppes, P.: 1991a, 'Indeterminism or Instability, Does It Matter?', in: G. G. Brittan, Jr. (Ed.), Causality, Method and Modality, Dordrecht: Kluwer, pp. 5-22.

Suppes, P.: 1991b, 'Can Psychological Software Be Reduced to Physiological Hard- ware?', in: E. Agazzi (Ed.), The Problem of Reduction in Science, Dordrecht: Kluwer, pp. 183-198.

Suppes, P.: 1993, 'The Transcendental Character of Determinism', Midwest Studies in Philosophy, 18, 242-257.

Suppes, P. and Ginsberg, R.: 1963, 'A Fundamental Property of All-or-None Models, Binomial Distribution of Responses Prior to Conditioning, with Application to Concept Formation in Children', Ps~chological Review, 70, 139-161.

EPILOGUE

PATRICK SUPPES

POSTSCRIPT

Writing these comments has been very much like having extended conversations with the various authors of the papers, most of whom are colleagues that I have known for a long time and many of whom were at one time graduate students at Stanford. First names are often used, as they would be in a conversation, and the discussion of even rather technical matters is left at an informal level, with just a few exceptions.

It is also characteristic of many of the papers that they set forth technical contributions to topics in which I have been interested. The papers are not really conducting an argument of an extended sort with my own views. Consequently, in many cases what I have to say is less argu- mentative and dialectical than is characteristic of much philosophical discourse. On the other hand, I do not think that I am known as being reluctant to engage in argument. In fact I rather like as much intensity as is compatible with not taking sharp points personally. In any case, a quick perusal will make it evident that many of my comments are expansions of points, not anything like expressions of disagreement. So, my aim has been a congenial spirit of conversation, which I know I have not succeeded in maintaining at a uniform level, but that is hardly to be expected.

In any case, I have enjoyed reading all of these papers on so many different subjects, and even more the opportunity to write about a variety of philosophical and scientific issues that matter to me, especially when raised by old friends.

Finally, my special thanks to two persons, to Maria Carla Galavotti for organizing the Venice Conference on my work in 1992 - a number of the papers in these two volumes were first presented there - and Paul Humphreys for his work in organizing and editing these volumes.

l? Humphreys (ed.), Patrick Suppes: Scientific Philosopher; Vol. 3, 273. @ 1994 Kluwer Academic Publislzers. Printed in the Netherlands.

BIBLIOGRAPHY OF PATRICK SUPPES


ABSTRACT. Each publication has been placed in one of five categories. Methodology, Probability and Measurement is the first, and the most inclusive. Any general publications on the philosophy of science are included here as well as those specifically on the foundations of probability, measurement or decision theory. Second is Psychology which includes both philosophical and scientific work. Physics is the third category and contains publications in the foundations of physics. Language and Logic includes any work on logic and the foundations of mathematics, as well as the philosophy of language. Some experimental studies of language are also included; they could just as easily have been classified under Psychology. Computers and Education is something of a catchall, since it includes articles on robotics which are not relevant to education, and some publications on education not relevant to computers, but most of the publications listed are about the use of computers in education.

METHODOLOGY, PROBABILITY AND MEASUREMENT

A set of independent axioms for extensive quantities. Portugaliae Mathematica, 195 1,10, 163-1 72.

PHYSICS

With J. C. C. MCKINSEY and A. C. SUGAR. Axiomatic foundations of classical particle mechanics. Journal of Rational Mechanics and Analysis, 2, 253-272. Spanish translation by A. G. de la Sienra: Fun- damentos axiomaticos para la mecinica de purtl'culas clisica. Mexico: Universidad Michoacana de San Nicolas de Hidalgo, 1978.

With J. C. C. MCKINSEY. Transformations of systems of classical particle mechanics. Journal of Rational Mechanics and Analysis, 2, 273-289.

P Humphreys (ed.), Patrick Suppes: Scientijic Philosopher, Vol. 3, 275-323. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

276 BIBLIOGRAPHY OF PATRICK SUPPES

With J. C. C. MCKINSEY. Philosophy and the axiomatic foundations of physics. Proceedings of the Eleventh International Congress of Philos- ophy, 6,49-54.


Some remarks on problems and methods in the philosophy of science. Philosophy of Science, 21,242-248.

PHYSICS

Descartes and the problem of action at a distance. Journal of the History of Ideas, 15, 146-152. Reprinted in Georges J. D. Moyal (Ed.) Rene Descartes: Critical Assessments. London: Routledge, 1992, 8 1-88.

With H. RUBIN. Transformations of systems of relativistic particle mechanics. PaciJic Journal of Mathematics, 4,563-601.


With D. DAVIDSON and J. C. C. MCKINSEY. Outlines of a formal theory of value, I. Philosophy of Science, 22, 140-160.

With H. RUBIN. A note on two-place predicates and fitting sequences of measure functions. Journal of Symbolic Logic, 20, 121-122.

With M. WINET. An axiomatization of utility based on the notion of utility differences. Journal of Management Science, 1,259-270. Reprinted in A. F. Veinott, Jr. (Ed.), Mathematical Studies in Management Science. Stanford: Stanford University Press, 1965,284-295.

PHYSICS

With J. C. C. MCKINSEY. On the notion of invariance in classical mechanics. British Journal for Philosophy of Science, 5, 290-302.


1956


Nelson Goodman on the concept of logical simplicity. Philosophy of Science, 23, 153-1 59.

The role of subjective probability and utility in decision-making. Pro- ceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1 955,5,61-73.

With D. DAVIDSON. A finitistic axiomatization of subjective probability and utility. Econometrics, 24, 264-275.


With D. DAVIDSON and S. SIEGEL. Decision Making: An experimental approach. Stanford, CA: Stanford University Press, 121 pp. Reprinted as Midway Reprint, 1977, Chicago: University of Chicago Press.

LANGUAGE AND LOGIC

Introduction to Logic. New York: Van Nostrand, 312 pp. Spanish translation by G. A. Carrasco: Introduccio'n a la lo'gica simbdlica. Mkxico: Compania Editorial Continental, SA., 1966, 378 pp. Chinese translation by Fu-Tseng Liu. Buffalo Book Co. Ltd., Taiwan, R.O.C.


With D. S C O ~ . Foundational aspects of theories of measurement. Journal of Symbolic Logic, 23, 1 13-128.


PSYCHOLOGY

With R. C. ATKINSON. An analysis of two-person game situations in terms of statistical leaming theory. Journal of Experimental Psycholo- gy, 55,369-378.


With H. L. ROYDEN and K. WALSH. A model for the experimental measurement of the utility of gambling. Behavioral Science, 4, 1 1-1 8.

With K. WALSH. A non-linear model for the experimental measurement of utility. Behavioral Science, 4, 204-21 1.

With J. LAMPERTI. Chains of infinite order and their application to leaming theory. Pacijic Journal of Mathematics, 9,739-754.

Measurement, empirical meaningfulness and three-valued logic. In C. W. Churchman and P. Ratoosh (Eds.), Measurement: DeJinitions and theories. New York: Wiley, 129-1 43.

PSYCHOLOGY

With R. C. ATKINSON. Applications of a Markov model to two-person noncooperative games. In R. R. Bush and W. K. Estes (Eds.), Studies in Mathematical Learning Theory. Stanford: Stanford University Press, 65-75.

With W. K. ESTES. Foundations of linear models. In R. R. Bush and W. K. Estes (Eds.), Studies in Mathematical Learning Theory. Stanford: Stanford University Press, 137-1 79.

A linear model for a continuum of responses. In R. R. Bush and W. K. Estes (Eds.), Studies in Mathematical Learning Theory. Stanford: Stanford University Press, 400414.

Stimulus sampling theory for a continuum of responses. In K. J. Arrow, S. Karlin, and P. Suppes (Eds.), Mathematical Methods in the Social Sciences, Stanford: Stanford University Press, 348-365.


PHYSICS

Axioms for relativistic kinematics with or without parity. In L. Henkin, P. Suppes, and A. Tarski (Eds.), The Axiomatic Method with Special Reference to Geometry and Physics. Proceedings of an international symposium held at the University of California, Berkeley, December 16, 1957-January 4, 1958. Amsterdam: North-Holland, 291-307.

COMPUTERS AND EDUCATION

With N. HAWLEY. Geometry in the first grade. American Mathematical Monthly, 66, 505-506.


Some open problems in the foundations of subjective probability. In R. E. Macho1 (Ed.), Information and Decision Processes. New York: McGraw-Hill, 162-1 69.

With J. LAMPERTI. Some asymptotic properties of Luce's beta learning model. Psychometrika, 25, 233-241.

PSYCHOLOGY

With R. C. ATKINSON. Markov Learning Models for Multiperson Inter- actions. Stanford: Stanford university Press, 296 pp.

LANGUAGE AND LOGIC

Axiomatic set theory. New York: Van Nostrand, 265 pp. Spanish translation by H. A. Castillo, Teorfa Axiomdtica de Conjuntos. Cali, Colombia: Editoriol Norma, 1968, 17 1 pp. Slightly revised edition published by Dover, New York, 1972, 267 pp.

Problem analysis and ordinary language. Proceedings of the Twelfth


International Congress of Philosophy, 4, 33 1-337.

A comparison of the meaning and uses of models in mathematlcs and the empirical sciences. Synthese, 12, 287-301. Czechoslovakian translation: Srovnini vyznamu a pouziti modelu v matematice a v empirickych vedich. In K. Berka and L. Tondl (Eds.), Teorie modelu a modelovdnl: Dordrecht: Reidel, 1967,208-222.


With N. HAWLEY. Geometry for Primary Grades. Book 1. San Fran- cisco: Holden-Day, 127 pp. Spanish translation: Geometria para 10s Grados Primarios. Libra 1. San Juan, Puerto Rico: Editorial Depar- tamento de Instruccion Publica, 1964. 126 pp. French translation: Geometrie pour Classes Elementaires. Livre 1. Montreal, Canada: Gontran Trottier, 1965.

With N. HAWLEY. Geometry for Primary Grades. Book 2. San Fran- cisco: Holden-Day, 126 pp. Spanish translation: Geometria para 10s Grados Primarios. Libra 2. San Juan, Puerto Rico: Editorial Departa- mento de Instruccion Publica, 1966, 126 pp.


Behavioristic foundations of utility. Econometrics, 29, 186-202.

The philosophical relevance of decision theory. Journal of Philosophy, 58,605-614.

PSYCHOLOGY

With F. KRASNE. Applications of stimulus sampling theory to situations involving social pressure. Psychological Review, 68,46-59.

With R. W. FRANKMANN. Test of stimulus sampling theory for a continuum of responses with unimodal noncontingent determinate reinforcement. Journal of Experimental Psychology, 61, 122-132.

BIBLIOGRAPHY OF PATRICK SUPPES 28 1

With J. ZINNES. Stochastic learning theories for a response continuum with non-determinate reinforcement. Psychometrika, 26, 373-390.

PHYSICS

Probability concepts in quantum mechanics. Philosophy of Science, 28, 378-389.


With B. MCKNIGHT. Sets and numbers in grade one, 1959-1 960. The Arithmetic Teacher, 8, 287-290.


Models of data. In E. Nagel, P. Suppes, and A. Tarski (Eds.), Logic, Methodology and Philosophy of Science: Proceedings of the 1960 Inter- national Congress. Stanford: Stanford university Press, 252-261. Czechoslovakian translation: Modely dat. In K. Berka and L. Tondl (Eds.), Teorie modelu a modelovani. Dordrecht: Reidel, 1967, 223- 235. German translation: Modelle von Daten. In M. Balzer and W. Heidelberger (Eds.), Zur Logik empirischer Theorien. Berlin: Walter de Gruyter, 1983, 191-204.

Recent developments in utility theory. Recent Advancements in Game Theory, Princeton University Conference, 61-72.

PSYCHOLOGY

With J. M. CARLSMITH. Experimental analysis of a duopoly situation from the standpoint of mathematical learning theory. International Eco- nomic Review, 3, 60-78.

With M. SCHLAG-REY. Test of some learning models for double contingent reinforcement. ~ s ~ c h o l o ~ i c a ~ Reports, 10, 259-268.


With R. GINSBERG. Application of a stimulus sampling model to children's concept formation with and without overt correction responses. Journal of Experimental Psychology, 63, 330-336.

With R. GINSBERG. Experimental studies of mathematical concept formation in young children. Science Education, 46,230-240.

With M. SCHLAG-REY. Analysis of social conformity in terms of generalized conditioning models. In J. H. Criswell, H. Solomon, and P. Suppes (Eds.), Mathematical Methods in Small Group Processes. Stan- ford, CA: Stanford University Press, 334-361.


With S. HILL. The concept of set. The Grade Teacher, 79,51,86-90.

Mathematical logic for the schools. The Arithmetic Teacher, 9, 396- 399. Reprinted in J. J. Gallagher (Ed.), Teaching Gifted Students: A book of readings. Boston: Allyn & Bacon, 1965.


With J. L. ZINNES. Basic measurement theory. In R. D. Luce, R. R. Bush, and E. H. Galanter (Eds.), Handbook of Mathematical Psycholo- gy, Vol. I. New York: Wiley, 3-76. Russian translation: Psychologich- eski izmerenia. In L. D. Meshalki (Ed.), Matematica. Moscow: Mir, 1967.

PSYCHOLOGY

With R. GINSBERG. A fundamental property of all-or-none models, binomial distribution of responses prior to conditioning, with application to concept formation in children. Psychological Review, 70, 139-161.


PHYSICS

The role of probability in quantum mechanics. In B. Baumrin (Ed.), Philosophy of Science: The Delaware seminar. New York: Wiley, 319-337.


Set theory in the primary grades. New York State Mathematics Teacher S Journal, 13,46-53. Reprinted in J. J. Gallagher (Ed.), Teaching Gifted Students: A book of readings. Boston: Allyn & Bacon, 1965.

PSYCHOLOGY

Some current developments in models of learning for a continuum of responses. The Institute of Electrical and Electronics Engineers Trans- actions on Applications and Industry, 83,297-305.

With H. ROUANET. A simple discrimination experiment with a continuum of responses. In R. C. Atkinson (Ed.), Studies in Mathematical Psychology. Stanford, CA: Stanford University Press, 3 17-357.

With H. ROUANET, M. LEVINE, and R. W. FRANKMANN. Empirical comparison of models for a continuum of responses with noncontingent bimodal reinforcement. In R. C. Atkinson (Ed.), Studies in Mathemati- cal Psychology. Stanford, CA: Stanford University Press, 358-379.

Problems of optimization in learning a list of simple items. In M. W. Shelly, 11, and G. L. Bryan (Eds.), Human Judgments and Optimality. New York: Wiley, 1 1 6-1 26.

On an example of unpredictability in human behavior. Philosophy of Science, 31, 143-148.

With E. KARSH. Probability learning of rats in continuous-time experiments. Psychonomic Science, 1,36 1-362.


LANGUAGE AND LOGIC

With S. HILL. First Course in Mathematical Logic. New York: Blais- dell, 274 pp. Spanish translation by E. L. Escardo, Introduccidn a La Ldgica Matematica. Barcelona: Editorial, S. A., 1968,283 pp.

With E. CROTHERS and R. WEIR. Latency phenomena in prolonged learning of visual representations of Russian sounds. International Review of Applied Linguistics, 2, 205-21 7.


The ability of elementary-school children to learn the new mathematics. Theory into Practice, 3, 57-6 1.

Modern learning theory and the elementary-school curriculum. Ameri- can Educational Research Journal, 1,79-93. Reprinted in H. C. Lind- gren (Ed.), Readings in Educational Psychology. New York: Wiley, 207-222. Reprinted also in R. Ripple (Ed.), Readings in Learning and Human Abilities. New York: Harper & Row, 197 1. Reprinted also in H. C. Lindgren and F. Lindgren (Eds.), Current Readings in Educational Psychology, 2nd edition. New York: Wiley, 197 1,216-230. Reprinted also in the Bobbs-Merrill Reprint Series in Psychology, P-810, Prod. No. 69065. Japanese translation in W. H. Holtzman (Ed.), Computer- assisted Instruction, Testing, and Guidance. New York: Harper & Row, 1970.

The formation of mathematical concepts in primary-grade children. In A. H. Passow and R. R. Leeper (Eds.), Papers from the ASCD Eighth Curriculum Research Institute, 99-1 19.


With R. D. LUCE. Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, and E. H. Galanter (Eds.), Handbook of Mathematical Psychology, Vol. 3. New York: Wiley, 249-410.


PSYCHOLOGY

With M. SCHLAG-REY and G. GROEN. Latencies on last error in paired- associate learning. Psychonomic Science, 2, 1 5-1 6.

With M. SCHLAG-REY. Observable changes of hypotheses under positive reinforcement. Science, 148, 661-662.

Towards a behavioral foundation of mathematical proofs. In K. Ajdukiewicz (Ed.), The Foundations of Statements and Decisions. Pro- ceedings of the International Colloquium on Methodology of Science, September 18-23, 1961. Warsaw: PWN - Polish Scientific Publishers, 327-34 1.

On the behavioral foundations of mathematical concepts. Monographs of the Society for Research in Child Development, 30, 60-96.

The kinematics and dynamics of concept formation. In Y. Bar-Hillel (Ed.), Proceedings for the 1964 International Congress for Logic, Methodology and Philosophy of Science. Amsterdam: North-Holland, 405-414.

LANGUAGE AND LOGIC

Logics appropriate to empirical theories. In J. W. Addison, L. Henkin, and A. Tarski (Eds.), Theory of Models. Amsterdam: North-Holland, 364-375. Reprinted in C. A. Hooker (Ed.), The Logico-algebraic Approach to Quantum Mechanics. Dordrecht: Reidel, 1975, 329-340. Rumanian translation in Gh. Enescu and C. Popa (Eds.), Logicas, tiint, ei. Bucharest: Editura Politics, 1970, 233-247.


Computer-based mathematics instruction. Bulletin of the International Study Group for Mathematics Learning, 3,7-22.

Learning the new mathematics. New Directions in Mathematics, Mem- bership Service Bulletin 16-A, Association for Childhood Education International, 57-64.

With F. BINFORD. Experimental teaching of mathematical logic in the


elementary school. The Arithmetic Teacher, 12, 187-1 95.

With D. HANSEN. Accelerated program in elementary-school mathematics: The first year. Psychology in the Schools, 2, 195-203.


Concept formation and Bayesian decisions. In J. Hintikka and P. Suppes (Eds.), Aspects of Inductive Logic. Amsterdam: North-Holland, 21-48.

Probabilistic inference and the concept of total evidence. In J. Hintikka and P. Suppes (Eds.), Aspects of Inductive Logic, Amsterdam: North- Holland, 49-65. Rumanian translation in Gh. Enescu and C. Popa (Eds.), Logicas, tiint, ei, Editura Politics, 1970,288-309.

A Bayesian approach to the paradoxes of confirmation. In J. Hintikka and P. Suppes (Eds.), Aspects of Inductive Logic. Amsterdam: North- Holland, 198-207.

Some formal models of grading principles. Synthese, 16,284-306.

PSYCHOLOGY

Mathematical concept formation in children. American Psychologist, 21, 139-150.

With J. L. ZINNES. A continuous-response task with nondeterminate, contingent reinforcement. Journal of Mathematical Psychology, 3, 197-216.

With G. GROEN and M. SCHLAG-REY. A model for response latency in paired-associate learning. Journal of Mathematical Psychology, 3, 99-128.

Towards a behavioral psychology of mathematical thinking. In J. Bruner (Ed.), Learning about Learning, a conference report. Washington, DC: US Government Printing Office, 226-234.


PHYSICS

The probabilistic argument for a non-classical logic of quantum mechanics. Philosophy of Science, 33, 14-21. French translation: L'argument probabiliste pour une logique non classique de la mkcanique quantique, Synthese, 1966, 16, 74-85. Reprinted in J. L. Destouches (Ed.), E. W Beth Memorial Colloquium: Logic and Foundations of Science, Paris, Institut Henri Poincar6, May 19-21, 1964. Reprinted also in C. A. Hooker (Ed.), The Logico-algebraic Approach to Quantum Mechanics. Dordrecht: Reidel, 1975,341-350.


Sets and Numbers. (Teacher's Edition, Books K-6). New York: Ran- dom House.

Sets and Numbers. (Books K-2). New York: Random House. Revised edition, 1968.

Sets and Numbers. (Books 3-6). New York: Random House. Revised edition, 1969.

Tomorrow's education. Education Age, 1966,2,4-11.

With M. JERMAN and G. GROEN. Arithmetic drills and review on a computer-based teletype. The Arithmetic Teacher, 13, 303-309.

Plug-in instruction. Saturday Review, 49(30), 25,29, 30.

The axiomatic method in high school mathematics. In The Role of Axiomatics and Problem Solving in Mathematics. The Conference Board of the Mathematical Sciences. Washington, DC: Ginn, 69-76.

The uses of computers in education. Scientific American, 215,206-220. Reprinted in Information: A ScientiJic American book. San Francisco: Freeman, 1966, 157-1 74. German translation: Anwendungen elektro- nischer Rechenanlagen in Unterricht. In Information Computer und kunstliche Intelligenz. Frankfurt am Main: Umschau, 1967, 157-1 72. Reprinted also in Mathematical Thinking in Behavioral Sciences: Read- ings from ScientiJic American. San Francisco and London: Freeman, 1968, 213-222. Russian translation in Informatsia. Moscow: Mir, 1968, 165-1 82. Japanese translation in Information, Scientific Ameri-


can Book, 1969. Polish translation: Zastosowania maszyn cyfrowych w nauczaniu-tlum. Tadeusz Wiewiorowski. In Dzis i jutro maszyn cyfrowych. Warsaw: 1969, 231-256. Reprinted also in Computers and Computation: Readings from Scientific American. San Francisco: Freeman, 1 97 1, 249-259. Reprinted also in Contemporary Psycholo- gy: Readings from Scientific American. San Francisco: Freeman, 197 1, 257-267.

Accelerated program in elementary-school mathematics: The second year. Psychology in the Schools, 3, 294-307. Rumanian translation in Gh. Enescu and C. Popa (Eds.), Logicas, tiint, ei, Editura Politica, 1970,233-247.

The psychology of arithmetic. In J. Bruner (Ed.), Learning abo~lt Learn- ing, a conference report. Washington, DC: US Government Printing Office, 235-242.

Applications of mathematical models of learning in education. In H. 0. A. Wold (Scientific Organizer), Model Building in the Human Sciences. Session of Entretiens de Monaco en Sciences Humaines, 1964. Monaco: Union Europienne d'Editions, 39-49.


Decision theory. In Encyclopedia of Philosophy, Vol. 2. New York: Macmillan and Free Press, 3 10-3 14.

What is a scientific theory? In S. Morgenbesser (Ed.), Philosophy of Science Today. New York: Basic Books, 55-67.

PSYCHOLOGY

With E. CROTHERS. Experiments in Second-language Learning. New York: Academic Press, 37 1 pp.

The psychological foundations of mathematics. In Les modkles et la formalisation du comportement. International colloquium of the Cen- tre National de la Recherche Scientifique. Paris: Editions du Centre


National de la Recherche Scientifique, 21 3-242.

Conclusion (of Colloquium) and Discussion. In Les mod2les et la formalisation du comportement. International colloquium of the Cen- tre National de la Recherche Scientifique. Paris: Editions du Centre National de la Recherche Scientifique, 4 13-42 1.

With J. DONIO. Foundations of stimulus-sampling theory for continuous-time processes. Journal of Mathematical Psychology, 4,202-225.

With F. S. ROBERTS. Some problems in the geometry of visual perception. Synthese, 17, 173-201.

PHYSICS

Some extensions of Randall's interpretation of Kant's philosophy of science. In J. P. Anton (Ed.), Naturalism and Historical Understanding: Essays on the philosophy of John Herman Randall, Jr., New York: State University of New York Press, 108-1 20.


Some theoretical models for mathematics learning. Journal of Research and Development in Education, 1,5-22.

With L. HYMAN and M. JERMAN. Linear structural models for response and latency performance in arithmetic on computer-controlled termi- nals. In J. P. Hill (Ed.), Minnesota Symposia on Child Psychology. Minneapolis: University of Minnesota Press, 160-200.

With G. GROEN. Some counting models for first-grade performance data on simple addition facts. In J. M. Scandura (Ed.), Research in Mathematics Education. Washington, DC: National Council of Teach- ers of Mathematics, 3 5 4 3 .

With C. IHRKE. Accelerated program in elementary-school mathematics: The third year. Psychology in the Schools, 4, 293-309. Reprinted in R. B. Ashlock and W. L. Herman, Jr. (Eds.), Current Research in Elementary School Mathematics. New York: Macmillan, 1970, 359- 374.


On using computers to individualize instruction. In D. D. Bushnell and D. W. Allen (Eds.), The Computer in American Education. New York: Wiley, 1 1-24.

The case for information-oriented (basic) research in mathematics education. In J. M. Scandura (Ed.), Research in Mathematics Education. Washington, DC: National Council of Teachers of Mathematics, 1-5. Reprinted in J. A. McIntosh (Ed.), Perspectives on Secondary Math- ematics Education. Englewood Cliffs, NJ: Prentice-Hall, 197 1, 233- 236.

The teacher and computer-assisted instruction. National Education Association Journal, 56(2), 15-32.

Computer-based instruction. Electronic Age, 1967,26, 2-6. Reprinted in: The Education Digest, 33(2), 8-1 0.

The teaching machine. Christian Science Monitor, August 10, p. 11.

The computer and excellence. Saturday Review, 50(2), 46-50.

1968


The desirability of formalization in science. Journal of Philosophy, 65, 65 1-664. Romanian translation: Dezirabilitatea formalizarii in stiinta. In I. P2rvu (Ed.), Epistemologie: Orientari contemporane. Bucharest: Editura Politics, 1974,268-283. German translation: In M. Balzer and W. Heidelberger (Eds.), Zur Logik empirischer Theorien. Berlin and New York: Walter de Gruyter, 1983,24-39.

Information processing and choice behavior. In I. Lakatos and A. Musgrave (Eds.), Problems in the Philosophy of Science. Amsterdam: North-Holland, 278-299.

PSYCHOLOGY

With R. R. BUSH and R. D. LUCE. Models, mathematical. In Inter- national Encyclopedia of the Social Sciences, Vol. 10. New York:


Macmillan and Free Press, 378-386. Reprinted in International Ency- clopedia of Statistics. New York: Free Press, 1978.

With M. SCHLAG-REY. Higher-order dimensions in concept identification. Psychonomic Science, 11, 141-1 42.

With I. ROSENTHAL-HILL. Concept formation by kindergarten children in a card-sorting task.Journa1 of Experimental Child Psychology, 6, 21 2-230.

LANGUAGE AND LOGIC

With R. D. LUCE. Mathematics. In International Encyclopedia of the Social Sciences, Vol. 10. New York: Macmillan and Free Press, 65-76. Reprinted in International Encyclopedia of Statistics. New York: Free Press.

With N. MOLER. Quantifier-free axioms for constructive plane geometry. Compositio Mathematica, 1968,20, 143-1 52.


With M. JERMAN and D. BRIAN. Computer-assisted Instruction: Stan- ford's 1965-66 arithmetic program. New York: Academic Press, 385 pp.

Computer technology and the future of education. Phi Delta Kappan, 44, 420-423. Reprinted in R. C. Atkinson and H. A. Wilson (Eds.), Computer-assisted Instruction: A Book of Readings. New York: Aca- demic Press, 1969, 41-47. Also reprinted in I. Taviss (Ed.), The Com- puter Impact. Englewood Cliffs, NJ: Prentice-Hall, 1970, 203-209. Also reprinted in R. A. Weisgerber (Ed.), Perspectives in Individualized Learning. Itasca, IL: Peacock, 1971, 391-398. Also reprinted in K. Hoover (Ed.), Readings on Learning and Teaching in the Secondary School, 2nd ed., Boston: Allyn & Bacon, 1971, 354-361.

Discussion-educational technology: New myths and old realities. Har- vard Educational Review, 38,730-735.

Computer-assisted instruction: An interview with Patrick Suppes. Nation's Schools, 82(4), 52,53,96. Reprinted in The Education Digest,


Can there be a normative philosophy of education? In G. L. Newsome, Jr. (Ed.), Philosophy of Education. Studies in Philosophy and Educa- tion series, Proceedings of the 24th annual meeting of the Philosophy of Education Society, Santa Monica, April 7-10, 1968, Edwardsville, IL: Southern Illinois University, 1-12. Reprinted in J. P. Strain (Ed.), Modern Philosophies of Education. New York: Random House, 197 1, 277-288.


Studies in the Methodology and Foundations of Science: Selected Papers from 1951 to 1969. Dordrecht: Reidel, 473 pp.

PSYCHOLOGY

Stimulus-response theory of finite automata. Journal of Mathematical Psychology, 6, 327-355. German translation: In M. Balzer and W. Heidelberger (Eds.), Zur Logik empirischer Theorien. Berlin and New York: Walter de Gruyter, 1983,245-280.

Stimulus-response theory of automata and TOTE hierarchies: A reply to Arbib. Psychological Review, 76, 51 1-514. Reprinted in J. M. Scandura (Ed.), Structural Learning: II. Issues and Approaches. New York: Gordon and Breach, 1976.

LANGUAGE AND LOGIC

Nagel's lectures on Dewey's logic. In S. Morgenbesser, P. Suppes, and M. White (Eds.), Philosophy, Science and Method: Essays in Honor of Ernest Nagel. New York: St. Martin's Press, 2-25.


With M. MORNINGSTAR. Computer-assisted instruction. Science, 166, 343-350. Reprinted in D. A. Erickson (Ed.), Educational Organization

BIBLIOGRAPHY OF PATRICK SUPPES 293

and Administration. Berkeley, CA: McCutchan, 1977,236-254.

Computer-assisted instruction: An overview of operations and problems. In A. J. H. Morrell (Ed.), Information Processing 68, Vol. 2. Proceedings of IFIP Congress 2968, Edinburgh. Amsterdam: North- Holland, 1 103-1 1 13.

With E. Lornus and M. JERMAN. Problem-solving on a computer-based teletype. Educational Studies in Mathematics, 2, 1-15. Romanian translation: Rezolvarea problemelor la un telescriptor conectat cu un calculator electronic. In E. Fischbein and E. Rusu (Eds.), Invatamintul matematic in lumea contemporana. Bucharest: Editura Didactica si Pedagogica, 1 97 1,276-296.

With M. JERMAN. A workshop on computer-assisted instruction in elementary mathematics. The Arithmetic Teacher, 16, 193-1 97.

With M. JERMAN. Computer-assisted instruction at Stanford. Educa- tional Technology, 9(1), 22-24.

With M. JERMAN. Some perspectives on computer-assisted instruction. Educational Media, 1(4), 4-7.

With M. JERMAN. Computer-assisted instruction at Stanford. Educa- tional Television International, 3(3), 176-1 79.

With M. JERMAN. Individualized Mathematics, Drill Kits AA-DD. New York: Random House.

With B. MESERVE and P. SEARS. Sets, Numbers, and Systems. Books 1 and 2. New York: Random House.

With B. MESERVE and P. SEARS. Sets, Numbers, and Systems, Teacher's Edition, Book 1. New York: Random House.


A Probabilistic Theory of Causality. Acta Philosophica Fennica, 24. Amsterdam: North-Holland, 1 30 pp.


PSYCHOLOGY

With D. JAMISON and D. LHAMON. Learning and the structure of information. In J. Hintikka and P. Suppes (Eds.), Information and Inference. Dordrecht: Reidel, 197-259.

LANGUAGE AND LOGIC

Probabilistic grammars for natural languages. Synthese, 22, 95-1 16. Reprinted in D. Davidson and G. Harman (Eds.), Semantics of Natural Language. Dordrecht: Reidel, 741-762.


With M. MORNINGSTAR. Four programs in computer-assisted instruction. In W. H. Holtzman (Ed.), Computer Assisted Instruction, Testing, and Guidance. New York: Harper & Row, 233-265.

With D. JAMISON and C. BUTLER. Estimated costs of computer assisted instruction for compensatory education in urban areas. Educational Technology, 10, 49-57. Reprinted in D. L. Roberts (Ed.), Planning Urban Education. Englewood Cliffs, NJ: Educational Technology Pub- lications, 28 1-301.

With M. MORNINGSTAR. Technological innovations: Computer- assisted instruction and compensatory education. In F. Korten, S. Cook, and J. Lacey (Eds.), Psychology and the Problems of Society. Washing- ton, DC: American Psychological Association, 221-236.

With C. IHRKE. Accelerated program in elementary-school mathematics: The fourth year. Psychology in the Schools, 7, 1 1 1 - 126.

Systems analysis of computer-assisted instruction. In G. J.Kelleher (Ed.), The Challenge to Systems Analysis. New York: Wiley, 98-1 10.

With M. JERMAN. Computer-assisted instruction. The Bulletin of the National Association of Secondary School Principals, 54(343), 27-40.

With B. MESERVE and P. SEARS. Sets, Numbers and Systems, Teacher's Edition, Book 2. New York: Random House.



With D. H. KRANTZ, R. D. LUCE, and A. TVERSKY. Foundations of Measurement, Vol. I: Additive and Polynomial Representations. New York: Academic Press, 577 pp.

LANGUAGE AND LOGIC

With S. FELDMAN. Young children's comprehension of logical connectives. Journal of Experimental Child Psychology, 12, 304-3 17.


Computer assisted instruction for deaf students. American Annals of the Deaf, 116,500-508.

Alternatives through computers. In B. Rusk (Ed.), Alternatives in Edu- cation. Toronto: General, 57-70.

Technology in education. In S. M. Brownell (Ed.), Issues in Urban Education. New Haven, CT: Yale University Press, 119-146.

With B. SEARLE. The computer teaches arithmetic. School Review, 79(2), 213-225.


Finite equal-interval measurement structures. Theoria, 38,45-63.

Veroyatnostnaya teoria prichinosti. Voprosi FilosoJi, 4, 90-102.

On the problems of using mathematics in the development of the social sciences. In Mathematics in the Social Sciences in Australia. Canberra: Australian Government Publishing Service, 3-1 5.

Measurement: Problems of theory and application. In Mathematics


in the Social Sciences in Australia. Canberra: Australian Government Publishing Service, 61 3-622.

PSYCHOLOGY

With E. F. LORUS. Structural variables that determine the speed of retrieving words from long-term memory. Journal of Verbal Learning and Verbal Behavior, 11,770-777.

Stochastic models in mathematical learning theory. In Mathematics in the Social Sciences in Australia. Canberra: Australian Government Publishing Service, 265-273.

PHYSICS

Some open problems in the philosophy of space and time. Synthese, 24,298-316. Reprinted in P. Suppes (Ed.), Space, Time and Geometry. Dordrecht: Reidel, 1973,383-401.


With M. MORNINGSTAR. Computer-assisted Instruction at Stanford, 1966-68: Data, Models, and Evaluation of the Arithmetic Programs. New York: Academic Press, 533 pp.

Computer-assisted instruction at Stanford. In Man and Computer. Pro- ceedings of international conference, Bordeaux 1970. Basel: Karger, 298-330. Reprinted in K. L. Zinn and A. Romano (Eds.), Computers in the Instructional Process: Report of an Internatiorzal School. Ann Arbor, MI: Extend. 1974,57-85.

With E. F. LOFTUS. Structural variables that determine problem-solving difficulty in computer-assisted instruction. Journal of Educational Psy- chology, 63, 531-542.

With A. GOLDBERG. A computer-assisted instruction program for exercises on finding axioms. Educational Studies in Mathematics, 4, 429- 449.


With J. D. FLETCHER. Computer assisted instruction in reading: Grades 4-6. Educational Technology, 12(8), 45-49.

Computer-assisted instruction. In W. Handler and J. Weizenbaum (Eds.), Display Use for Man-Machine Dialog. Munich: Hanser, 155- 185.


New foundations of objective probability: Axioms for propensities. In P. Suppes, L. Henkin, G. C. Moisil, and A. Joja (Eds.), Logic, Method- ology, and Philosophy of Science I E Proceedirzgs of the Fourth Inter- national Congress for Logic, Methodology and Philosophy of Science, Bucharest, 1971. Amsterdam: North-Holland, 5 15-529.

The concept of obligation in the context of decision theory. In J. Leach, R. Butts, and G. Pearce (Eds.), Science, Decision and Value. Proceed- ings of the Fifth University of Western Ontario Philosophy Colloquium, 1969. Dordrecht: Reidel, 1-1 4.

PSYCHOLOGY

Theory of automata and its application to psychology. In G. J. Dalenoort (Ed.), Process Models for Psychology. Lecture notes of the NUFFIC International Summer Course, 1972. Rotterdam: Rotterdam University Press, 78-1 23.

LANGUAGE AND LOGIC

Semantics of context-free fragments of natural languages. In K. J. J. Hintikka, J. M. E. Moravcsik, and P. Suppes (Eds.), Approaches to Natural Language. Dordrecht: Reidel, 370-394.

Congruence of meaning. Proceedings and Addresses of the American Philosophical Association, 46, 21-38.



Facts and fantasies of education. Phi Delta Kappa Monograph. Reprint- ed in M. C. Wittrock (Ed.), Changing Education: Alternatives from Educational Research. Englewood Cliffs, NJ: Prentice-Hall, 6-45.


The measurement of belief. Journal of the Royal Statistical Society (Series B) , 36, 160-191.

The structure of theories and the analysis of data. In F. Suppe (Ed.), The Structure of ScientiJic Theories, 2nd ed. Urbana, IL: University of Illinois Press, 266-283. (Originally published, 1974.) Spanish translation: La estructura de las teorias y el anailisis de datos. In F. Suppe (Ed.), La estructura de las teorias cient@cas, translated by P. Castrillo and E. Rada. Madrid: Editorial National, 1979.

With R. D. LUCE. Theory of Measurement. In Encyclopaedia Britan- nica, Vol. 11, 15th edition, 739-745.

The essential but implicit role of modal concepts in science. In K. F. Schaffner and R. S. Cohen (Eds.), PSA 1972. Dordrecht: Reidel, 305- 3 14.

The axiomatic method in the empirical sciences. In L. Henkin et al. (Eds.), Proceedings of the Tarski Symposium, Proceedings of Symposia in Pure Mathematics, 25. Providence, RI: American Mathematical Society, 465-479.

PSYCHOLOGY

With W. K. ESTES. Foundations of stimulus sampling theory. In D. H. Krantz, R. C. Atkinson, R. D. Luce, and P. Suppes (Eds.), Contemporary Developments in Mathematical Psychology, Vol. 1 : Learning, Memory, and Thinking. San Francisco: Freeman, 163-1 83.

Cognition: A survey. In J. A. Swets and L. L. Elliott (Eds.), Psychol- ogy and the Handicapped Child. Washington, DC: U.S. Government


Printing Office, 109-1 26.

A survey of cognition in handicapped children. Review of Educational Research, 44, 145-176. Reprinted in S. Chess and A. Thomas (Eds.), Annual Progress in Child Psychiatry and Child Development, 1975. New York: Brunner/Mazel, 1975,95-129.

With W. ROITMAYER. Automata. In E. C. Carterette and M. P. Friedman (Eds.), Handbook of Perception, Vol. 1 : Historical and Philosophical Roots of Perception. New York: Academic Press, 335-362.

PHYSICS

Popper's analysis of probability in quantum mechanics. In P. A. Schilpp (Ed.), The Philosophy of Karl Popper,Vol. 2. La Salle, IL: Open Court, 760-774.

Aristotle's concept of matter and its relation to modern concepts of matter. Synthese, 28, 27-50.

With M. Z A N O ~ I . Stochastic incompleteness of quantum mechanics. Synthese, 29, 3 11-330. Reprinted in P. Suppes (Ed.), Logic and Probability in Quantum Mechanics. Dordrecht: Reidel, 303-322.

LANGUAGE AND LOGIC

The semantics of children's language. American Psychologist, 29,103- 114.

Model-theoretic semantics for natural language. In C. H. Heidrich (Ed.), Semantics and Communication. Amsterdam: North-Holland, 285-344.

With R. SMITH and M. L ~ V E I L L ~ . The French syntax of a child's noun phrases. Archives de Psychologie, 42,207-269.

On the grammar and model-theoretic semantics of children's noun phrases. Colloques Internationaux du C.N.R.S. Probl2mes Actuels en Psycholinguistique, 206,49-60.



With D. JAMISON and S. WELLS. The effectiveness of alternative instructional media: A survey. Review of Educational Research, 44, 1-67.

The place of theory in educational research. Educational Researcher, 3(6), 3-10.

With B. SEARLE and P. LORTON, JR. Structural variables affecting CAI performance on arithmetic word problems of disadvantaged and deaf students. Educational Studies in Mathematics, 5, 371-384.

Mathematical models of learning and performance in a CAI setting. In K. L. Zinn, M. Refice, and A. Romano (Eds.), Computers in the Instructional Process: Report of an International School. Ann Arbor, MI: Extend, 339-353.

The promise of universal higher education. In S. Hook, P. Kurtz, and M. Todorovich (Eds.), The Idea of a Modern University. Buffalo, NY: Prometheus, 21-32.

With J. D. FLETCHER. Computer-assisted instruction in mathematics and language arts for deaf students. In AFIPS Conference Proceedings, Vol. 43. 1974 National Computer Conference. Montvale, NJ: AFIPS Press, 127-131.

With J. B. CARROLL. The Committee on Basic Research in Education: A four year tryout of basic science funding procedures. Educational Researcher, 3(2), 7-10.


A probabilistic analysis of causality. In H. M. Blalock, A. Aganbegian, F. M. Borodkin, R. Boudon, and V. Capecchi (Eds.), Quantitative Soci- ology. New York: Academic Press, 49-77.

Approximate probability and expectation of gambles. Erkenntnis, 9, 153-161.


PSYCHOLOGY

From behaviorism to neobehaviorism. Theory and Decision, 6, 269- 285.

With H. WARREN. On the generation and classificatlon of defence mechanisms. The Internatiorzal Journal of Psycho-Analysis, 56, 405- 414. Reprinted in R. Wollheim and J. Hopkins (Eds.), Philosophical Essays on Freud. Cambridge: Cambridge University Press, 1982,163- 179.


The school of the future: Technological possibilities. In L. Rubin (Ed.), The Fut~lre of Education: Perspectives on Tomorrow's School- ing. Boston: Allyn & Bacon, 145-157.

Impact of computers on curriculum in the schools and universities. In 0. Lecarme and R. Lewis (Eds.), Computers in Education, IFIe Part 1. Amsterdam: North-Holland, 173-1 79.

With J. D. FLETCHER and M. ZANOTTI. Performance models of Amer- ican Indian students on computer-assisted instruction in elementary mathematics. Instructional Science, 4, 303-3 13.

With B. SEARLE. The Nicaragua radio mathematics project. Educa- tional Broadcasting International, September, 1 17-1 20.


Testing theories and the foundations of statistics. In W. L. Harper and C. A. Hooker (Eds.), Foundations of Probability Theory, Statistical Infer- ence, and Statistical Theories of Science,VoL. 2. Dordrecht: Reidel, 437-455.

Archimedes' anticipation of conjoint measurement. In Role and Impor- tance of Logic and Methodology of Science in the Study of the History of Science. Colloquium presented at the XI11 International Congress of


the History of Science, Moscow, 1971. Moscow: Nauka.

With M. ZANOTTI. Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualitative probability ordering. Journal of Philosophical Logic, 5,431-438.

PSYCHOLOGY

Syntax and semantics of children's language. In W. R. Harnad, H. D. Steklis, and J. Lancaster (Eds.), Origins and Evolution of Language and Speech, Annals of the New York Academy of Sciences, 280,227-237. New York: New York Academy of Sciences.

With M. L ~ V E I L L ~ . La comprkhension des marques d'appartenance par les enfants. Enfance, 3, 309-3 18.

PHYSICS

With M. ZANOTTI. On the determinism of hidden variable theories with strict correlation and conditional statistical independence of observable~. In P. Suppes (Ed.), Logic and Probability in Quantum Mechanics. Dordrecht: Reidel, 445455.

LANGUAGE AND LOGIC

Elimination of quantifiers in the semantics of natural language by use of extended relation algebras. Revue Internationale de Philosophie, 117-118,243-259.


With B. SEARLE and J. FRIEND. The Radio Mathematics Project: Nicaragua 1974-1975. Stanford, CA: Stanford University, Institute for Mathematical Studies in the Social Sciences, 261 pp.

With J. D. FLETCHER and M. ZANOTTI. Models of individual trajectories in computer-assisted instruction for deaf students. Journal of


Educational Psychology, 68, 1 17-1 27.

With E. MACKEN, R. VAN DEN HEUVEL, and T. SUPPES. Home Based Education: Needs and Technological Opportunities, U.S. Department of Health, Education, and Welfare, National Institute of Education, April 1976,130.

With A. GOLDBERG. Computer-assisted instruction in elementary logic at the university level. Educational Studies in Mathematics, 6,447474.

With J. D. FLETCHER. The Stanford project on computer-assisted instruction for hearing-impaired students. Journal of Computer-Based Instruction, 3, 1-1 2.

With B. SEARLE. Survey of the instructional use of radio, television, and computers in the United States. Journal of the Society of Instrument and Control Engineers, 15,7 12-720.

With B. SEARLE. The Radio Mathematics Project. The Mathematics Teacher (India), 11A, 47-5 1.

With D. T. JAMISON, J. D. FLETCHER, and R. C. ATKINSON. Cost and performance of computer-assisted instruction for education of disadvantaged children. In J. Froomkin, D. T. Jamison, and R. Radner (Eds.), Education a s an Industry. Cambridge, MA: NBER, Ballinger, 201-240.

With E. MACKEN. Evaluation studies of CCC elementary-school cur- riculums, 197 1-1 975. CCC Educational Studies, l, 1-37.


The distributive justice of income inequality. Erkenntnis, 11,233-250. Reprinted in H. W. Gottinger and W. Leinfellner (Eds.), Decision Theory and Social Ethics. Dordrecht: Reidel, 1977, 303-320.

Some remarks about complexity. In PSA, Vol. 2. Philosophy of Science Association, 543-547.

With M. Z A N O ~ I . On using random relations to generate upper and lower probabilities. Synthese, 36,427440.


PSYCHOLOGY

Learning theory for probabilistic automata and register machines, with applications to educational research. In H. Spada and W. F. Kempf (Eds.), Structural Models of Thinking and Learning, Proceedings of the 7th IPN-Symposium on Formalized Theories of Thinking and Learning and their Implications for Science Instruction. Bern: Hans Huber Pub- lishers, 57-79.

Is visual space Euclidean? Synthese, 35, 397-421.

A survey of contemporary learning theories. In R. E. Butts and J. Hintik- ka (Eds.), Foundational Problems in the Special Sciences, Part 2 of the Proceedings of the Fifth International Congress of Logic, Methodology and Philosophy of Science, London, Ontario, Canada, 1975. Dordrecht: Reidel, 21 7-239.


With R. SMITH and M. BEARD. University-level computer-assisted instruction at Stanford: 1975. Instructional Science, 6, 15 1-1 85.

With B. SEARLE and J. FRIEND. The Nicaragua Radio Mathematics Project. In P. L. Spain, D. T. Jamison, and E. McAnany (Eds.), Radio for Education and Development: Case studies, Vol. 1. Washington, DC: Education Department of the World Bank, 2-32.

With B. SEARLE. Computer usage in the Nicaragua Radio Mathematics Project. In J. A. Jordan, Jr., and K. Malaivongs (Eds.), Proceedings of the International Conference on Computer Applications in Developing Countries, Vol. 1. Bangkok: Asian Institute of Technology, 361-374.

PSYCHOLOGY

A philosopher as psychologist. In T. S. Krawiec (Ed.), The Psycholo- gists: Autobiographies of Distinguished Living Psychologists, Vol. 3. Brandon, VT.: Clinical Psychology Publishing Co., 261-288.


With H. WARREN. Psychoanalysis and American elementary education. In P. Suppes (Ed.), Impact of Research on Education: Some case studies. Washington, DC: National Academy of Education, 319-396.

LANGUAGE AND LOGIC

With E. MACKEN. Steps toward a variable-free semantics of attributive adjectives, possessives, and intensifying adverbs. In K. E. Nelson (Ed.), Children's Language, Vol. 1. New York: Gardner Press, 81-1 15.


The future of computers in education. In Computers and the Learning Society. (Hearings before the Subcommittee on Domestic and Interna- tional Scientific Planning, Analysis and Cooperation, of the Committee on Science and Technology, U.S. House of Representatives, Ninety-fifth Congress, First Session, October 4, 6, 12, 13, 18, and 27, 1977 [No. 471.) Washington: U.S. Government Printing Office, 548-569.

With E. MACKEN and M. ZANOTTI. The role of global psychological models in instructional technology. In R. Glaser (Ed.), Advances in Instructional Psychology, Vol. 1. Hillsdale, NJ: Erlbaum, 229-259.

With I. LARSEN and L. 2. MARKOSIAN. Performance models of under- graduate students on computer-assisted instruction in elementary logic. Instructional Science, 7, 15-35.

La informhtica en la educaci6n. Nonotza, Revista de Dzfisio'n Cierzt$ca, Tecnolbgica y Cultural, 13, IBM de Mexico.

With B. SEARLE. Achievement levels of students learning primary- school mathematics by radio in Nicaragua. Studies in Science and Mathematics Education (India), 1, 63-80.

With B. SEARLE, P. MATTHEWS, and J. FRIEND. Formal evaluation of the 1976 first-grade instructional program. In P. Suppes, B. Searle, and J. Friend (Eds.), The Radio Mathematics Project: Nicaragua 1976-1 977. Stanford: Institute for Mathematical Studies in the Social Sciences, Stanford University, 97-1 24.


With E. MACKEN. The historical path from research and development to operational use of CAI. Educational Technology, 18(4), 9-12.


El estudio de las revoluciones cientificas: teoria y metodologia. In La filosofa y las revoluciones cient$cas: Teoria y praxis. Mtxico, DF: Editorial Grijalbo, S A, 295-306.

The role of formal methods in the philosophy of science. In P. D. Asquith and H. E. Kyburg, Jr. (Eds.), Current Research in Philosophy of Science. East Lansing, MI: Philosophy of Science Association, 16- 27.

Self-profile. In R. J. Bogdan (Ed.), Patrick Suppes. Dordrecht: Reidel, 3-56.

Replies. In R. J. Bogdan (Ed.), Patrick Suppes. Dordrecht: Reidel, 207-232.

PSYCHOLOGY

The logic of clinical judgment: Bayesian and other approaches. In H. T. Engelhardt, Jr., S. F. Spicker, and B. Towers (Eds.), Clinical Judgment: A Critical Appraisal. Dordrecht: Reidel, 145-1 59.

With M. L ~ V E I L L ~ and R. SMITH. Probabilistic modelling of the child's productions. In P. Fletcher and M. Garman (Eds.), Language Acquisi- tion. Cambridge: Cambridge University Press, 397-41 7.

LANGUAGE AND LOGIC

Variable-free semantics for negations with prosodic variation. In E. Saarinen, R. Hilpinen, I. Niiniluoto, and M. P. Hintikka (Eds.), Essays in Honour of Jaakko Hintikka. Dordrecht: Reidel, 49-59.

Logical inference in English: A preliminary analysis. Studia Logica, 38,375-391.



Current trends in computer-assisted instruction. In M. C. Yovits (Ed.),Advances in Computers, Vol. 18. New York: Academic Press, 173-229.

Past, present and future educational technologies. In M.E.A. El Tom (Ed.), Developing Mathematics in Third World Countries. Amsterdam: North-Holland, 53-66.

With T. W. MALONE, E. MACKEN, M. ZANOTTI, and L. KANERVA. Projecting student trajectories in a computer-assisted instruction curriculum. Journal of Educational Psychology, 71,74-84.

With T. W. MALONE and E. MACKEN. Toward optimal allocation of instructional resources: Dividing computer-assisted instruction time among students. Instructional Science, 8, 107-1 20.

The future of computers in education. Journal of Computer-Based Instruction, 6, 5-10.


Some remarks on statistical explanations. In G. H. von Wright (Ed.), Logic and Philosophy. The Hague: Martinus Nijhoff, 53-58.

With J. SACHAR. Estimating total-test scores from partial scores in a matrix sampling design. Educational and Psychological Measurement, 40,687-699.

Probabilistic empiricism and rationality. In R. Hilpinen (Ed.), Ratio- nality in Science. Dordrecht: Reidel, 17 1-1 90.

PHYSICS

With M. ZANOTTI. A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distribution. In P. Suppes (Ed.), Studies in the Foundations of Quan- tum Mechanics. East Lansing, MI: Philosophy of Science Association.,


Limitations of the axiomatic method in ancient Greek mathematical sciences. In J. Hintlkka, D. Gruender, and E. Agazzi (Eds.), Pisa Con- ference Proceedings, Vol. 1, Dordrecht: Reidel, 197-21 3.

LANGUAGE AND LOGIC

Procedural semantics. In R. Haller and W. Grass1 (Eds.), Language, Logic, and Philosophy, Proceedings of the 4th International Wittgen- stein Symposium, Kirchberg am Wechsel, Austria 1979. Vienna: Holder-Pichler-Tempsky, 27-3 5.

In J. Speck (Ed.), Handbuch wissenschaftstheoretischer Begriffe. Gottingen, Germany: Vandenhock and Ruprecht: 'Definition', Vol. 1, 124-1 29; 'Grosse', Vol. 2, 268-269; 'Mengenlehre', Vol. 2, 41 1-41 5; 'Messung', Vol. 2,415-423; 'Observable', Vol. 2, p. 464.


Computer-assisted instruction in logic at Stanford. Newsletter on Teach- ing Philosophy, 6-9.

With E. MACKEN and M. ZANO~TI. Considerations in evaluating individualized instruction. Journal of Research and Development in Edu- cation, 14, 79-83.

METHODOLOGY. PROBABILITY AND MEASUREMENT

Logique du Probable. Paris: Flammarion, 136 pp. Italian translation by Alberto Artosi, La logica del probabile, un approccio bayesiano alla razionalitti. Bologna, Italy: Cooperativa Libraria Universitaria Editrice Bologna, 1984, 145 pp.

With M. Z A N O ~ I . When are probabilistic explanations possible? Syn- these, 48, 191-199.


Scientific causal talk: A reply to Martin. Theory and Decision, 13, 363-380.

The limits of rationality. Grazer Philosophische Studien, 12/13, 85- 101.

The plurality of science. In P. Asquith and I. Hacking (Eds.), PSA 1978, Vol. 2. Lansing, MI: Philosophy of Science Association, 3-16. Reprinted in J. A. Kourany (Ed.), Scientific Knowledge: Basic Issues in the Philosophy of Science. Belmont, CA: Wadsworth, 1987,317-325.

PSYCHOLOGY

With D. G. DANFORTH and D. R. ROGOSA. Application of learning models to speech recognition over a telephone. In P. Suppes (Ed.), University-level Computer-assisted Instruction a t Stanford: 1968- 1980. Stanford, CA.: Stanford University, Institute for Mathematical Studies in the Social Sciences, 589-600.

PHYSICS

Probability in relativistic particle theory. Erkenntnis, 16, 299-305.

Some remarks on hidden variables and the EPR paradox. Erkenntnis, 16,311-314.

Causal analysis of hidden variables. In P. Asquith and R. Giere (Eds.), PSA 1980, Vol. 2. East Lansing, MI: Philosophy of Science Association, 563-571.

LANGUAGE AND LOGIC

Direct inference in English. Teaching Philosophy, 4,405-41 8.

With J. SHEEHAN. CAI course in axiomatic set theory. In P. Sup- pes (Ed.), University-level Computer-assisted Instruction at Stanford: 1968-1980. Stanford, CA: Stanford University, Institute for Mathemat- ical Studies in the Social Sciences, 3-80.


With J. SHEEHAN. CAI course in logic. In P. Suppes (Ed.), University-level Computer-assisted Instruction at Stanford: 1968- 1980. Stanford, CA: Stanford University, Institute for Mathematical Studies in the Social Sciences, 193-226.


With W. R. SANDERS and C. GRAMLICH. Data compression of linear- prediction (LP) analyzed speech. In P. Suppes (Ed.), University-level Computer-assisted Instruction at Stanford: 1968-1 980. Stanford, CA: Stanford University, Institute for Mathematical Studies in the Social Sciences, 503-538.

Future educational uses of interactive theorem proving. In P. Sup- pes (Ed.), University-level Computer-assisted Instruction at Stanford: 1968-1980. Stanford, CA: Stanford University, Institute for Mathemat- ical Studies in the Social Sciences, 165-182.

With R. LADDAGA and A. LEVINE. Studies of student preference for computer-assisted instruction with audio. In P. Suppes (Ed.), University- level Computer-assisted Instruction at Stanford: 1968-1 980. Stanford, CA: Stanford University, Institute for Mathematical Studies in the Social Sciences, 399430.

With R. LADDAGA and W. R. SANDERS. Testing intelligibility of computer-generated speech with elementary-school children. In P. Sup- pes (Ed.), University-level Computer-assisted Instruction at Stanford: 1968-1 980. Stanford, CA: Stanford University, Institute for Mathemat- ical Studies in the Social Sciences, 377-397.


With M. ZANOITI. Necessary and sufficient qualitative axioms for conditional probability. Zur Wahrscheinlichkeitstheorie venvandte Gebi- ete, 60, 163-169.

Rational allocation of resources to scientific research. In L. J. Cohen, J. Los, H. Pfeiffer, and K.-P. Podewski (Eds.), Logic, Methodology and


Philosophy of Science, VI. Amsterdam: North-Holland, 773-789.

With M. ZANOTTI. Alcuni risultati positivi e negativi sulla esistenza di cause. Laboratorio di Science dell'Uomo, 3-4, June, Uso e prospettive delle scienze umane, 21 9-225.

Problems of causal analysis in the social sciences. Epistemologia, 5, 239-250.

PSYCHOLOGY

With M. COHEN, R. LADDAGA, J. ANLIKER and R. FLOYD. Research on eye movements in arithmetic performance. In R. Groner and P.Fraisse (Eds.), Cognition and Eye Movements. Amsterdam: North-Holland, 57-73.

LANGUAGE AND LOGIC

Variable-free semantics with remarks on procedural extensions. In T. W. Simon and R. J. Scholes (Eds.), Language, Mind, and Brain. Hillsdale, NJ: Erlbaum, 21-34.


Historical perspective on educational technology. In R. M. Bossone (Ed.), What Works in Urban Schools, Proceedings of the Second Con- ference of the UniversityIUrban Schools National Task Force. New York: Center for Advanced Study in Education, The Graduate School and University Center of the City University of New York, 30-57.

Sur les experiences d'enseignement assist6 par ordinateur dans divers autres pays. In Actes du Colloque, Le Mariage du Si2cle: Education et informatique. Paris: Minist2re de 17Education, 9-20.

On the effectiveness of educational research. In D. B. P. Kallen, G. B. Kosse, H. C. Wagenaar, J. J. J. Kloprogge, and M. Vorbeck (Eds.), Social Science Research and Public Policy-making: A reappraisal. Windsor, B erks., England: NFER-Nelson, 255-270. First published by Founda- tion for Educational Research in the Netherlands (SVO), 1982.



The meaning of probability statements.Erkenntnis, 19, 397403.

Heuristics and the axiomatic method. In R. Groner, M. Groner, and W. F. Bischof (Eds.), Methods of Heuristics. Hillsdale, NJ: Erlbaum, 79-88.

Arguments for randomizing. In P. D. Asquith and T. Nickles (Eds.), PSA 1982. Lansing, MI: Philosophy of Science Association, 464475.

Probability and information. (CommentaryIDretske: Knowledge and the flow of information.) The Behavioral and Brain Sciences, 6, 81-82.

Procedure scientifiche e razionalith. Nuova Civilta delle Macchine, Autunno, Anno I(4), 30-37.

PSYCHOLOGY

With M. COHEN, R. LADDAGA, J. ANLIKER, and R. FLOYD. A procedural theory of eye movements in doing arithmetic. Journal of Mathematical Psychology, 27,341-369.

LANGUAGE AND LOGIC

Language learning in the limit. (ReviewIK. Wexler and P. W. Culi- cover: Formal Principles of Language Acquisition.) Contemporary Psychology, 28,5-6.


Probabilistic Metaphysics. Oxford, England: Blackwell, 25 1 pp. Romanian translation: MetaJizica Probabilista. Bucharest, Romania: Humanitas, 1990,384 pp.


Conflicting intuitions about causality. Midwest Studies in Philosophy, IX, 151-168.

PHYSICS

With M. ZANOTI. Causality and symmetry. In S. Diner, D. Fargue, C. Lochak, and W. Sellers (Eds.), The Wave-particle Dualism. Dordrecht: Reidel, 33 1-340.

LANGUAGE AND LOGIC

A puzzle about responses and congruence of meaning. Synthese, 58, 39-50.

The next generation of interactive theorem provers. In R. E. Shostak (Ed.), Proceedings of the 7th International Conference on Automated Deduction, Napa, California, May 14-1 6, 1984, Lecture Notes in Com- puter Science, No. 170. New York: Springer-Verlag, 303-3 15.

With J. MCDONALD. Student use of an interactive theorem prover. In W. W. Bledsoe and D. W. Loveland (Eds.), Automated Theorem Prov- ing: After 25 Years. Providence, RI: American Mathematical Society. Contemporary Mathematics, 1984,29,3 15-360.


Observations about the application of artificial intelligence research to education. (Excerpt with slight modifications/Current trends in computer-assisted instruction.) In D. F. Walker and R. D. Hess (Eds.), Instructional Sofhyare: Principles and Perspectives for Design and Use. Belmont, CA: Wadsworth, 298-308.

Computers: Past, present and future. In R. M. Bossone and J. H. Pol- ishook (Eds.), Proceedings: The Fzfth Conference of the University Urban Schools National Task Force. New York: The Graduate SchooI School of the City University of New York, 112-124.

With R. E. MAAS. A note on discourse with an instructable robot. Theoretical Linguistics, 11, 5-20.



Explaining the unpredictable. Erkenntnis, 22, 187-1 95.

PSYCHOLOGY

Davidson's views on psychology as a science. In B. Vermazen and M. B. Hintikka (Eds.), Essays on Davidson: Actions & events. Oxford, England: Clarendon Press, 183-1 94.

Some general remarks on the cognitive sciences. In W. Kintsch, J. R. Miller, and P. G. Polson (Eds.), Method and Tactics in Cognitive Science. Hillsdale, NJ: Erlbaum, 297-304.


With R. E. MAAS. Natural-language interface for an instructable robot. International Journal of Man-Machine Studies, 22,2 15-240.

With R. F. FORTUNE. Computer-assisted instruction: Possibilities and problems. NASSP Bulletin, 69(480), 30-34.


Non-Markovian causality in the social sciences with some theorems on transitivity. Synthese, 68, 129-140. Italian translation: La causalit2 non-Markoviana nelle scienze sociali con alcuni teoremi sulla transi- tivita. In M. C. Galavotti and G. Gambetta (Eds.), Epistemologia ed Economia, 149-1 6 1. Bologna, Italy: Cooperativa Libraria quniversi- taria Editrice Bologna, 1988.

Comment on 'The Axioms of Subjective Probability', by Peter C. Fish- burn. Statistical Science, 1, 347-350.

Philosophy of science and public policy. In P. D. Asquith and P. Kitcher


(Eds.), PSA 1984, Vol. 2. East Lansing, MI: Philosophy of Science Association, 3-1 3.

LANGUAGE AND LOGIC

The primacy of utterer's meaning. In R. E. Grandy and R. Warner (Eds.), Philosophical Grounds of Rationality: Intentions, Categories, Ends. Oxford: Clarendon Press, 109-1 29.

Congruency theory of propositions. In Me'rites et limites des Me'thodes Logiques en Philosophie, Colloque international organiskpar la Fonda- tion Singer-Polignac en juin 1984. Paris: Librairie Philosophique J. Vrin, 279-299.


Computers and education in the 21st century. In Proceedings of an International Conference on Social and Technological Change, The University into the 21st Century, May 2-5, Victoria, B.C., Canada. Reprinted in W.A.W. Neilson and C. Gaffield (Eds.), Universities in Crisis: A Mediaeval Institution in the Twenty-First Century. Toronto: The Institute for Research on Public Policy, 137-15 1.


Propensity representations of probability. Erkenntnis, 26, 335-358.

Some further remarks on propensity: Reply to Maria Carla Galavotti. Erkenntnis, 26,369-376.

Maximizing freedom of decision: an axiomatic analysis. In G. R. Feiwel (Ed.), Arrow and the Foundations of the Economic Policy. Washington Square, New York: New York University Press, 243-254.

Axiomatic theories. In J. Eatwell, M. Milgate, and P. Newman (Eds.), The New Palgrave: A Dictionary of Economics, Vol. 1. New York: Stockton Press, 163-1 65.



With C. CRANGLE and S. MICHALOWSKI. Types of verbal interaction with instructable robots. In G. Rodriguez (Ed.), Proceedings of the Workshop on Space Telerobotics, Vol. II (JPL Publication 87-1 3). Pasadena, CA: NASA, Jet Propulsion Laboratory, California Institute of Technology, July 1, 393402.

With C. CRANGLE. Context-fixing semantics for instructable robots. International Journal of Man-Machine Studies, 27, 371-400.


Estudios deJilosoJia y metodologia de la ciencia. Alianza Universidad, S.A., Madrid, 250 pp.

Lorenz curves for various processes: A pluralistic approach to equity. Social Choice and Welfare, 5, 89-101. Reprinted in W. Gaertner and P.K. Pattanaik (Eds.), Distributive Justice and Inequality. Berlin Hei- delberg: Springer-Verlag, 1-1 3.

Empirical structures. In Erhard Scheibe (Ed.), The Role of Experience in Science, Proceedings of the 1986 Conference of the Academic Interna- tionale de Philosophie des Sciences (Bruxelles). Held at the University of Heidelberg. Berlin, New York: Walter de Gruyter, 23-33.

Advice to graduate students. In Proceedings and Addresses of The American Philosophical Association, 62, 266-268.

Comment: Causality, Complexity and Determinism. Statistical Sci- ence, 3, 398-403.

Representation theory and the analysis of structure. Philosophia Natu- ralis, 25: 3-4, 254-268.

PHYSICS

Probabilistic causality in space and time. In B. Skyrms and W. L. Harper (Eds.), Causation, Chance, and Credence, Dordrecht: Kluwer, 135-151.


LANGUAGE AND LOGIC

Philosophical implications of Tarski's work. Journal of Symbolic Logic, 53,80-91.

With C. CRANGLE. Context-fixing semantics for the language of action. In J. Dancy, J.M.E. Moravcsik, and C. C. W. Taylor (Eds.), Human Agency: Language, Duty, and Value Stanford, CA: Stanford University Press, 47-76,288-290.


With C. CRANGLE, L. LIANG, and M. BARLOW. Using English to instruct a robotic aid: an experiment in an office-like environment. In Proceedings of the International Conference for the Association for the Advancement of Rehabilitation Technology, 25-30 June 1988,466-467, Montreal.

Lo sviluppo dell'apprendimento computerizzato. In Vittorio Pranzini and Donatella Mazza (Eds.), Prof Computer va a scuola. progetto del comune di Ravenna per l'introduzione dell'informatica nella scuola. Ravena: Editrice Diamond Byte, 64-7 1.

Computer-assisted instruction. In Derick Unwin and Ray McAleese (Eds.), The Encyclopaedia of Educational Media Comnzunications and Technology (2nd Edition), New York: Greenwood Press, 107-1 16.


With D. H. KRANTZ, R. D. LUCE, and A. TVERSKY. Foundations of Measurement, Vol. II. Geometrical, Threshold, and Probabilistic Representations. New York: Academic Press, 493 pp.

With M. Z A N O ~ I . Conditions on upper and lower probabilities to imply probabilities. Erkenntnis, 31, 323-345.

Philosophy and the Sciences. In W. Sieg (Ed.) Acting and Reflecting Dordrecht: Kluwer Academic Publishers, 3-30.


Review of Duncan, O.D., Notes on social measurement: Historical and critical. Journal of Oficial Statistics, 5, 299-3 1 1.

PSYCHOLOGY

Problems of axiomatics and complexity in studying numerical compe- tence in animals. (CommentaryIDavis and Pkrusse: Animal counting.) Behavioral and Brain Sciences, 11,599.

Current directions in mathematical learning theory. In E. E. Roskam (Ed.) Mathematical Psychology in Progress. Berlin, Heidelberg: Springer-Verlag, 3-28.

LANGUAGE AND LOGIC

With S. TAKAHASHI. An interactive calculus theorem-prover for conti- nuity properties. Journal of Symbolic Computation, 7, 573-590.

With C. CRANGLE. Geometrical semantics for spatial prepositions. In P.A. French, T.E. Uehling, Jr., and H. K. Wettstein (Eds.) Midwest Studies in Philosophy, XIV, Notre Dame, IN: University of Notre Dame Press, 399-422.

Commemorative Meeting for Alfred Tarski, Stanford University, November 7, 1983. In Peter Duren (Ed.), A Century of Mathematics in America, Part III. Providence, RI: American Mathematical Society, 395-396.


Computers at Stanford: An Overview. In Dr. C. Calude, Dr. D. Chitoran and Dr. M. Malitza (Eds.), New Information Technologies in Higher Education Bucharest: European Centre for Higher Education, 97-111.



With D. H. KRANTZ, R. D. LUCE, and A. TVERSKY. Foundations of Measurement,Vol. 111: Representation, Axiomatization, and Invariance. New York: Academic Press, 356 pp.

On deriving models in the social sciences. Journal of Mathematical and Computer Modelling, 14, 21-28.

PSYCHOLOGY

Eye-movement models for arithmetic and reading performance. In E. Kowler (Ed.), Reviews of Oculomotor Research, Vol. IV Eye Movements and Their Role in Visual and Cognitive Processes, New York, Elsevier, 455-475.

Problems of extension, representation, and computational irreducibility. (Commentary1 Hanson and Burr: What connectionist models learn: Learning and representation in connectionist networks.) Behavioral and Brain Sciences, 13,507-508.

PHYSICS

Probabilistic causality in quantum mechanics. Journal of Statistical Planning and Inference, 25, 293-302.

LANGUAGE AND LOGIC

With R. CHUAQUI. An equational deductive system for the differential and integral calculus. In P. Martin-Lof and G. Mints (Eds.) Lecture Notes in Computer Science, Proceedings of COLOG-88 International Conference on Computer Logic, held in Tallinn, USSR. Berlin Heidel- berg: Springer Verlag, 25-49.



With C. CRANGLE. Robots that learn: A test of intelligence. Revue Internationale de Philosophie, 44, No. 172, 5-23.

With C. CRANGLE. Instruction dialogues: Teaching new skills to a robot. In G. Rodriguez and H. Seraji (Eds.), Proceedings of the NASA Conference on Space Telerobotics, Vol. V , JPL Publication 89-7, Vol. V. Pasadena, CA: NASA, Jet Propulsion Laboratory, California Institute of Technology, January 31,91-101.

Three current tutoring systems and future needs. In C. Frasson, and G. Gauthier (Eds.), Intelligent Tutoring Systems: At the Crossroads of ArtiJicial Intelligence and education. Norwood, NJ: Ablex Publishing Corporation, 25 1-265.

Uses of artificial intelligence in computer based instruction. In V. Marik, 0. Stepankova and Z. Zdrahal (Eds.), Artijicial Intelligence in Higher Education. Springer Verlag, 206-225.

Intelligent tutoring, but not intelligent enough. (ReviewIH. Mandl and A. Lesgold (Eds.): Learning issues for intelligent tutoring systems; and J. Psotka, L. D. Massey, and S. A. Mutter (Eds.): Intelligent tutoring systems: Lessons learned.) Contemporary Psychology, 35,648-650.


Metaphysics V: Probabilistic Metaphysics. In H. Burkhardt and B. Smith (Eds.), Handbook of Metaphysics and Ontolog-y, Vol. 2: L-2. Munich; Philadelphia; Vienna: Philosophia (Analytica), 546-548.

Rules of proportion in architecture. Midwest Studies in Philosophy, 16, 352-358.

PSYCHOLOGY

Can psychological software be reduced to physiological hardware? In E. Agazzi (Ed.), The Problem of Reductionism in Science. Dordrecht:


Kluwer Academic Publishers, 183-198.

The principle of invariance with special reference to perception. In 9. Doignon and J. Falmagne (Eds.), Mathematical Psychology: Current Developments. New York: Springer Verlag, 35-53.

PHYSICS

Indeterminism or instability, Does it matter? In G. G. Brittan, Jr. (Ed.), Causality, Method, and Modality. Kluwer Academic Publishers, 5-22.

With M. ZANOTTI. New Bell-type inequalities for N > 4 necessary for existence of a hidden variable. Foundations of Physics Letters, 4, 1, 101-107.

With M. Z A N O ~ I . Existence of hidden variables having only upper probabilities. Foundations of Physics, 21, 12, 1479-1499.

LANGUAGE AND LOGIC

Language for Humans and Robots. Oxford: Blackwell, 4 17 pp.

Definition 11: Rules of definition. In H. Burkhardt and B. Smith (Eds.), Handbook of Metaphysics and Ontology, Vol. 1: A-K. Munich; Philadelphia; Vienna: Philosophia (Analytica), 204-208.


With M. ZANO'ITI. Qualitative axioms for random-variable representation of extensive quantities. In C. W. Savage and P. Ehrlich (Eds.), Philosophical and Foundational Issues in Measurement Theory. Hills- dale, NJ, Lawrence Erlbaum, 39-52.

Axiomatic methods in science. In Marc E. Carvallo (Ed), Nature, Cognition and System 11, Dordrecht: Kluwer Academic Publishers, 205-232.


PSYCHOLOGY

Estes' statistical learning theory: Past, present, and future. In A.F. Healy, S.M. Kosslyn and R.M. Shiffrin (Eds.), From Learning Theory to Connectionist Theory: Essays in Honor of William K. Estes, Volume 1, Hillsdale, NJ: Lawrence Erlbaum, 102-1 27.

Problem spaces, language and connectionism: Issues for cognition. (Commentary1 A. Newell: Unified theories of cognition). Behavioral and Brain Sciences, 15, 3 ,457458.

LANGUAGE AND LOGIC

With L. LIANG and M. B O E ~ N E R . Complexity issues in robotic machine learning of natural language. In L. Lam and V. Naroditsky (Eds.), Modelling Complex Phenomena, New York: Springer Verlag, 102-127.

With M. B O E ~ N E R and L. LIANG. Comprehension grammars generated from machine learning of natural language. In P. Dekker and M. Stokhof (Eds), Proceedings of the Eighth Amsterdam Colloquium, Institute for Logic, Language and Computation, University of Amsterdam, 93-1 12.


Instructional computers: Past, present, and future. International Jour- nal of Educational Research, 17,5-17.


Models and Methods in the Philosophy of Science: Selected Essays. Dordrecht: Kluwer Academic Publishers, 525 pp.

PHYSICS

The transcendental character of determinism. Midwest Studies in Philosophie, 18, 242-257.


TO APPEAR


With N. ALY OSHINA. The definability of the qualitative independence of events in terms of extended indicator functions. Journal of Mathe- matical Psychology.

PSYCHOLOGY

With S. TAKAHASHI. Hierarchical Learning of Boolean Functions. Syn- these. Vladimir Smirnov Festschrift.

PHYSICS

With A. DE BARROS. A random walk approach to interference. Inter- national Journal of Theoretical Physics.

LANGUAGE AND LOGIC

With C. CRANGLE. Instructable Robots. Stanford, CA: Center for the Study of Language and Information.

With R. CHUAQUI. A finitarily consistent free-variable positive fragment of infinitesimal analysis. Proceeding of the 9th Latin American Symposium on Mathematical Logic, held at Bahia Blanca, Argentina, August 1992.

With R. CHUAQUI. Variable-free axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof. Journal of Symbolic Logic.

NAME INDEX

Alyoshina, N. 323 Anderson, J. 90, 102 enliker, J. 31 1, 312 Aqvist, L. 217 Aristotle 100 Atkinson, R.C. 174, 176, 188, 278,279,303

Barlow, M. 317 Barros, de, A. 323 Bayes 260 Beard, M. 304 Belnap, N.D. 90, 102, 204, 205, 212, 2 13 ff.,

217 Beltrametti, E. 149, 163 Bentham, J. 260 Benthem, van, J. 30, 34 Binford, F. 285 Biro 13 Boettner, M. 322 Bogdan, R. 262 Bolzano 6 Boole, G.H. 24, 34 Bottner, M. 19-39,68,70 Bower, G.H. 178, 188 Bratman, M. 227,238 Brian, D. 291 Brink, C. 20, 21, 34 Brooks, R.A. 234, 238 Brouwer, L.E.J. 159 Busch, P. 149, 163 Bush, R.R. 172, 176, 188,290 Butler, C. 294

Cain, J. 99, 118, 120 Campbell, J.A. 189 Carlsmith, J.M. 281 Carnap, R. 8,93, 106, 118, 120 Carroll, J.B. 300 Cartwright, N. 258 Cassidy, D. 140 Cassinelli, G. 149, 163 Cattaneo, G. 149, 163 Chellas, B.F. 215, 217 Chiara, dalla, M.L. 90, 1 18, 120, 147-167 Chierchia, G. 63, 67 Chisholm, R. 217 Chuaqui, R. 319,323

Church, A. 6, 14, 15 Cleave, J.P. 108, 109, 119, 120 Cohen, M. 31 1,312 Cornrie, B. 228, 238 Crangle, C. 32, 34, 35, 85, 87, 223-25 1, 316,

3 17,318, 320,323 Crothers, E. 284, 288 Curry 69 Czermak, J. 119, 120

Damnjanovic, J. 99, 1 18, 120 Danforth, D.G. 309 Davidson, D. 71, 84,227, 238,276, 277 Davies, E.B. 149, 163 Dewey, J. 248 Dirac, P.A.M. 140 Dishkant, H. 163 Donio, J. 289 Dorn, G. 120, 121 Dunn, M. 105, 119, 120

Eells, E. 258 Eijck, van, J. 63, 67 Estes, W.K. 171-192,278,298 Euclid 7 Evans, G . 54,58, 67

F@llesdal, D. 3-1 8 Faltz, L.M. 23, 24, 34 Feldman, S. 295 Finetti, de, B. 252, 254, 255 Fletcher, J.D. 297, 300, 301, 302, 303 Floyd, R. 31 1, 312 Fortune, R.F. 314 Frankmann, R. W. 280,283 Freed, A.F. 228,238 Frege, G. 5, 6, 8, 76 Fresnel265 Friend, J. 302,304,305

Gabbay, D.M. 72, 84 Galavotti, M.C. 245-270 Gillet, E. 120 Ginsberg, R. 266,270,282 Giuntini, R. 147-1 67 Givant, S. 69, 70 Gluck, M.A. 189

NAME INDEX

Gochet, P. 120 Gochet-Gillet 1 18 Godel, K. 99, 100 Goldberg, A. 296, 303 Goldblatt, R.I. 164 Goldman, A. 218 Good, I.J. 257 Goodman, N. 94, 106, 1 18, 120 Goos, G. 6 1,68 Gramlich, C. 310 Greechie 148 Grice 12 Groen, G. 285,286,289 Groenendijk, J. 41, 63, 67 Gudder 148, 154 Guthrie 190

Halonen, I. 125-145 Hansen, D. 286 Hatsopoulis, N. 189 Hawley, N. 279,280 Heisenberg, W. 140, 25 1 Hempel, G. 94, 11 8, 120, 129 Henkin, L. 99, 118, 120 Hesse, M. 93, 106, 118, 120 Heuvel, van den, R. 303 Hilbert, D. 7, 41, 156 Hill, S. 282, 284 Hintikka, J. 10, 15, 35, 125-145,263 Holmstrom-Hintikka, G. 21 8 Hull, J. 190 Hume, D. 191 Hurwitz, J.B. 189 Hyman, L. 289

Ihrke, C. 289, 294

James, W. 248 Jamison, D. 294,300,303 JaSkowski 104 Jauch 149 Jerman, M. 289,291,293, 294 Jordan, E. 140

Kanerva, L. 307 Kanger, S. 218 Kant, I. 15, 101 Karsh, E. 283 Keenan, E.L. 21, 23, 30,34 Kleene, St.C. 99, 100, 1 18, 120 Klein, F. 7 Korner, S. 108 f., 119, 120 Krantz, D.H. 270,295, 317,3 19

Krasne, F. 280 Kraus, K. 149, 164 Kreisel, G. 1 18, 120 Kripke, S. 9, 15, 100, 118, 120 Laddaga, R. 310, 31 1, 312 Lahti, P. 149, 163 Lamperti, J. 278, 279 Larsen, I. 305 Latombe, J. 238 Laudisa, F. 163 Leblanc, H. 119, 120 Leibniz, von, G.F.W. 101, 149 LkveillC, M. 299, 302, 306 Levine, M. 283 Levine, A. 310 Lhamon, D. 294 Liang, L. 37,317,322 Lindenbaum, A. 103 Link, G. 34 Loftus, E.F. 293, 296 Lorton, P. 300 Lowenheim 69 Luce, R.D. 181, 189,270,284,290,291,

295,298,317,319 Ludwig, G. 149, 164

Maas, R.E. 313,314 Macken, E. 21,303,305,306,307,308 Mackey, G.W. 144, 145, 148,155, 157, 164 Malone, T.W. 307 Marino, G. 163 Markosian, L.Z. 305 Mates, B. 9, 15 Matthews, P. 305 McDonald, J. 3 13 McKinsey, J.C.C. 275,276 McKnight, B. 281 Medin, D.L. 180, 189 Meserve, B. 293,294 Michalowski, S. 238, 316 Miller, D. 98, 118, 120 Mittelstaedt, P. 119, 120, 121, 149, 163 Moler, N. 291 Montague, R. 71, 85 Moravcsik, J. 7 1-87 Morgan, de, A. 21 Morningstar, M. 292, 294, 296 Morscher, E. 84 Mosteller, F. 172, 176, 188 Moulines, C.U. 263

Nagel, E. 248 Neumann, von, J. 142, 143

NAME INDEX

Nida, E. 58, 67 Nilsson, N.J. 238 Niniiluoto, I. 98 Nistico, G. 163 Nosofsky, R.M. 180, 183, 189

Parry, W.T. 105, 119, 121 Peano, G. 27 Peirce, C.S. 21, 69,248 Pelletier, F.J. 72, 85 Perloff, M. 204, 205, 212, 2 13 ff., 2 17 Piron 149 Poisson 265 Popper, K. 90, 94, 98, 106, 107, 114, 118,

121 Porn, I. 2 18 Prior, A.N. 1 18, 12 1 PrugoveEki 149 Pt&, P. 149, I64 PulmannovS, S. 149, 164 Purdy, W.C. 34,41 Putnam, H. 81

Quine, W.V.O. 6, 9, 15

Ramsey, F.P. 255 Reichenbach, H. 247,257 Restle, F. 176, 189 Riguet, J. 21, 34 Robbins, D. 176, 190 Roberts, F.S. 289 Rogosa, D.R. 309 Rosenthal-Hill, I. 29 1 Ross, A. 97, 106, 118, 121 Rottmayer, W. 299 Rouanet, H. 283 Royden, H.L. 278 Rubin, H. 276

Sachar, J. 307 Salmon, W.C. 257 Sanders, W.R. 310 Sandu, G. 193453 Schaffer, M.M. 180, 189 Schlag-Rey, M. 281, 282, 285, 286, 291 Schoeffler, M.S. 176, 190 Schonfinkel69 Schroder 69 Schurz, G. 105,113,114, 118, 119, 121 Schiitte, K. 1 16, 121 Scott, D. 277 Searle, B. 300, 301, 302, 303, 304, 305 Sears, P. 293,294

Sheehan, J. 309,310 Siegel, S. 277 Slater, B.H. 55, 56, 57,60, 66,67 Smith, R. 299, 304, 306 Sneed, J. 263 Sommers, M. 54. 55 Spanos, A. 263 Spohn, W. 258 Stokhof, M. 41,58,63,67 Straughan, J.H. 172, 189 Sugar, A.C. 275 Suppes, P. 19,21, 22, 23, 27, 28, 31, 32, 34,

41, 67, 78, 1 18, 263, 303 attitude to logic 89 comments on Bottner 335 comments on Crangle 238 comments on Dalla Chiara and Giuntini

164 comments on Estes 190 comments on F@ilesdal 3 15 comments on Galavotti 264 comments on Hintikka and Halonen 143 comments on Moravcsik 85 comments on Purdy 68 comments on Tuomela and Sandu 218 comments on Weingartner 12 1

computers and education 279, 280, 28 1, 282, 283, 284, 285, 287, 289,291, 292, 294, 295, 296, 298, 300, 301, 302, 304,305,307,308,310,311,312, 313, 314, 315, 316, 317, 318, 320, 322

language and logic 270,279,284,285, 292, 294, 295, 297, 299, 302, 305, 306, 308, 309, 311, 312, 313, 3.15, 317,318,319,321,322,323

logical concepts 125 methodology, probability and

measurement 275, 276,277, 278, 279, 280, 281, 282, 284, 286, 288, 290, 292, 293, 295, 297, 298, 300, 301, 303, 306, 307, 308, 310, 312, 314, 315, 316, 317, 319, 320, 321, 322,323

philosophy of language, contribution to 3-18

philosophy of science 245-270 physics 275, 276, 279, 281, 283, 287,

289, 296, 299, 302, 307, 309, 313, 316,319,321,323

postscript 273 psychology 278,279,280,281,282,

283, 285, 286, 288, 290, 292, 294,

328 NAME INDEX

296, 297, 298, 301, 302, 304, 306, 309, 311, 312, 314, 318, 319, 320, 322,323

award to 3 quantum probability views 147

Takahashi, S. 3 18, 323 Takeuti, G. 1 18, 120 Tarski, A. 5, 16,20, 35, 69, 70, 72, 75,78,

81, 83, 85 Thorndike 190 Tichy, P. 94, 107, 1 18, 121 Tolman 190 Tuomela, R. 193453 Tversky, A. 270,295,317,319

Uhl, C.N. 176, 190 Underwood, B.J. 178, 190

Varadarajan 148 Vendler, Z. 228, 238

Vries, de, F.-J. 63, 67

Waite, W. 6 1, 68 Walsh, K. 278 Warren, H. 30 1,305 Weingartner, P. 89-1 24 Weir, R. 284 Wells, S. 300 Wilson, G.M. 218 Winet, M. 276 Woods, W.A. 21,22,35 Wright, von, G.H. 218 Wronski, A. 9 1

Zadeh, L. 159 Zanotti, M. 2 1, 22, 35, 166, 167, 252,299,

301, 302, 303, 305, 307, 308, 310, 31 1,313,317,321

Zinnes, J.L. 28 1, 282, 286

SUBJECT INDEX

A-criterion 102 shortcomings 1 13

A-relevance 10 1 A-restriction, implication 103 A* criterion 105 action, generated 198 ff.

language of 5 performing 194 ff. property and execution 227 seeing-to-it-that-something-is-done

193-453. See also stit semantics of 37

activity, intentional 193 addition principle 93 additivity 150 agency 194 ff. algebra, event 148

extended relation 20 quantum mechanical 148

algorithm, translation 60-63 analysandum 196 analysis, cut free 99 analyticity 9 anaphora, logic for, variable free 4 1-70 argument, negative 72-76 array similarity model 180-1 84

theory 171-192 assent, profile 11 avoidability 208

Bayesianism 252-254 behavior, primitive 234 belief, coherent 10

context, personal 11 defining vs. factual 9

CanStitA Q 210 cardinality 27 Carnap paradox 106 causality 25 1, 257-260

experiments and 264 probabilistic 257 f.

cause, determinant 251 certainty domain 152 change, objective 255 characteristic function, fuzzy 158 child, language 5

closure under substitution 102 under transitivity of implication 103

coding, logic results depend on 99 cognitive capacity, language and 71 collectivity 27 command satisfaction, habits and 223-251 commitm~nt paradox 97 complementarity 154 computer, language for 5 conditional sentence 56 f. confirmation paradox (Hesse) 93, 106 congruence, extensional 8

geometrical and meaning 7 intensional 8 latency 10 M 8 meaning and 122

conjunction, dynamic 64 continuation, possible 64 contraposition, nonclosure under 103 contrast dependency 75 control group, experiments with 265 covering law theorem 128 f., 132, 133, 136,

139, 141

deduction theorem 49 definability theory 126 denotation, meaning and 81 deontic logic, paradoxes in 96 description 41

generic 42, 59, 61 proper 42,61 representative 42,59, 61

determinism 25 1 indeterminism vs. 269 weak 151

discontinuity, identification and 139 f. disposition predicates paradox (Carnap) 93 distribution 58 DO 193 DO,A (8R) 200 DO,AR 197 DOA(P,R) 198 DOA R 196

effect, semitransparent 159 empiricism 246-248

SUBJECT INDEX

logical 250 probabilistic 245 f.

epistemic logic, paradoxes 96 paradox 106

epsilon-operator (Hilbert) 41,55, 68 Erlanger program 7 event, compatibility between 153

orthogonal 148 probability, restriction on 185 sharp and unsharp 152 single, probability of 268

exfalso quodlibet 93, 96, 97 experiment, causality and 264

control group 265 measurement 266 observational 267 testing, theoretical model 265

explanation 253 explanatory schema, meaning as 85 extensionality 149

feedback, command 236 fuzzification (Mackey) 157

generic indefinite 56 geometry, meaning vs. 17 global similarity I 8 1 Godel number 99, 100 Goodman paradox 94, 106 grammar, attribute 61

relational, definitions 20-22 problems in 19-39 transformational 23

habit 233 f. command satisfaction and 223-25 1

Henkin sentence 99 hidden variable problem 140-143

theorem 14 1 HiIbert space I56

fuzzy 158 ff.

identifiability 130 theory, quantum logic and 130 f.

identification, discontinuity and 139 f. logic of 131-137

implication, analytic 105 indeterminism 255, 256 f.

determinism vs. 269 index, relative, semantic 77 inquiry, interrogative model 127 f. intention, expected 219

varieties of 2 19

invariance 247 isomorphism 247

K-criterion, shortcomings 1 13 K-restriction 108 ff.

applications 1 12 ff.

language, acquisition, robot 5 semantics vs. syntax 5

language, children's 5 expressive power of 7 1 learning 86 logic results depend on 99 philosophy of, Suppes contribution to

3-1 8 C N , abbreviations 44 f. C N , anaphoric pronouns in 53-60 C N , axiomatization 46-53 .C N , definition 4 2 4 6 C N , metavariables 44 f. C N , proper interpretation 46 C N , semantics 43 f. C N , singular expressions 46 . C N , syntax 42 f. natural, nature of 7 1-87 passim

lattice, orthomodular, quantum logic as 139 learning 185, 233 f.

model, linear, probability matching and 172 f.

parameter 185 theory, mathematical 17 1

Lindenbaum lemma 50 LISP 37 localization, macroscopic 157 logic, applied, Aristotelean 100 ff.

classical, limitations to 89-124 problems with 92-1 00 restriction, reasons for 92 deontic, paradoxes in 96 epistemic, paradoxes 96 event 148 first order, skepticism about 336 identification 13 1-137 in quantum mechanics 147-167 language dependency of 99 quantum 90, 108, 137 f., 148, 160-163 identification logic 125-1 45 identifiability theory and 130 f. orthomodular lattice 139 weakness of 220

logical equivalence of utterances 8

mass term 72

SUBJECT INDEX

matrix, finite, logic representation by 103 meaning, congruence and 4 f., 122

denotation and 8 1-84 explanatory schema 85 intersubjective 6 primacy of 16

measurement 248,251,252,266 mereology, set theory vs. 72 mind, formal semantics and 78-81 modal operator 41 model 247,259

learning 17 1 plurality 249 theoretic, experiment test 265

modus ponens, closure under 105 nonclosure under 103

monotonicity 66 Montague grammar 22 f.

dynamic 41,63 Morgan, de, law 97 multireferentiality 77

necessity operation 152 non-contradiction, logical principle 91, 15 1 noun, common, universally quantified 58 f.

proper 22-26 number 26-29

obligation, derived, paradox 97 observable, compatibility and

complementarity between 154 joint (Gudder) 154 primitive 155 spectral 153

observation, experimental 267 operator, Boolean 21,21 orthogonality 150 ortholattice, orthomodular 151 orthoposet, orthomodular 148, 15 1 outcome, specific 229, 240

paradox, logical 92 ff. pattern model 186 philosophy of language, Suppes contribution

to 3-18 plural, bare 28

singular 27 pluralism 248, 250, 251, 259 poser, Brouwer-Zadeh 159 positivism, logical 245 f., 261 pragmatism 246, 248-25 1 predicate, set theoretical 247 probabilistic approach 25 1

probability 246, 252 distribution, joint 154 in quantum mechancis 147-167 matching, linear learning model and

172 f. nonmonotonic upper 166 objective 254-256 qualitative 164 single event 268 statement 253 theory, subjective 252

problem solving, science as 268 process judgment 23 1 f.

semantics 225 ff. product rule 180 pronoun 29-3 1

A-type 55 f. anaphoric 41,53-60 B-type 54, 59 f., 60 E-type 54 f., 60 EP-type 58 mixed type 60

propensity 269 proposition 85

synonymy and 15 propositional calculus,

Aristotelean limitation 101, 105 applications 106 ff.

prosody 11

quantification, universal 58 f. quantum mechanics, logic and probability in

147-167 quantum theory, logician's approach 126 question 155

R 1 -criterion 1 14 R1 -restiction, generalization 1 15 R 1 g-restriction 1 I7 R 1 G-restriction 1 16 randomness 251,252,254,256, 257 rationality 250, 260 f. reductionism 250, 252 redundancy (Korner) 108 representation, skepticism 240

theorem 247 f., 254 response probability 185 result semantics 228 ff. robot 223

intelligent, design 234 language for 4 verbal instruction understanding 5

Ross paradox 97, 106

332 SUBJECT INDEX

Samaritan paradox 97 sampling 185 satisfaction condition 224, 233 f. satisfiability theorem 50 science, philosophy, logical paradoxes in 95

problem solving 268 semantics, formal, mind and 78-81

repairs to 76-78 variable free 5

semisingular 57 sentence, conditional 6 1 set theory, mereology vs. 72

variable free 69 ZF 122

sigma-additivity 15 1 similarity 19 1

global 18 1 singular, plural and 27 snow, constituents of 73

mass term 72 non-white 74 whiteness of 7 1-87

soundness theorem 47 stability 256 state-event structure 149-1 60 stimulus sampling theory 17 1-192

revision of 176-180 variants 174

stit, seeing-to-it-that 193-453, esp. 201 ff. See also action

game theoretical approach 207

StitA Q 2 10 StitAR 214 structure, congruence relations for 12 syllogistics, Aristotelean 100 synonymy 9

propositions and 15 syntax 5

tableau 127 task planner 234 tautology, surface 10 theory 247 f., 249

confirmation, deductive 93 descriptive vs. normative 21 8 physical 254 scientific 246, 255

transformation, in science 107 translation 60-63

uncertainty principle (Heisenberg) 14 1 , 25 1 universal generalization theorem 49 utterance, analysis of meaning 224

equiprobable 7

validity 91 value theory, paradoxes of 97 verb 31-33 verisimilitude 106

definitions 98 paradox 94

TABLE OF CONTENTS

Volume 1 : Probability and Probabilistic Causality

PAUL HUMPHREYS / Introduction

PART I: PROBABILITY

KARL POPPER and DAVID MILLER / Some Contributions to Formal Theory of Probability / Comments by Patrick Suppes

PETER J. HAMMOND 1 Elementary Non-Archimedean Representations of Probability for Decision Theory and Games / Comments by Patrick Suppes

ROLANDO CHUAQUI / Random Sequences and Hypotheses Tests / Comments by Patrick Suppes

ISAAC LEVI / Changing Probability Judgements / Comments by Patrick Suppes

TERRENCE L. FINE / Upper and Lower Probability / Comments by Patrick Suppes

xiii

PHILIPPE MONGIN 1 Some Connections between Epistemic Logic and the Theory of Nonadditive Probability / Comments by Patrick Suppes 135

WOLFGANG SPOHN / On the Properties of Conditional Independence / Comments by Patrick Suppes 173

ZOLTAN DOMOTOR / Qualitative Probabilities Revisited / Comments by Patrick Suppes 197

JEAN-CLAUDE FALMAGNE / The Monks' Vote: A Dialogue on Unidimensional Probabilistic Geometry / Comments by Patrick Suppes 239

334 TABLE OF CONTENTS TO VOLUME 1

PART 11: PROBABILISTIC CAUSALITY

PAUL W. HOLLAND 1 Probabilistic Causation Without Probability / Comments by Patrick Suppes

I. J. GOOD / Causal Tendency, Necessitivity and Sufficientivity: An Updated Review / Comments by Patrick Suppes

ERNEST W. ADAMS / Practical Causal Generalizations / Comments by Patrick Suppes

CLARK GLYMOUR, PETER SPIRTES, and RICHARD SCHEINES / In Place of Regression / Comments by Patrick Suppes

D. COSTANTINI / Testing Probabilistic Causality / Comments by Patrick Suppes

PAOLO LEGRENZI and MARIA SONINO 1 Psychologistic Aspects of Suppes's Definition of Causality / Comments by Patrick Suppes

Name Index

Subject Index

Table of Contents to Volumes 2 and 3

TABLE OF CONTENTS

Volume 2: Philosophy of Physics, Theory Structure,

and Measurement Theory

PART 111: PHILOSOPHY OF PHYSICS

BARRY LOEWER 1 Probability and Quantum Theory / Comments by Patrick Suppes

ARTHUR FINE / Schrodinger's Version of EPR, and Its Problems / Comments by Patrick Suppes

JULES VUILLEMIN / Classical Field Magnitudes / Comments by Patrick Suppes

BRENT MUNDY / Quantity, Representation and Geometry / Comments by Patrick Suppes

PAUL HUMPHREYS / Numerical Experimentation / Comments by Patrick Suppes

PART IV: THEORY STRUCTURE

RYSZARD WOJCICKI 1 Theories and Theoretical Models / Comments by Patrick Suppes 125

N. C. A. DA COSTA and F. A. DORIA / Suppes Predicates and the Construction of Unsolvable Problems in the Axiomatized Sciences / Comments by Patrick Suppes 151

JOSEPH D. SNEED 1 Structural Explanation / Comments by Patrick Suppes 195

PART V: MEASUREMENT THEORY

R. DUNCAN LUCE and LOUIS NARENS / Fifteen Problems Concerning the Representational Theory of Measurement /

336 TABLE OF CONTENTS TO VOLUME 2

Comments by Patrick Suppes 219

FRED S. ROBERTS and ZANGWILL SAMUEL ROSENBAUM / The Meaningfulness of Ordinal Comparisons for General Order Relational Systems 1 Comments by Patrick Suppes 25 1

C. ULISES MOULINES and Jose A. D ~ E Z 1 Theories as Nets: The Case of Combinatorial Measurement Theory / Comments by Patrick Suppes 275

Name Index 301

Subject Index 305

Table of Contents to Volumes 1 and 3 311

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65. H. E. Kyburg, Jr., The Logical Foundations of Statistical Inference. 1974 ISBN 90-277-0330-2; Pb 90-277-0430-9

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73. J. Hintikka (ed.), Rudolf Carnap, Logical Empiricist. Materials and Perspectives. 1975" ISBN 90-277-0583-6

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122. J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic. 1979 ISBN 90-277-0879-7

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129. M. W. Wartofsky, Models. Representation and the Scientific Understanding. [Boston Studies in the Philosophy of Science, Vol. XLVIII] 1979

ISBN 90-277-0736-7; Pb 90-277-0947-5 130. D. Ihde, Technics and Praxis. A Philosophy of Technology. [Boston Studies in the

Philosophy of Science, Vol. XXIV] 1979 ISBN 90-277-0953-X; Pb 90-277-0954-8 13 1. J. J. Wiatr (ed.), Polish Essays in the Methodology of the Social Sciences. [Boston

Studies in the Philosophy of Science, Vol. XXIX] 1979 ISBN 90-277-0723-5; Pb 90-277-0956-4

132. W. C. Salmon (ed.), Hans Reichenbach: Logical Empiricist. 1979 ISBN 90-277-0958-0

133. P. Bieri, R.-P. Horstmann and L. Kriiger (eds.), Transcendental Arguments in Science. Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5

134. M. MarkoviC and G. PetroviC (eds.), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of Science, Vol. XXXVI] 1979 ISBN 90-277-0727-8; Pb 90-277-0968-8

135. R. Wdjcicki, Topics in the Formal Methodology of Empirical Sciences. Translated from Polish. 1979 ISBN 90-277- 1004-X

136. G. Radnitzky and G. Andersson (eds.), The Structure and Development of Science. [Boston Studies in the Philosophy of Science, Vol. LIX] 1979

ISBN 90-277-0994-7; Pb 90-277-0995-5 137. J. C. Webb, Mechanism, Mentalism and Metamathematics. An Essay on Finitism.

1980 ISBN 90-277- I 046-5 138. D. F. Gustafson and B. L. Tapscott (eds.), Body, Mirzd and Method. Essays in Honor

of Virgil C. Aldrich. 1979 ISBN 90-277- 1013-9 139. L. Nowak, The Structure of Idealization. Towards a Systematic Interpretation of the

Marxian Idea of Science. 1980 ISBN 90-277- 1014-7 140. C. Perelman, The New Rhetoric and the Humanities. Essays on Rhetoric and Its

Applications. Translated from French and German. With an Introduction by H. Zyskind. 1979 ISBN 90-277- 101 8-X; Pb 90-277- 1019-8

141. W. Rabinowicz, Universalizability. A Study in Morals and Metaphysics. 1979 ISBN 90-277- 1020-2

142. C. Perelman, Justice, Law and Argument. Essays on Moral and Legal Reasoning. Translated from French and German. With an Introduction by H.J. Berman. 1980

ISBN 90-277- 1089-9; Pb 90-277- 1090-2 143. S. Kanger and S. 0hman (eds.), Philosophy and Grammar. Papers on the Occasion

of the Quincentennial of Uppsala University. 1981 ISBN 90-277- 109 1-0 144. T. Pawlowski, Concept Formation in the Humanities and the Social Sciences. 1980

ISBN 90-277- 1096- 1 145. J. Hintikka, D. Gruender and E. Agazzi (eds.), Theory Change, Ancient Axiomatics

and Galileo 's Methodology. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume I. 198 1 ISBN 90-277- 1 126-7

146. J. Hintikka, D. Gruender and E. Agazzi (eds.), Probabilistic Thinking, Ther- modynamics, and the Interactiorz of the History and Philosophy of Science. Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science, Volume 11. 1981 ISBN 90-277- 1 127-5

147. U. Monnich (ed.), Aspects of Philosophical Logic. Some Logical Forays into Central Notions of Linguistics and Philosophy. 198 1 ISBN 90-277-1201-8

148. D. M. Gabbay, Semantical Investigations in Heyting 's Intuitionistic Logic. 1981 ISBN 90-277-1202-6

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149. E. Agazzi (ed.), Modern Logic - A Suwey. Historical, Philosophical, and Mathemati- cal Aspects of Modern Logic and Its Applications. 1981 ISBN 90-277- 1 137-2

150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for Explanatory Principles below the Level of Physics. 1981 ISBN 90-277- 1214-X

151. J. C. Pitt, Pictures, Images, and Coriceptual Change. An Analysis of Wilfrid Sellars' Philosophy of Science. 198 1 ISBN 90-277- 1276-X; Pb 90-277- 1277-8

152. R. Hilpinen (ed.), New Studies in Deontic Logic. Norms, Actions, and the Founda- tions of Ethics. 1981 ISBN 90-277- 1 278-6; Pb 90-277- 1346-4

153. C. Dilworth, Scient$c Progress. A Study Concerning the Nature of the Relation between Successive Scientific Theories. 2nd, rev, and augmented ed., 1986. 3rd rev. ed., 1994

ISBN 0-7923-2487-0; Pallas Pb 0-7923-2488-9 154. D. Woodruff Smith and R. McIntyre, Husserl and Intentionality. A Study of Mind,

Meaning, and Language. 1982 ISBN 90-277- 1 392-8; Pb 90-277- 1730-3 155. R. J. Nelson, The Logic of Mind. 2nd. ed., 1989

ISBN 90-277-28 19-4; Pb 90-277-2822-4 156. J. F. A. K. van Benthem, The Logic of Time. A Model-Theoretic Investigation into

the Varieties of Temporal Ontology, and Temporal Discourse. 1983; 2nd ed., 199 1 ISBN 0-7923- 108 1-0

157. R. Swinburne (ed.), Space, Time and Causality. i983 ISBN 90-277- 1437- 1 158. E. T. Jaynes, Papers on Probability, Statistics and Statistical Physics. Ed. by R. D.

Rozenkrantz. 1983 ISBN 90-277- 1448-7; Pb (1989) 0-7923-0213-3 159. T. Chapman, Time: A Philosophical Analysis. 1982 ISBN 90-277- 1465-7 160. E. N. Zalta, Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983

ISBN 90-277- 1474-6 161. S. Harding and M. B. Hintikka (eds.), Discovering Reality. Feminist Perspectives on

Epistemology, Metaphysics, Methodology, and Philosophy of Science. 1983 ISBN 90-277- 1496-7; Pb 90-277- 1538-6

162. M. A. Stewart (ed.), Law, Morality and Rights. 1983 ISBN 90-277-15 19-X 163. D. Mayr and G. Siissmann (eds.), Space, Time, and Mechanics. Basic Structures of a

Physical Theory. 1983 ISBN 90-277- 1525-4 164. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. I:

Elements of Classical Logic. 1983 ISBN 90-277- 1542-4 165. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. 11:

Extensions of Classical Logic. 1984 ISBN 90-277-1604-8 166. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. 111:

Alternative to Classical Logic. 1986 ISBN 90-277- 1605-6 167. D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic. Vol. IV:

Topics in the Philosophy of Language. 1989 ISBN 90-277- 1606-4 168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic.

1983 ISBN 90-277- 1543-2 169. M. Fitting, Proof Methods for Modal and Intuitionistic Logics. 1983

ISBN 90-277- 1573-4 170. J. Margolis, Culture and Cultural Entities. Toward a New Unity of Science. 1984

ISBN 90-277- 1574-2 17 1. R. Tuomela, A Theory of Social Action. 1984 ISBN 90-277- 1703-6 172. J. J. E. Gracia, E. Rabossi, E. Villanueva and M. Dascal (eds.), Philosophical

Analysis in Latin America. 1 984 ISBN 90-277- 1749-4 173. P. Ziff, Episternic Analysis. A Coherence Theory of Knowledge. 1984

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ISBN 90-277- 175 1 -7 174. P. Ziff, Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984

ISBN 90-277-1773-7 175. W. Balzer, D. A. Pearce, and H.-J. Schmidt (eds.), Reduction in Science. Structure,

Examples, Philosophical Problems. 1984 ISBN 90-277- 18 1 1-3 176. A. Peczenik, L. Lindahl and B. van Roermund (eds.), Theory of Legal Science.

Proceedings of the Conference on Legal Theory and Philosophy of Science (Lund, Sweden, December 1983). 1984 ISBN 90-277- 1834-2

1 77. I. Niiniluoto, Is Science Progressive? 1984 ISBN 90-277- 1835-0 178. B. K. Matilal and J. L. Shaw (eds.), Analytical Philosophy in Comparative

Perspective. Exploratory Essays in Current Theories and Classical Indian Theories of Meaning and Reference. 1985 ISBN 90-277- 1870-9

179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985 ISBN 90-277- 1894-6

1 80. J. H. Fetzer, Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1

18 1. L. Haaparanta and J. Hintikka (eds.), Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. 1986 ISBN 90-277-21 26-2

182. M. Detlefsen, Hilbert's Program. An Essay on Mathematical Instrumentalism. 1986 ISBN 90-277-2 15 1-3

183. J. L. Golden and J. J. Pilotta (eds.), Practical Reasoning in Human AfSairs. Studies in Honor of Chaim Perelman. 1986 ISBN 90-277-2255-2

184. H. Zandvoort, Models of Scientific Development and the Case of Nuclear Magnetic Resonnnce. 1986 ISBN 90-277-2351-6

185. I. Niiniluoto, Truthlikeness. 1987 ISBN 90-277-2354-0 186. W. Balzer, C. U. Moulines and J. D. Sneed, An Architectonic for Science. The

Structuralist Program. 1987 ISBN 90-277-2403-2 187. D. Pearce, Roads to Commensurability. 1987 ISBN 90-277-2414-8 188. L. M. Vaina (ed.), Matters of Intelligence. Conceptual Structures in Cognitive Neuro-

science. 1987 ISBN 90-277-2460- 1 189. H. Siegel, Relativism Rehted. A Critique of Contemporary Epistemological

Relativism. 1987 ISBN 90-277-2469-5 190. W. Callebaut and R. Pinxten, Evolutionary Epistemology. A Multiparadigm

Program, with a Complete Evolutionary Epistemology Bibliograph. 1987 ISBN 90-277-2582-9

19 1. J. Kmita, Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8 192. J. H. Fetzer (ed.), Probability and Causality. Essays in Honor of Wesley C. Salmon,

with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2 193. A. Donovan, L. Laudan and R. Laudan (eds.), Scrutinizing Science. Empirical

Studies of Scientific Change. 1988 ISBN 90-277-2608-6 194. H.R. Otto and J.A. Tuedio (eds.), Perspectives on Mind. 1988

ISBN 90-277-2640-X 195. D. Batens and J.P. van Bendegem (eds.), Theory and Experiment. Recent Insights

and New Perspectives on Their Relation. 1988 ISBN 90-277-2645-0 196. J. Osterberg, Selfand Others. A Study of Ethical Egoism. 1988

ISBN 90-277-2648-5 197. D.H. Helman (ed.), Analogical Reasoning. Perspectives of Artificial Intelligence,

Cognitive Science, and Philosophy. 1988 ISBN 90-277-27 11-2

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198. J. Wolenski, Logic and Philosophy in the Lvov- Warsaw School. 1989 ISBN 90-277-2749-X

199. R. Wbjcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6

200. J. Hintihka and M.B. Hintikka, The Logic of Epistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041 -6

201. E. Agazzi (ed.), Probability in the Sciences. 1988 ISBN 90-277-2808-9 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 ISBN 90-277-28 14-3 203. R.L. Tieszen, Mathematical Intuition. Phenomenology and Mathematical

Knowledge. 1989 ISBN 0-7923-01 31 -5 204. A. Melnick, Space, Time, and Thought in Kant. 1989 ISBN 0-7923-01 35-8 205. D.W. Smith, The Circle of Acquaintance. Perception, Consciousness, and Empathy.

1989 ISBN 0-7923-0252-4 206. M.H. Salmon (ed.), The Philosophy of Logical Mechanism. Essays in Honor of

Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work. 1990 ISBN 0-7923-0325-3

207. M. Kusch, Language as Calculus vs. Language as Universal Medium. A Study in Husserl, Heidegger, and Gadamer. 1989 ISBN 0-7923-0333-4

208. T.C. Meyering, Historical Roots of Cognitive Science. The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989

ISBN 0-7923-0349-0 209. P. Kosso, Obsewability and Observation in Physical Science. 1989

ISBN 0-7923-0389-X 21 0. J. Kmita, Essays on the Theory of Scientific Cognition. 1990 ISBN 0-7923-0441 - 1 2 1 1. W. Sieg (ed.), Acting and Reflecting. The Interdisciplinary Turn in Philosophy. 1990

ISBN 0-7923-05 12-4 21 2. J. Karpin'ski, Causality in Sociological Research. 1990 ISBN 0-7923-0546-9 2 13. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters. 199 1

ISBN 0-7923-0823-9 214. M. Ter Hark, Beyond the Inner and the Outer. Wittgenstein's Philosophy of

Psychology. 1990 ISBN 0-7923-0850-6 215. M. Gosselin, Nominalism and Contemporary Nominalism. Ontological and

Epistemological Implications of the Work of W.V.O. Quine and of N. Goodman. 1990 ISBN 0-7923-0904-9

216. J.H. Fetzer, D. Shatz and G. Schlesinger (eds.), Definitions and Definability. Philosophical Perspectives. 199 1 ISBN 0-7923- 1046-2

217. E. Agazzi and A. Cordero (eds.), Philosophy and the Origin and Evolution of the Universe. 199 1 ISBN 0-7923-1322-4

218. M. Kusch, Foucault's Strata and Fields. An Investigation into Archaeological and Genealogical Science Studies. 199 1 ISBN 0-7923- 1462-X

219. C.J. Posy, Kant's Philosophy of Mathematics. Modern Essays. 1992 ISBN 0-7923- 1495-6

220. G. Van de Vijver, New Perspectives on Cybernetics. Self-organization, Autonomy and Connectionism. 1992 ISBN 0-7923- 15 19-7

22 1. J.C. Nyiri, Tradition and Individuality. Essays. 1 992 ISBN 0-7923- 1566-9 222. R. Howell, Kant's Transcendental Deduction. An Analysis of Main Themes in His

Critical Philosophy. 1992 ISBN 0-7923- 1571 -5

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223. A. Garcia de la Sienra, The Logical Foundations of the Marxian Theory of Value. 1992 ISBN 0-7923-1778-5

224. D.S. Shwayder, Statement and Referent. An Inquiry into the Foundations of Our Conceptual Order. 1992 ISBN 0-7923-1803-X

225. M. Rosen, Problems of the Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human Reality. 1993 ISBN 0-7923-2047-6

226. P. Suppes, Models and Methods in the Philosophy of Science: Selected Essays. 1993 ISBN 0-7923-221 1-8

227. R. M. Dancy (ed.), Kant and Critique: New Essays in Honor of W. H. Werkmeister. 1993 ISBN 0-7923-2244-4

228. J. Wolen'ski (ed.), Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2 229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional

Logic. 1993 ISBN 0-7923-2342-4 230. B.K. Matilal and A. Chakrabarti (eds.), Knowing from Words. Western and Indian

Philosophical Analysis of Understanding and Testimony. 1994 ISBN 0-7923-2345-9

231. S.A. Kleiner, The Logic of Discovery. A Theory of the Rationality of Scientific Research. 1993 ISBN 0-7923-2371-8

232. R. Festa, Optimum Inductive Methods. A Study in Inductive Probability, Bayesian Statistics, and Verisimilitude. 1993 ISBN 0-7923-2460-9

233. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher, Vol. 1: Probability and Probabilistic Causality. 1994

ISBN 0-7923-2552-4; Set Vols. 1-3 ISBN 0-7923-2554-0 234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of

Physics, Theory Structure, and Measurement Theory. 1994 ISBN 0-7923-2553-2; Set Vols. 1-3 ISBN 0-7923-2554-0

235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and Psychology. 1994

ISBN 0-7923-2862-0; Set Vols. 1-3 ISBN 0-7923-2554-0 236. D. Prawitz and D. Westersttihl (eds.), Logic and Philosophy of Science in Uppsala.

Papers from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994 ISBN 0-7923-2702-0

237. L. Haaparanta (ed.), Mind, Meaning and Mathematics. Essays on the Philosophical Views of Husserl and Frege. 1994 ISBN 0-7923-2703-9

238. J. Hintikka (ed.): Aspects of Metaphor. 1994 ISBN 0-7923-2786- 1 239. B. McGuinness and G. Oliveri (eds.), The Philosophy of Michael Dummett. With

Replies from Michael Dummett. 1994 ISBN 0-7923-2804-3 240. D. Jamieson (ed.), Language, Mind, and Art. Essays in Appreciation and Analysis,

In Honor of Paul Ziff. 1994 ISBN 0-7923-28 10-8 241. G. Preyer, F. Siebelt and A. Ulfig (eds.), Language, Mind and Epistemology. On

Donald Davidson's Philosophy. 1994 ISBN 0-7923-28 11-6 242. P. Ehrlich (ed.), Real Numbers, Generalizations of the Reals, and Theories of

Continua. 1994 ISBN 0-7923-2689-X 243. G. Debrock and M. Hulswit (eds.): Living Doubt. Essays concerning the epistemol-

ogy of Charles Sanders Peirce. 1994 ISBN 0-7923-2898-1

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