hilary putnam - models and reality

8/3/2019 Hilary Putnam - Models and Reality

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REALISM AND REASON Ihow dowelearn them? We learn themjust the way Dummett thinksat le~s~in the case of the less theoretical parts of the language, byacqumng a practice. What Dummett misses, inmyview, isthat whatwe acquire is not a knowledge that can be applied as if it were analgorithm. Wedo learn that in certain circumstances weare supposedto accept' There is a chair in front of me' (normally). But we areexpected to use our heads. We can refuse to accept 'There is a chairin front of me' even when it looks to us exactlyas if there is a chairin front of us, if our general intelligence screams 'override'. Theimpossibility (in practice at least) of formalizing the assertibilityconditions for arbitrary sentences isjust the impossibility of formal-izing general intelligence itself.If assertibility (in the sense of warranted assertibility) is notformalizable, idealized warranted assertibility (truth) is even less so,for the notion of better and worse epistemic conditions (for aparticular judgment) upon which it depends is revisable as ourem~irical knowledge increases. That it is, nevertheless, ameaningfulnotion: that there are better and worse epistemic conditions for mostjudgments, and a fact of the matter as to what the verdict would beif the conditions were sufficiently good, a verdict to which opinionwould 'converge' if we were reasonable, is the heart of my own,realism'. It is a kind of realism, and I mean it to be a human kindofrealism, abeliefthat there isa factof the matter asto what isrightlyassertible for us, as opposed to what is rightly assertible from theGo~'s eye v~ewsodear to the classicalmetaphysical realist. (On this,see Reflections on Goodman's Ways of Worldmaking'.)

Models and reality"

In 1922 Skolem delivered an address before the Fifth Congress ofScandinavian Mathematicians inwhichhepointed out what hecalleda 'relativity of set theoretic notions'. This' relativity' has frequentlybeen regarded as paradoxical; but today, although one hears theexpression 'The Lowenheim-Skolem paradox', it seems to bethought of as only an apparent paradox, something the cognoscentienjoy but are not seriously troubled by. Thus van Heijenoort writes,'The existence of such a "relativity" is sometimes referred to as theLowenheim-Skolem paradox. But, of course, it is not a paradox inthe sense of an antinomy; it is a novel and unexpected feature offormal systems.' In this paper I want to takeup Skolem's arguments,not with the aimof refuting them but with the aimofextending themin somewhat the direction he seemed to be indicating. It is not myclaim that the "Lowenheim-Skolem paradox' is an antinomy informal logic; but I shall argue that it is an antinomy, or somethingclose to it, inphilosophy of language. Moreover, I shall argue that theresolution ofthe antinomy - the only resolution that I myself can seeas making sense - has profound implications for the great meta-physical dispute about realism which has always been the centraldispute in the philosophy of language.

The structure ofmy argument will be asfollows. I shall point outthat in many different areas there are three main positions onreference and truth: there is the extreme Platonist position, whichposits non-natural mental powers of directly' grasping' forms (it ischaracteristic of this position that 'understanding' or 'grasping' isitself an irreducible and unexplicated notion); there is the verifi-cationist position which replaces the classicalnotion of truth with thenotion of verification or proof, at least when it comes to describinghow the language is understood; and there is the moderate realistpositionwhich seeks to preserve the centrality of the classicalnotionsof truth and reference without postulating non-natural mental

President ial Address del ivered before the Winter Meet ing of the Association forSymbolic Logic inWashington, D.C., 29December 1977. I wish to thank Bas van Fraassenfor valuable comments on and criticisms of an earlier version.

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MODELS AND REALITYREALISM AND REASON

(i) '" (3R) (R is one-to-one. The domain of R cN. The range ofvalues of R is S)

where' N' isa formal term for the set of all whole numbers and thethree conjuncts in the matrix havethe obvious first-order definitions.

Replace'S' with the formal term for the set of all real numbersin your favorite formalized set theory. Then (i) will be a theorem(proved by Cantor's celebrated 'diagonal argument '). So yourformalized set theory says that a certain set (call it'S') is non-denumerable. So S must be non-denumerable in all models of yourset theory. So your set theory - say ZF (Zermelo-Fraenkel settheory) - has only non-denumerable models. But this is impossible!For, by the Lowenheim-Skolem theorem, no theory can have onlynon-denumerable models; if a theory has anon-denumerable modelit must have denumerably infinite ones as well. Contradiction. '

The resolution ofthis apparent contradiction isnot hard, asSkolempoints out (and it is not this apparent contradiction that I referredto as an antinomy, or close to an antinomy). For (i) only 'says' thatS is non-denumerable when the quantifier (3R) is interpreted asranging over all relations on N x S. But when we pick a denumerablemodel for the language of set theory, '(3R)' does not range over allrelations; it only ranges over relations in the model. (i) only' says' thatS isnon-denumerable in a relative sense: the sense that the membersof S cannot be put in one-to-one correspondence with a subset ofNby any R in the model. And a set S can be 'non-denumerable' in thisrelative sense and yet be denumerable 'in reality'. This happenswhen there are one-to-one correspondences between Sand N but allof them lie outside the given model. What is a 'countable' set fromthe point of view of one model may be an uncountable set from thepoi?t of view of another model. As Skolem sums it up, 'even thenotions "finite", "infinite", "simply infinite sequence", and soforth turn out to be merely relative within axiomatic set theory'.

The philosophical problemUp to a point all commentators agree on the significance of theexistence of 'unintended' interpretations, for example, models inwhich what are 'supposed to be' non-denumerable sets are 'inreality' denumerable. All commentators agree that the existence ofsuch models shows that the' intended' interpretation, or, as someprefer to speak, the 'intuitive notion of a set', is not 'captured' bythe formal system. But if axioms cannot capture the' intuitive notionof a set', what possibly could?A technical fact is of relevance here. The Lowenheim-Skolemtheorem has a strong form (the so-called 'downward Lowenheim-Skolemtheorem '), which requires the axiom ofchoice to prove, andwhich tells us that a satisfiable first-order theory (in a countablelanguage) has a countable model which is a submodel of any givenmodel. In other words ifwe are given a non-denumerable model Mfor a theory, then we can find a countable model M' of that sametheory in which the predicate symbols stand for the same relations(restricted to the smaller universe in the obvious way) as they did inthe original model. The only difference between M and M' is thatthe' universe' of M' - i.e., the totality that the variables of quanti-fication range over - is a proper subset of the' universe' of M.

Now the argument that Skolem gave, and that shows that 'theintuitive notion of a set' (if there is such a thing) is not' captured'by any formal system, shows that even aformalization of total science(if one could construct such a thing), or even a formalization of allour beliefs (whether they count as 'science' or not), could not ruleout denumerable interpretations, and, afortiori, such a formalizationcould not rule out unintended interpretations of this notion.

This shows that' theoretical constraints' , whether they comefromset theory itself or from 'total science', cannot fix the interpretationof the notion set in the 'intended' way. What of 'operationalconstraints' ?

Even if we allow that there might be a denumerable infinity ofmeasurable' magnitudes', and that eachof them might be measuredto arbitrary rational accuracy (which certainly seems a utopianassumption), it wouldn't help. For, by the' downward Lowenheim-Skolemtheorem' ,wecanfind acountable submodel ofthe' standard'model (if there is such a thing) in which countably many predicates(eachofwhich may havecountably many things in its extension) havetheir extensions preserved. In particular, we can fix the values ofcountably manymagnitudes atallrational space-time points, and still

powers. I shall argue that it is, unfortunately, the moderate realistposition which is put into deep trouble by the Lowenheim-Skolemtheorem and related model-theoretic results. Finally I will opt forverificationism as a way of preserving the outlook of scientific orempirical realism, which is totally jettisoned by Platonism, eventhough this means giving up metaphysical realism.

The Lowenheim-Skolem theorem saysthat a satisfiablefirst-ordertheory (ina countable language) has a countable model. Consider thesentence:

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An epistemological/logical digressionThe problemjust pointed out isa seriousproblem for anyphilosopheror philosophically minded logicianwho wishes to view set theory asthe description of a determinate independently existing reality. Butfrom a mathematical point of view, it may appear immaterial: whatdoes it matter if there are many different models of set theory, andnot a unique 'intended model' if they all satis fy the same sentences?What we want to know asmathematicians is which sentences of settheory are true; we don't want to have the sets themselves in ourhands.Unfortunately, the argument can be extended. First of all, the

theoretical constraints wehavebeenspeakingofmust, onanaturalisticview, come from only two sources: they must come from somethinglike human decision or convention, whatever the source of the'naturalness' of the decisions or conventions may be, or from humanexperience, both experience with nature (which is undoubtedly thesource of our most basic 'mathematical intuitions', even if it beunfashionable to say so), and experience with 'doing mathematics'.It is hard to believe that either or both ofthese sources together canevergiveusa complete set ofaxiomsforset theory (since, for one thing,a complete set of axioms would have to be non-recursive, and it ishard to envisagecoming to have a non-recursive set of axioms in theliterature or in our heads even in the unlikely event that the humanracewent onforever doing set theory). And ifa complete set ofaxiomsis impossible, and the intended models (in the plural) are singled outonly by theoretical plus operational constraints then sentences whichare independent of the axioms which we shall arrive at in the limitof set-theoretic inquiry really have nodeterminate truth value; theyare just true in some intended models and false in others.

To show what bearing this fact may have on actual set-theoreticinquiry, I shall have to digress for a moment into technical logic. In1938 Godel put forward a new axiom for set theory: the axiom,V=L'. Here L is the class of all constructible sets, i.e., the classof all sets which can be defined by a certain constructive procedureif we pretend to have names available for all the ordinals, howeverlarge. (Of course, this sense of 'constructible' would be anathemato constructive mathematicians.) V is the universe of all sets. So, V =L' just says all sets are constructible. Byconsidering the innermodel for set theory in which 'V =L' is true, Godel was able toprove the relative consistency of ZF and ZF plus the axiom ofchoiceand the generalized continuum hypothesis.

, V=L' is certainly an important sentence, mathematicallyspeaking. Is it true?

Godel briefly considered proposing that we add ' V =L' to theaccepted axioms for set theory, as a sort ofmeaning stipulation, buthe soon changed his mind. His later view was that' V = L' is reallyfalse,even though it isconsistent with set theory, ifset theory isitselfconsistent.Godel's intuition iswidely shared amongworkingset theorists. Butdoes this' intuition' make sense?

Let MAG bea countable set ofphysicalmagnitudes which includesall magnitudes that sentient beings in this physical universe canactuallymeasure (it certainly seems plausible that wecannot hope to

finda countable submodel which meets all the constraints. In short,therecertainly seems to bea countable model ofour entire body of beliefwhich meets all operational constraints.

The philosophical problem appears at just this point. If we aretold 'axiomatic set theory does not capture the intuitive notion of aset' then it is natural to think that something else - our'understanding' - does capture it. But what can our' understanding'come to, at least for a naturalistically minded philosopher, which ismore than the way we use our language? And the Skolem argument canbe extended, as we have just seen, to show that the total use of thelanguage (operational plus theoretical constraints) does not 'fix' aunique 'intended interpretation' any more than axiomatic set theoryby itself does. . .This observation can push a philosopher of mathematics 10 twodifferent ways. If he is inclined to Platonism, he will take this asevidence that themind hasmysterious facultiesof' graspingconcepts'(or 'perceiving mathematical objects ') which the naturalisticallyminded philosopher will never succeed in giving an account of. Butifhe is inclined to some speciesofverificationism (i.e., to identifyingtruth with verifiability, rather than with some classical 'correspon-dence with reality') he will say, 'Nonsense! All the "paradox"shows is that our understanding of "The real numbers are non-denumerable" consists in our knowingwhat it is for this tobeproved,and not in our "grasp" of a "model".' In short, the extremepositions - Platonism and verificationism - seem to receive comfortfrom the Lowenheim+Skolem paradox; it is only the 'moderate'position (which tries to avoid mysterious' perceptions' of 'm~th~-matical objects' while retaining a classical notion of truth) which ISin deep trouble.

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t Barwise (1971) has proved the much stronger theorem that every countable mo.del ofZF has a proper end extension which is a model of ZF plus V =L. The theorem Inthetext was proved by me before 1963.

ilz-sentence is true in L, then it must be true in V. So the abovesentence is true in V.0

What makes this theorem startling is the following reflection:suppose that Godel is right, and 'V =L' is false (' in reality').Suppose that there is, in fact, a non-constructible real number (asGodelalso believes). Since the predicate 'is constructible' is absolute inp-models - i.e., in models in which the' well orderings' relative to themodel are well orderings 'in reality' (recall Skolem's 'relativity ofset theoretic notions' I),nomodel containing such anon-constructibles can satisfy's is constructible' and be a p-model. But, by theabove theorem, a model containing s can satisfy's is constructible'(because it satisfies 'V =L " and 'V =L' says everything is con-structible) and be an co-model.

Now, suppose we formalize the entire language of science withinthe set theory ZF plus V=L. Any model for ZF which contains anabstract set isomorphic to OP can be extended to a model for thisformalized language of science which is standard with respect to OP;hence even if OP is non-constructible 'in reality', we can find amodel for the entire language of science which satisfies 'everything isconstructible' and which assigns the correct values to all the physicalmagnitudes in MAG at all rational space-time points.

The claim Godel makes is that ' V = L' is false 'in reality'. Butwhat on earth can this mean? Itmust mean, at the very least, thatin the case just envisaged, the model we have described in which, V = L' holds would not be the intended model. But why not? Itsatisfies all theoretical constraints; and wehave gone to great lengthto make sure it satisfies all operational constraints as well.

Perhaps someone will saythat' V * ' L' (orsomething which impliesthat Vdoes not equal L) should be added to the axioms of ZF as anadditional 'theoretical constraint'. (Godel often speaks of newaxiomssomeday becoming evident.) But, while this may be acceptablefrom a non-realist standpoint, it can hardly be acceptable from arealist standpoint. For the realist standpoint is that there is a fact ofthe matter - a fact independent of our legislation - as to whetherV=L or not. A realist like Godel holds that we have access to an'intended interpretation' of ZF, where the access is not simply bylinguistic stipulation.

What the above argument shows is that if the' intended interpre-tation' is fixed only by theoretical plus operational constraints, thenif ' V * ' L' does not follow from those theoretical constraints - if wedonot decide to make V =L true or tomake V =L false - then therewill be 'intended' models in which V =L is true. If I am right, then

measure more than a countable number of physical magnitudes). LetOP be the 'correct' assignment of values; i.e., the assignment whichassigns to each member of MAG the value that that. magnit~deactuallyhas at each rational space-time point. Then allthe information'operational constraints' might give us (and, in fact, infinitely more)is coded into OP.One technical term: an to-model for a set theory isa model inwhichthe natural numbers are ordered as they are 'supposed to be'; i.e.,the sentence of 'natural numbers' of the model is an w-sequence.

Now for a small theorem. tTheorem: ZF plus V = Lhas an to-model which contains any givencountable set of real numbers.Proof

Since a countable set of reals can be coded as a single real by well-known techniques, it sufficesto prove that For every real s, there isanM such that M is an to-model for ZFplus V =Lands isrepresented inM.

By the' downward Lowenheim-Skolem theorem', this statementis true if and only if the following statement is:

For every real s, there is a countable Msuch that Mis an co-modelfor ZF plus V=Land s is represented in M.Countable structures with the property that the' natural numbers'

ofthe structure form anw-sequence can becoded as reals by standardtechniques. When this isproperly done, the predicate' M isan to-modelfor ZF plus V = Land s is represented in M' becomes a two-placearithmetical predicate of reals M, s. The above sentence thus has thelogical form (For every real s) (There is a real M) (...M, s, ... ). Inshort, the sentence is a ilz-sentence.

Now, consider this sentence in the inner model V =L. For everys in the inner model- i.e., for every sin L - there is a model- namelyL itself - which satisfies ' V=L' and contains s. By the downwardLowenheim-Skolem theorem, there is a countable submodel whichis elementary equivalent to L and contains s. (Strictly speaking, weneed here not just the downward Lowenheim-Skolem theorem, butthe' Skolem hull' construction which isused to prove that theorem.)By Godel's work, this countable submodel itself lies in L, and as iseasilyverified, sodoes the real that codes it. So the above ilz-sentenceis true in the inner model V= L.But Schoenfield has proved that ilz-sentences are absolute: if a

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REALISM AND REASON MODELS AND REALITY

the 'relativity of set-theoretic notions' extends to a relativity of thetruth value of ' V =L' (and, by similar arguments, of the axiom ofchoice and the continuum hypothesis as well).

argument shows that a model exists which fits all the facts that willactually be registered or observed and fits our theoretical constraintsand this model induces an interpretation of the counterfactual idiom(a 'similarity metric on possible worlds', in David Lewis' theory)which renders true just the counterfactuals that are true accordingto some completion of our theory. Thus appeal to counterfactualobservations cannot rule out anymodels at allunless the interpretationof the counterfactual idiom itself is already fixed by somethingbeyond operational and theoretical constraints.

(A related point is made by Wittgenstein in his PhilosophicalInvestigations: talk about what an ideal machine - or God - couldcompute is talk within mathematics - in disguise - and cannot serveto fix the interpretation of mathematics. 'God', too, has manyinterpretations. )

Operational constraints and counterfactualsItmay seem to some that there isa major equivocation in the notionofwhat can bemeasured, orobserved, whichendangers the apparentlycrucial claim that the evidence we could have amounts to at mostdenumerably many facts. Imagine a measuring apparatus that simplydetects the presence of a particle within a finite volume dv aroundits own geometric center during each full minute on its clock.Certainly it comes up with at most denumerably many reports (eachyes or no) even if it is left to run forever. But how many are the facts~tcould rep?rt? Well, if it were jiggled a little, by chance let us say,Its geometric center would shift r centimeters in a given direction.Itwould then report totally different facts. Since for each number rit could be jiggled that way, the number of reports it could produceisnon-denumerable - and it does not matter to this that we and theapparatus itself, are incapable of distinguishing every real number rfrom every other one. The problem is simply one of scope for themodal word 'can'. In my argument, I must be identifying what Icall operational constraints, not with the totality of facts that couldbe registered by observation - i.e., ones that either will be registered,or would be registered ifcertain chance perturbations occurred - butwith the totality of facts that will in actuality be registered orobserved, whatever those be.

In reply, I would point out that even if the measuring apparatuswere jiggled r centimeters in a given direction, we could only knowthe real number r to some rational approximation. Now, if theintervals involved are all rational, there are only countably many factso_fthe form: if action A (an action described with respect to place,time, and character up to some finite' tolerance') were performed, thenthe result re (a result described up to some rational tolerance) wouldbe obtained with probability in the interval (a, b). To know all facts ofthis form would be to know the probability distribution of all possibleobservable results of all possible actions. And our argument showsthat a model could beconstructed which agrees with all ofthese facts.

There is a deeper point to be made about this objection, however.Suppose we 'first orderize' counterfactual talk, say, by includingevents in the ontology of our theory and introducing a predicate('subjunctively necessitates') for the counterfactual connectionbetween unactualized event types at a given place-time. Then our

'Decision' and 'convention'!have used the word 'decision' in connection with open questionsl~ set theory, and obviously this is a poor word. One cannot simplySit down 10 one's study and' decide' that' V=L' is to be true, orthat the axiom of choice is to be true. Nor would it be appropriatefor the mathematical community to call an International Conventionand legislate these matters. Yet, it seems tome that ifwe encountereda~ extra-terrestrial species of intelligent beings who had developed ahigh level of mathematics, and it turned out that they rejected the~xiom of choice (perhaps because of the Tarski-Banach theoremj+,It would be wrong to regard them as simply making a mistake. Todo that would, on my view, amount to saying that acceptance of theaxiom ofchoice is built in to our notion of rationality itself; and thatdoes not seem to me to be the case. To be sure, our acceptance ofchoice is not arbitrary; all kinds of 'intuitions' (based, most likely,on experience with the finite) support it; its mathematical fertilitysupports it; but none of this is so strong that we could say that anequally successful culture which based its mathematics on principlesincompatible with choice (e.g., on the so-called 'axiom of deter-minacy')t was irrational.t This is a very counterintuitive consequence of the axiom of choice. Call two objects

A'.B cong ruent by f in it e decompos it ion' i f t hey can be divided into f ini tel y many di sjo in tpo~t sets AI> ... An. BI~ ... Bn. such. that A =Al U A. U ... U An.B - BI U B. U ... U Bn. and (for I = I . Z .. n) Aj IS congruen t to Bj. Then Tarski andBanach showed that all spheres are congruent by finite decomposition.. t Thi~ axiom. f ir st s t~di ed ~y J. Mycie lski ~19~3)asserts tha t infin ite games with perfectinformation are determined. i.e. there IS a winning strategy for either the first or secondpl ayer . The axiom of det erminacy impl ies t he ex is tence of a non- tr iv ial count ably addi ti vetw~-valued ~easure on the real numbers. contradicting a well-known consequence of theaxiom of choice,

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MODELS AND REALITYREALISM AND REASONBut ifboth systems ofsettheory - ours and the extra-terrestrials' -

count as rational, what sense does it make to call one true and theothers false? From the Platonist's point of view there is no troublein answering this question. 'The axiom ofchoice is true - true in themodel', he will say (if he believesthe axiom ofchoice). 'We are rightand the extra-terrestrials are wrong.' But what is the model? If theintended model is singled out by theoretical and operationalconstraints, then, firstly, 'the' intended model is plural not singular(so the 'the' is inappropriate; our theoretical and operational con-straints fit many models, not just one, and so do those of theextra-terrestrials as we saw before). And, secondly, the intended~odels for us do satisfy the axiomofchoiceand the extra-terrestriallyIntended models do not; we are not talking about the same models,so there is no question of a 'mistake' on one side or the other.

The Platonist will reply that what this really shows isthat wehavesome mysterious faculty of 'grasping concepts' (or 'intuitingmathematical objects ') and it is this that enables us to fix a model asthe model, and not just operational and theoretical constraints butthis appeal to mysterious faculties seemsboth unhelpful asepistemo-logyandunpersuasive asscience.What neural process, after all, couldbe described as the perception of a mathematical object? Why of onemathematical object rather than another? I do not doubt that somemathematical axioms are built in to our notion of rationality ('everynumber has a successor '); but, if the axiom of choice and thecontinuum hypothesis are not, then, I am suggesting, Skolen'sargument, or the foregoing extension of it, casts doubt on the viewthat these statements have a truth value independent of the theoryin which they are embedded.

Now, suppose this is right, and the axiom of choice is true whentaken in the sense that it receives from our embedding theory, andfalse when taken in the sense that it receives from extra-terrestrialtheory. Urging this relativism isnot advocating unbridled relativism;I do not doubt that there are some objective (if evolving) canons ofrationality; I simply doubt that wewould regard them assettling thissort of question, let alone as singling out one unique 'rationallyacceptable set theory'. If this is right, then one is inclined to say thatthe extra-terrestrials have decided to let the axiom of choice be falseand we have decided to let it be true; or that we have different'conventions'; but, of course, none of these words is literally right.It may well be the case that the idea that statements have their truthvalues independent of embedding theory is so deeply built in to ourways of talking that there is simply no 'ordinary language' word or

short phrase which refers to the theory-dependence of meaning andtruth. Perhaps this is why Poincare was driven to exclaim 'Con-vention, yes!Arbitrary, no! ' when he was trying to express a similaridea in another context.

Is the problem a problem with the notion of a 'set'?Itwould be natural to suppose that the problem Skolem points out,the problem of a surprising' relativity' ofour notions, has to dowiththe notion of a 'set', given the various problems which are knownto surround that notion, or, at least, has to do with the problem ofreference to 'mathematical objects'. But this is not so.

To seewhy it is not so, let us consider briefly the vexed problemof reference to theoretical entities in physical science. Although thismay seem to be a problem more for philosophers of science orphilosophers of language than for logicians, it is a problem whoselo~ical aspects have frequently been of interest to logicians, as iswitnessed by the expressions 'Ramsey sentence', 'Craig trans-lation', etc. Here again, the realist - or, at least, the hard-coremetaphysical realist - wishes it to be the case that truth and rationalacceptability should be independent notions. Hewishes it to be the casethat what, for example, electrons are should be distinct (and possiblydifferent) from what we believe them to be or even what we wouldbelieve them to be given the best experiments and the epistemicallybest theory. Once again, the realist - the hard-core metaphysicalrealist - holds that our intentions single out 'the' model, and thatour beliefs are then either true or false in 'the' model whether we canfind out their truth values or not.To see the bearing of the Lowenheim-Skolem theorem (or of theintimately related Godel completeness theorem and its model-theoretic generalizations) on this problem, let us again do a bit ofmodel construction. This time the operational constraints have to behandled a little more delicately, since we have need to distinguishoperational concepts (concepts that describe what we see, feel, hear,etc., as we perform various experiments, and also concepts thatdescribe our acts of picking up, pushing, pulling, twisting, lookingat, sniffing, listening to, etc.) from non-operational concepts..To describe our operational constraints weshall need three things.

First, we shall have to fix a sufficiently large 'observational voca-bulary '. Like the 'observational vocabulary' of the logical empiri-cists, we shall want to include in this set - call it the set of,O-terms ' - such words as 'red', 'touches', 'hard', 'push', 'look

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at', etc. Second, we shall assume that there exists (whether we candefine it or not) a set of S which can be taken to be the set ofmacroscopically observable things and events (observable with thehuman sensorium, that means). The notion of an observable thingor event is surely vague; sowe shall want Sto be a generous set, i.e.,God isto err in the direction ofcounting too many things and eventsas 'observable for humans' when he defines the set S (if it isnecessary to err ineither direction), rather than to err in the directionof leaving out some things that might be counted as borderline'observables '. If one is a realist, then such a set Smust exist, ofcourse, even ifour knowledge of theworld and the human sensoriumdoesnot permit us to defineit at the present time. The reasonweallowSto contain events (and not just things) is that, asRichard Boyd haspointed out; someofthe entitieswecandirectly observe areforces - wecanfeel forces- and forces are not objects. But I assume that forcescan be construed as predicates of either objects, e.g. our bodies, orof suitable events.

The third thing we shall assume given is a valuation (call it, onceagain 'OP ') which assigns the correct truth value to each n-placeO-term (forn = I, 2, 3, ... )on eachn-tuple ofelements ofSonwhichit is defined. O-terms are in general also defined on things not in S;for example, two molecules too small to see with the naked eye maytouch, a dust-mote too small to see may be black, etc. Thus OP isa partial valuation in a double sense; it is defined on only a subsetof the predicates of the language, namely the O-terms, and even onthese it only fixes a part of the extension, namely the extension ofTtS (the restriction of T to S), for each O-term T.

Once again, it is the valuation OP that captures our 'operationalconstraints'. Indeed, it captures these' fromabove', since itmaywellcontain more information than we could actually get by using ourbodies and our senses in the world.

What shall we do about 'theoretical constraints'? Let us assumethat there exists a possible formalization ofpresent-day total science,call it ' T', and also that there exists a possible formalization of idealscientific theory, call it ' 1 1 '. 1 1 isto be 'ideal' in the sense of beingepistemically ideal for humans. Ideality, in this sense, is a rather vaguenotion; but we shall assume that, when God makes up 1 1 , Heconstructs a theory which it would be rational for scientists to accept,or which is a limit of theories that it would be rational to accept, asmore and more evidence accumulates, and also that he makes up atheory which is compatible with the valuation OP.

Now, the theory T is, we may suppose, well confirmed at thepresent time, and hence rationally acceptable on the evidence we now

have' but there is a clear sense in which it may be false. Indeed, itmay~elliead to false predictions, and thus conflictwith OP. But 1 1 ,by hypothesis, does not lead to any false predictions. Still, themetaphysical realist claims - and it isjust this claim that makes hima metaphysical asopposed to an empirical realist - that 1 1 maybe, inreality, false.What isnot knowable astrue maynone the lessbe true;what is epistemically most justifiable to believe may none the lessbefalse, on this kind of realist view. The striking connection betweenissuesand debates in the philosophy ofscience and issues and debatesin the philosophy of mathematics - is that this sort of realism runsinto precisely the samedifficulties that wesawPlatonism run into. Letus pause to verify this.

Since the ideal theory 1 1 must, whatever other properties it mayormaynot have, have the property ofbeing consistent, it follows fromthe Godel completeness theorem (whoseproof, as all logiciansknow,is intimately related to one of Skolem's proofs of the Lowenheim-Skolem theorem), that 1 1 has models. We shall assume that 1 1contains a primitive or defined term denoting eachmember ofS, theset of 'observable things and events'. The assumption that wemade,that 1 1 agreeswith OP, means that allthose sentences about membersofSwhich OP requires to be true are theorems of 1 1 . Thus if Misany model of 1 1 , M has to have a member corresponding to eachmember of S. We can even replace each member of M whichcorresponds to a member ofSby that member ofSitself, modifyingthe interpretation of the predicate letters accordingly, and obtain amodel M' in which each term denoting a member of S in the'intended' interpretation does denote that member of S. Then theextension of each O-term in that model will be partially correct tothe extent determined by OP: i.e., everything that OP 'says' is inthe extension of P is in the extension of P, and everything that OP,says' is in the extension of the complement of P is in the extensionof the complement of P, for each O-term P, in any such model. Inshort, such a model is standard with respect to P t S(P restricted toS) for each O-term P.

Now, suchamodel satisfiesalloperational constraints, sinceitagreeswith OP. And it satisfies those theoretical constraints we wouldimpose in the ideal limit of inquiry. So, once again, it looksas if anysuch model is 'intended' - for what else could single out a model as,intended' than this? But if this is what it is to be an 'intendedmodel', 1 1 must be true: true in all intended modelsIThe meta-physical realist's claim that even the ideal theory 1 1 might be false'in reality' seems to collapse into unintelligibility.

Of course, it might be contended that' true' does not follow from1312


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'true in all intended models'. But 'true' is the same as 'true in theintended interpretation' (or' in all intended interpretations', if theremaybemore than one interpretation intended - orpermitted - bythespeaker), on any view. So to follow this line - which is, indeed, ~heright one, in my view- one needs to develop a theory on whichinterpretations are specified other than by specifying m~del~.

Once again, an appeal to mysterious powers of the mind is madeby some. Chisholm (following the tradition of Bre~tano) contendsthat the mind has a faculty of referring to external objects (or perhapsto external properties) which he calls by the good old name' inten-tionality'. And once againmost naturalistically minded philosophers(and, of course, psychologists), find the postulation of une~plainedmental faculties unhelpful epistemology and almost certainly badscience as well.

There are two main tendencies in the philosophy of science (Ihesitate to call them 'views', because each tendency is representedby many different detailed views) about the way in which thereference of theoretical terms gets fixed. According to one tendency,which wemay call the Ramsey tendency, and whose various versionsconstituted the received viewfor many years, theoretical terms comeinbatches orclumps. Each clump - for example, the clump consistingof the primitives of electromagnetic theory - is defined by a theory,in the sense that all the models of that theory which are standard onthe observation terms count asintended models. The theory is 'true'just incaseit has such amodel. (The' Ramsey sentence' ofthe theoryisjust the second-order sentence that asserts the existence of such amodel.) A sophisticated version of this view, which amounts torelativizing the Ramsey sentence to an open set of' intended appli-cations' has recently been advanced by Joseph Sneed.

The other tendency is the realist tendency. While realists differamong themselves even more than proponents of the (former)received view do, realists unite in agreeing that a theory may havea true Ramsey sentence and not be (in reality) true.

The first of the two tendencies I described, the Ramsey tendency,represented in the United States by the school of Rudolf Carnap,accepted the 'relativity of theoretical notions', and abandonedthe realist intuitions. The second tendency is more complex. Its, soto speak, conservative wing, represented byChisholm, jo~nsPlato~ndthe ancients in postulating mysterious powers wherewith the ~md,grasps' concepts, aswehave already said. Ifwehave more ~vatlab!ewith which to fix the intended model than merely theoretical andoperational constraints, then the problem disappears. The radical

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pragmatist wing, represented, perhaps, by Quine, is willing to giveup the intuition that 1 1 might be false"in reality'. This radical wingis 'realist' in the sense of being willing to assert that present-dayscience, taken more or less at face value (i.e., without philosophicalreinterpretation) is at least roughly true; 'realist' in the sense ofregarding reference as trans-theoretic (a theory with a true Ramseysentence may be false, because later inquiry may establish anincompatible theory as better); but not metaphysical realist. It is themoderate' center' of the realist tendency, the center that would liketo hold on to metaphysical realism without postulating mysteriouspowers of the mind that is once again in deep trouble.

Pushing the problem back: the Skolemization of absolutelyeverything

We have seen that issues in the philosophy of science having to dowith reference of theoretical terms and issues in the philosophy ofmathematics having to do with the problem of singling out a unique'intended model' for set theory are both connected with theLowenheim-Skolem theorem and its near relative, the Godelcompleteness theorem. Issues having to do with reference also ariseinphilosophy inconnection with sense data and material objects, and,once again, these connect with the model-theoretic problems wehavebeen discussing. (In some way, it really seems that the Skolemparadox underlies the characteristic problems of twentieth-centuryphilosophy. )

Although the philosopher John Austin and the psychologist FredSkinner both tried to drive sense data out of existence, it seems tomethat most philosophers and psychologists think that there are suchthings assensations, or qualia. These may not beobjects ofperception,aswas once thought (it is becoming increasingly fashionable to viewthem as states or conditions of the sentient subject, as Reichenbachlong ago urged weshould); wemay not have incorrigible knowledgeconcerning them; they may be somewhat ill-defined entities ratherthan the perfectly sharp particulars they were once taken to be; butit seems reasonable to hold that they are part ofthe legitimate subjectmatter ofcognitivepsychology andphilosophy, andnotmere pseudo-entities invented by bad psychology and bad philosophy.

Accepting this, and taking the operational constraint this time tobe that we wish the ideal theory to correctly predict all sense data,it iseasily seen that the previous argument can be repeated here, thistime to showthat (if the' intended' models are the oneswhich satisfy

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the operational and theoretical constraints wenow have, or even theoperational and theoretical constraints we would impose in somelimit) then, either the present theory is 'true', in the sense of being,true in all intended models' , provided it leads to no falsepredictionsabout sense data, or else the ideal theory is 'true'. The firstalternative corresponds to taking the theoretical constraints to berepresented by current theory; the second alternative corresponds totaking the theoretical constraints to be represented by the idealtheory. This time, however, it will be the case that even termsreferring to ordinary material objects - terms such as 'cat' and'dog' - are interpreted differently in the different 'intended'models. It seems, this time, as if we cannot even refer to ordinarymiddle-sized physical objects except as formal constructs variouslyinterpreted in various models.

Moreover, ifweagreewith Wittgenstein that the similarity relationbetween sense data wehave at different times isnot itself somethingpresent to my mind - that' fixing one's attention' on a sense datumand thinking' by "red" I mean whatever is like this' doesn't reallypick out any relation of similarity at all - and make the natural moveof supposing that the intended models ofmy language when I nowand inthe future talkof the sense data I had at some past time taresingled out by operational and theoretical constraints, then, again, itwill turn out that mypast sense data aremere formal constructs whichare interpreted differently in various models. And ifwefurther agreewith Wittgenstein that the notion of truth requires a public language(or requires at least states of the self at more than one time: that a'private language for one specious present' makes no sense), theneven mypresent sense data are in this same boat... In short, one can,Skolemize' absolutely everything. It seems to be absolutelyimpossible to fix a determinate reference (without appeal to non-natural mental powers) for any term at all. And if we apply theargument to the very metalanguage we use to talk about thepredicament ... ?

The same problem has even surfaced recently in the field ofcognitive psychology. The standard model for the brain/mind in thisfield is the modern computing machine. This computing machine isthought of as having something analogous to a formalized languagein which it computes. (This hypothetical brain language has evenreceived a name: 'mentalese'.) What makes the model of cognitivepsychology a cognitive model is that 'mentalese' is thought to be amedium whereby thebrain constructs an internal representation oftheexternal world. This idea runs immediately into the following

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problem: if 'mentalese' is to be a vehicle for describing the externalworld, then the various predicate letters must have extensions whichare sets ofexternal things (or sets of n-tuples ofexternal things). Butif the way 'mental ese' is 'understood' by the deep structures in thebrain that compute, record, etc. in this 'language' is via whatArtificial Intelligence people call' procedural semantics' - i.e., if thebrain's program for using 'mentalese' comprises its entire 'under-s.tanding' of 'mentalese', where the program for using' mentalese' ,hke any program, refers only to what is inside the computer - thenhowdo extensions evercome into the picture atall ? In the terminologyI have been employing in this paper, the problem is this: if theextension of predicates in 'mentalese' is fixedby the theoretical andoperational constraints 'hard wired in' to the brain, or even bytheoretical and operational constraints that it evolves in the courseof inquiry, then these will not fix a determinate extension for anypredicate. If thinking is ultimately done in 'mentalese', then noconcept we have will have a determinate extension. Or so it seems.

The bearing of causal theories of referenceThe term 'causal theory of reference' was originally applied to mytheory of the reference of natural kind terms and to Kripke's theory(seepp. 70-5 for an account). These theories did not attempt to definereference, but rather attempted to saysomething about how referenceis fixed, if it is not fixed by associating definite descriptions with theterms and names in question. Kripke and I argued that the intentionto preserve reference through a historical chain of uses and theintention to cooperate socially in the fixing of reference make itpossible to use terms successfully to refer although no one definitedescription is associated with any term by all speakers who use thatterm. These theories assume that individuals can be singled out forthe purpose of a 'naming ceremony' and that inferences to theexistence of definite theoretical entities (to which names can then beattached) can be successfully made. Thus these theories did notaddress the question as to how any term can acquire a determinatereference (or any gesture, such aspointing - ofcourse, the' reference'ofgestures isjust asproblematic asthe reference ofterms, ifnot moreso). Recently, however, it has been suggested by various authors thatsome account can be given of how at least some basic sorts of termsrefer interms ofthe notion ofa 'causal chain'. In one version (Evans1973), a version strikingly reminiscent of the theories ofOckham andother fourteenth-century logicians, it isheld that a term refers to 'the

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dominant source' of the beliefs that contain the term. Assuming wecan circumvent the problem that the dominant cause of our beliefsconcerning electrons may well be textbooks, t it is important to noticethat even if a correct view of this kind can be elaborated, it will donothing to resolve the problem wehave been discussing.

The problem isthat adding toour hypothetical formalized languageof science a body of theory entitled 'Causal theory of reference' isjust adding more theory. But Skolem's argument, and our extensionsof it, are not affected by enlarging the theory. Indeed, you can eventake the theory to consist ofall true sentences, and there will bemanymodels - models differing on the extension of every term not fixedby OP (or whatever you take OP to be in a given context) - whichsatisfy the entire theory. If' refers' can be defined in terms of somecausalpredicate or predicates inthe metalanguage ofour theory, then,since each model of the object language extends in an obvious wayto a corresponding model of the metalanguage, it will turn out that,in each model M, re/erenceM is definable in terms of causesa;but,unless the word 'causes' (or whatever the causal predicate orpredicates may be) is already glued to one definite relation withmetaphysical glue, this does not fix a determinate extension for'refers' at all.

This is not to say that the construction of such a theory would beworthless as philosophy or as natural science. The program ofcognitive psychology already alluded to, the program of describingour brains as computers which construct an 'internal representationofthe environment' , seems to require that' mentalese ' utterances be,in some cases at least, describable as the causal product of devices inthe brain and nervous system which 'transduce' information fromthe environment, and suchadescription might wellbewhat thecausaltheorists are lookingfor. And the programofrealismin thephilosophyof science - of empirical realism, not metaphysical realism - is toshow that scientific theories can be regarded as better and betterrepresentations of an objective world with which weare interacting;and if such a view is to be part of science itself, asempirical realistscontend it should be, then the interactions with the world bymeansofwhich this representation is formed and modifiedmust themselvesbe part of the subject matter ofthe representation. But the problemas to how the whole representation, including the empirical theory ofknowledge that isa part ofit, can determinately refer isnot a problemthat can be solved by developing more and better empirical theory.

Ideal theories and truthOne reaction to the problem I have posed would be to say: there aremany ideal theories in the sense of theories which satisfy theoperational constraints, and inaddition haveallthe virtues (simplicity,coherence, containing the axiom of choice, whatever) that humanslike to demand. But there are no 'facts of the matter' not reflectedin constraints on ideal theories in this sense. Therefore, what is reallytrue iswhat iscommon to all such ideal theories; what is really falseis what they all deny; and all other statements are neither true norfalse.

Such a reactionwould lead to too fewtruths, however. It may wellbethat there arerational beings - evenrational human species - whichdo not employ our color predicates, or who do not employ thepredicate 'person', or who do not employ the predicate 'earth-quake'. t I see no reason to conclude from this that our talk of redthings, or of persons, or of earthquakes, lacks truth value. If thereare many ideal theories (and if' ideal' is itself a somewhat interest-relative notion), if there are many theories which (given appropriatecircumstances) it is perfectly rational to accept, then it seems betterto say that, in so far as these theories say different (and sometimes,apparently incompatible) things, some facts are 'soft' in the senseof depending for their truth value on the speaker, the circumstancesofutterance, etc. This iswhat we have to say in any caseabout casesof ordinary vagueness, about ordinary causal talk, etc. It is what wesay about apparently incompatible statements of simultaneity in thespecial theory of relativity. To grant that there ismore than one trueversion of reality is not to deny that some versions are false.Itmay be, of course, that there are some truths that any species

of rational inquirers would eventually acknowledge. (On the otherhand, the set of these may be empty, or almost empty.) But to saythat by definition these are all the truths there are is to redefine thenotion in a highly restrictive way. (It alsoassumes that the notion ofan 'ideal theory' is perfectly clear, an. assumption which seemsplainly false.)

t Evans handles this case bysaying that there are appropriateness conditions onthe typeof causal chain which must exist between the item referred to and the speaker's body ofinformation.

IntuitionismIt is a striking fact that this entire problem does not arise for thestandpoint ofmathematical intuitionism. This would not beasurpriseto Skolem: it was precisely his conclusion that' most mathematicianswant mathematics to deal, ultimately, with performable computing

t For a discussion of this very point , see David Wiggins (1976).18 19


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operations and not to consist of formal propositions about objectscalled this or that'.

In intuitionism, knowing the meaning of a sentence or predicateconsists in associating the sentence or predicate with a procedurewhichenablesone to recognizewhenone has aproof that thesentenceis constructively true (i.e., that it is possible to carry out theconstructions that the sentence asserts can be carried out) or that thepredicate applies to a certain entity (i.e., that a certain full sentenceofthe predicate isconstructively true). The most striking thing aboutthis standpoint isthat the classical notion of truth isnowhere used - thesemantics is entirely given in terms of the notion of 'constructiveproof', including the semantics of 'constructive proof' itself.

Of course, the intuitionists do not think that' constructive proof'can be formalized, or that 'mental constructions' can be identifiedwith operations in our brains. Generally, they assume a stronglyintentionalist and aprioristic posture in philosophy; i.e., they assumethe existence of mental entities called 'meanings' and of a specialfacultyofintuiting constructive relations betweenthese entities. Theseare not the aspects of intuitionism I shall be concerned with. RatherI wish to look on intuitionism as an example of what MichaelDummett has called' non-realist semantics', i.e., a semantic theorywhichholds that a language is completely understood when a verificationprocedure is suitably mastered, and not when truth conditions (in theclassical sense) are learned.

The problemwith realist semantics - truth-conditional semantics -asDummett has emphasized, isthat ifwehold that the understandingof the sentences of, say, set theory consists in our knowledge of their'truth conditions', then howcanwepossibly saywhat that knowledgein turn consists in? (It cannot, as we have just seen, consist in theuse of language or 'mentalese' under the control of operational plustheoretical constraints, be they fixed or evolving, since suchconstraints are too weak to provide a determinate extension for theterms, and it is this that the realist wants.)If, however, the understanding of the sentences of a mathematicaltheory consists in the mastery of verification procedures (which neednot be fixed once and for all: we can allow a certain amount of'creativity '), then a mathematical theory can be completely under-stood, and this understanding does not presuppose the notion of a'model' at all, let alone an 'intended model'. \

Nor does the intuitionist (or, more generally, the 'non-realist'semanticist) have to foreswear forever the notion of a model. He hasto foreswear reference tomodels in his account ofunderstanding; but,

once he has succeeded in understanding a rich enough language toserveasa metalanguage for sometheory T(which may itselfbe simplyasublanguageofthe metalanguage, in the familiarway), he can define'true in T' Ii la Tarski, he can talk about' models' for T, etc. Hecan even define' reference' (or' satisfaction') exactly as Tarski did.

Does the whole' Skolemparadox' arise again to plague him at thisstage? The answer is that it does not. To see why it does not, onehas to realize what the 'existence of a model' means in constructivemathematics.,Objects' in constructive mathematics are given through descrip-tions. Those descriptions do not have to be mysteriously attached tothose objects by somenon-natural process (or bymetaphysical glue).Rather the possibility of proving that a certain construction (the,sense', so to speak, of the description of the model) has certainconstructive properties iswhat is asserted and all that is asserted bysaying the model' exists'. In short, reference is given through sense,and sense isgiven through verification procedures and not through truthconditions. The 'gap' between our theory and the 'objects' simplydisappears - or, rather, it never appears in the first place.

Intuitionism liberalizedIt isnotmyaim,however, to try toconvertmyreaders to intuitionism.Set theory may not be the 'paradise' Cantor thought it was, but itisn't such a bad neighborhood that I want to leaveofmy own accord,either. Can we separate the philosophical idea behind intuitionism,the idea of 'non-realist' semantics, from the restrictions andprohibitions that the historic intuitionists wished to impose uponmathematics?The answer is that we can. First, asto set theory: the objection toimpredicativity, which is the intuitionist ground for rejecting muchof classical set theory, has little or no connection with the insistenceupon verificationism itself. Indeed, intuitionist mathematics is itself'impredicative', in asmuch asthe intuitionist notion of constructiveproof presupposes constructive proofs which refer to the totality ofall constructive proofs.

Second, as to the propositional calculus: it is well known that theclassical connectives can be reintroduced into an intuitionist theoryby reinterpretation. The important thing is not whether one uses'classical propositional calculus' or not, but how one understands thelogic ifone does use it. Using classical logic as an intuitionist wouldunderstand it,means, forexample, keepingtrackofwhenadisjunction

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is selective (i.e., one of the disjuncts is constructively provable), andwhen it is non-selective; but this doesn't seem too bad an idea.

In short, while intuitionism may go with a greater interest inconstructive mathematics, a liberalized version of the intuitioniststandpoint need not rule out 'classical' mathematics as eitherillegitimate or unintelligible.

What about the language of empirical science? Here there aregreater difficulties. Intuitionist logic is given in terms of a notion ofproof, and proof issupposed to be apermanent feature of statements.Moreover, proof isnon-holistic; there is such a thing as the proof (ineither the classical or the constructive sense) of an isolated mathe-matical statement. But verification in empirical science is a matter ofdegree, not a 'yes-or-no' affair; even if we made it a 'yes-or-no'affair in some arbitrary way, verification is a property of empiricalsentences that can be lost; and in general the 'unit of verification'in empirical science is the theory and not the isolated statement.

These difficulties show that sticking to the intuitionist standpoint,however liberalized, would beabad idea in the context of formalizingempirical science. But they are not incompatible with 'non-realist'semantics. The crucial question is this: do we think of the under-standing of the language asconsisting in the fact that speakers possess(collectively if not individually) an evolving network of verificationprocedures, or as consisting in their possession of a set of 'truthconditions'? If we choose the first alternative, the alternative of'non-realist' semantics, then the 'gap' between words and world,between our use of the language and its 'objects', never appears.tMoreover, the' non-realist' semantics is not inconsistent with realistsemantics; it issimply prior to it, inthe sense that it isthe' non-realist'

semantics that must be internalized if the language is to beunderstood.Even if it is not inconsistent with realist semantics, taking thenon-realist semantics asour picture ofhowthe language isunderstoodundoubtedly will affect the way weviewquestions about reality andtruth. For one thing, verification in empirical science (and, to a lesserextent, inmathematics aswell, perhaps) sometimes depends onwhatwebefore called 'decision' or 'convention'. Thus facts may, on thispicture, depend on our interests, saliencies, and decisions. There willbemany' soft facts'. (Perhaps whether V =L or not isa 'soft fact' .)I cannot, myself, regret this. Ifappearance and reality end up beingendpoints on a continuum rather than being the two halves of amonster Dedekind Cut in allweconceive and don't conceive, it seemsto me that philosophy will be much better off. The search for the'furniture of the universe' will have ended with the discovery thatthe universe is not a furnished room.

t To the suggestion that we identify truth with being verif ied, or accepted, or acceptedinthe long run, itmay be objected that aperson could reasonably, and possibly truly. makethe assertion:

A but itcould have been the case that A and our scientific development differ in sucha way to make Apart ofthe ideal theory accepted inthe long run; inthat circumstance,i t would have been the case that A but it was not true that A.

This argument is fallacious, however, because the different .scientific development'means here the choice of a different version; we cannot assume the sentence rA' has a fixedmeaning independent of what version we accept. . ~ .. . .

In fact the same problem can confront a metaphysical realist . Real ists also have torecognize that there are cases in which the reference of a term depends on which theoryone accepts, so that A can be atrue sentence if r. isaccepted and a false one if T o is accepted,where r. and T o are both true theories. But then imagine someone saying, /

A but itcould have been the case that A and our scientific development differ so thatT o ' was accepted; inthat case itwould have been the case that A but A would not havebeen true.

Where did we go wrong? - The problem solvedWhat Skolemreally pointed out is this: no interesting theory (in thesense of first-order theory) can, in and of itself, determine its ownobjects up to isomorphism. And Skolem's argument can be extendedas we saw, to show that if theoretical constraints don't determinereference, then the addition of operational constraints won't do iteither. It is at this point that reference itself begins to seem' occult';it begins to seemthat one can't be any kind ofa realist without beinga believer in non-natural mental powers. Many moves have beenmade in response to this predicament, aswenoted above. Some haveproposed that second-order formalizations are the solution, at least formathematics; but the 'intended' interpretation of the second-orderformalism isnot fixedby theuse ofthe formalism(the formalism itselfadmits so-called' Henkin models', i.e., models inwhich the second-order variables fail to range over the full power set of the universeof individuals), and it becomes necessary to attribute to the mindspecial powers of 'grasping second-order notions'. Some haveproposed to accept the conclusion that mathematical language isonlypartly interpreted, and likewise for the language we use to speak of'theoretical entities' in empirical science; but then are 'ordinarymaterial objects' any better off? Are sense data better off? BothPlatonism and phenomenalism have run rampant at different timesand in different places in response to this predicament.

The problem, however, lies with the predicament itself. The23

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predicament only is a predicament because wedid two things: first,we gave an account of understanding the language in terms ofprograms and procedures for using the language (what else?); andthen, secondly, weaskedwhat the possible' models' for the languagewere, thinking of the models as existing 'out there' independent ofany description. At this point, something really weird had alreadyhappened, had westopped to notice. On anyview, the understandingofthe languagemust determine the reference of the terms, or, rather,must determine the reference given the context ofuse. Iftheuse, evenin a fixed context, doesn't determine reference, then use isn'tunderstanding. The language, on the perspective wetalked ourselvesinto, has a full program of use; but it still lacks an interpretation.

This is the fatal step. To adopt a theory of meaning according towhich a language whose whole use is specifiedstill lacks something -namely its 'interpretation' - is to accept a problem which can onlyhave crazy solutions. To speak as if this were my problem, 'I knowhow to use my language, but, now, how shall I single out aninterpretation?' is to speak nonsense. Either the use already fixesthe'interpretation' or nothing can.

Nor do 'causal theories of reference', etc., help. Basically, tryingto get out of this predicament by these means ishoping that the worldwillpickone definite extension for eachofour terms even ifwe can't.But the world doesn't pick models or interpret languages. Weinterpret our languages or nothing does.

We need, therefore, a standpoint which links use and reference injust the way that the metaphysical realist standpoint refuses to do.The standpoint of' non-realist semantics' isprecisely that standpoint.From that standpoint, it istrivial to say that a model in which, as itmight be, the set of cats and the set of dogs are permuted (i.e., 'cat'is assigned the set of dogs as its extension, and 'dog' isassigned theset of cats) is 'unintended' even if corresponding adjustments in theextensions of all the other predicates make it end up that the oper-ational and theoretical constraints of total science or total belief areall 'preserved'. Such a model would be unintended because we don'tintend the word' cat ' to refer to dogs. From the metaphysical realiststandpoint, this answer doesn't work; itjust pushes the question backto the metalanguage. The axiom of the metalanguage, '" cat" refersto cats' can't rule out such anunintended interpretation ofthe objectlanguage, unless the metalanguage itself already has had its intendedinterpretation singled out; but we are in the same predicament with "'.respect to the metalanguage that weare inwith respect to the objectlanguage, from that standpoint, so all is in vain. However, from the

viewpoint of' non-realist' semantics, the metalanguage is completelyunderstood, and so is the object language. So we can say andunderstand, '" cat" refers to cats'. Even though the model referredto satisfies the theory, etc., it is 'unintended'; and werecognize thatit is unintended from the description through which it isgiven (as in theintuitionist case). Models are not lost noumenal waifs looking for~omeone to name them; they are constructions within our theoryitself, and they have names from birth.

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hilary putnam - models and reality

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