indeterminacy of identity of objects and sets

Indeterminancy of Identity of Objects and SetsAuthor(s): Peter W. Woodruff and Terence D. ParsonsSource: Noûs, Vol. 31, Supplement: Philosophical Perspectives, 11, Mind, Causation, and World(1997), pp. 321-348Published by: Blackwell PublishingStable URL: http://www.jstor.org/stable/2216136 .Accessed: 28/12/2010 02:41

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Philosophical Perspectives, 11, Mind, Causation, and World, 1997

INDETERMINACY OF IDENTITY OF OBJECTS AND SETS

Peter W. Woodruff and Terence D. Parsons University of California, Irvine

1 Might Identity be Indeterminate?'

The purpose of this essay is to explore the idea that identity might be indeterminate. That is, that certain questions of identity have no answer, not because of an inadequacy in the language in which they are framed, but because of genuine indeterminacy in the world. In section 1 we describe this idea in general terms, in section 2 we give a classical "picturing" of the situation designed to allay fears that the idea is incoherent, in section 3 we discuss Evans-like attempts to disprove the view, in section 4 we extend the account to sets of objects, and in section 5 we discuss a classical alternative to the view of indeterminacy described in the first three sections.

1.] Examples of indeterminate identity

The hypothesis of indeterminacy of identity arises naturally as a plausible solution to puzzles in which identity questions seem to have no answers. Exam- ples in the literature fall into three classes. First, there is a question of identity over time when there is a simple disruption of some kind. For example, a person receives a new brain having the old memories, or a new set of memories is in- serted into an existing brain, or...There seems to be a single person under discussion before the disruption, and a single person under discussion after the disruption, and the question arises as to whether we are dealing with one person or two. Put in terms of identity, is person a (the person before the disruption) identical with person b (the person after the disruption)? Cleverly designed cases will under- mine any definite answer to the question, and one possible reaction to this is to hypothesize that this is because the world leaves the answer indeterminate (while leaving determinate the claim that there is exactly one person before the disruption, and exactly one after the disruption).

Additional complexities arise in cases of splitting across time. If a ship is repaired by having its parts replaced one at a time, and if the replaced parts are reassembled, is the repaired ship b identical to the original ship a, or is the reas-

322 / Peter W. Woodruff and Terence D. Parsons

sembled ship c identical to a? Here we have a non-identity to complicate the picture:

btc & aMb & a=c.

Again, there is the option that the question marks do not indicate uncertainty about a definite answer, but rather the lack of an answer, and again not because of any deficiency in language, but rather because of indeterminacy in the world.

A third popular sort of case arises at-a-time. Cats have indeterminate boundaries; there may be no answer to the question whether a molecule loosely attached to the tip of a hair that is midway in the process of being shedded is a part of the cat or not. Consider a cat, and consider the various cat-like objects that overlap it and that have determinate boundaries. (E.g. a catlike thing that definitely includes the molecule, a catlike thing that definitely excludes it, a cat-like thing that includes that molecule but excludes a certain other one, and so on.) Call these "p-cats." Then the p-cats are definitely distinct from one another, but there may be no answer to the question of whether the cat itself is identical with p-cat number 9, and so on for all the others. In this case the pattern is:

PI P2 & P 1P3 &. ..& cat pI & cat p2 &...

These three sorts of examples arise from different puzzles and motivations; what they have in common is the possibility that identity claims may have no answer. We will address what they have in common, by exploring the view that the ques- tionable identities are ordinary identities that are not made determinate by the world, while the obvious non-identities and identities are made determinate by the world.

1.2 Indeterminacy in the world

What would it be like for there to be genuine indeterminacies in the world, not due to indeterminacy of how our language relates to the world? Here is a description:

The world consists of some objects, and some properties and relations, with the objects possessing (or not possessing) properties and standing in (or not standing in) relations. Call these possessings and standings-in "states of affairs." Then some but not all such states of affairs are determined by how the world is.

Many people are willing to accept this possiblity with one reservation; they hold that identity is an exception to the claim that states of affairs can be indeterminate. If an identity sentence lacks truth value, then that must be because there is some indeterminacy in what its singular terms refer to. For identity in the world could not possibly be indeterminate; that would be incoherent. The purpose of this

Indeterminacy of Identity / 323

paper is to oppose this view. There is no incoherency in the view that identity itself is sometimes indeterminate.

We can anchor the debate somewhat if we accept the traditional Leibnizian definition of identity between a and b as indistinguishability of a from b in terms of the actual properties and relations that they have or stand in. Whether this holds in a particular case may be undetermined by the facts. If so, a and b will actually be neither determinately identical nor determinately distinct. This will be a case of indeterminacy of identity. The framework of properties and relations sketched above is neutral about whether this ever actually happens. It will happen if a definitely has every property that b definitely has (and vice versa), and a definitely lacks every property that b definitely lacks (and vice versa), but there is some property that a definitely has or definitely lacks when it is indeterminate whether b has it (or vice versa).

2 Describing the Theory

Many people sincerely doubt the cogency of indeterminate identity. They are confused by the notion, and descriptions such as those above, while helpful, do not completely clarify things, because the descriptions sound as incoherent as the view being defended.

In general, there are two ways to defend a view whose coherence is at issue. One way is to describe the view in its own terms, and to produce sufficient discussion of problem cases from this point of view that it begins to become clear that the view suffers from no internal incoherence. This is frus- trating to those who genuinely have difficulty in comprehending it. The second way is to picture the view in neutral terms. Below, we give a way to do this for indeterminate identity: to picture indeterminate identity within a classical bi- valent conceptual scheme that makes no use of indeterminate identity, just as a possible worlds modelling of modal logic can be stated in language containing no modalities. This picturing proves that the proposed view is coherent, because it can be coherently modelled by the classical view. Incoherencies in the view under discussion, if they existed, would show up as inconsistencies in the picturing.

2.1 Picturing indeterminacy

We can picture mundane indeterminacy using diagrams similar to Euler or Venn diagrams, in which objects are represented not by points but by small filled regions. We use circles to represent the extensions of properties, as in Venn diagrams. Then an object is pictured as having a property if its image is wholly inside the property's extension, and it is pictured as lacking a property if its image is wholly outside a property' s extension. When its image overlaps the boundary of the property's extension, that means that it is indeterminate whether the object pictured has the property:


Fig-ext of p

Lacksp p

/ ~~~~~Has p Neither has p nor lacks p

It is easy to extend this kind of picturing to identity of objects; objects a and b are pictured as identical if they are represented by the same image, and they are pictured as distinct if they are represented by disjoint images. If their images properly overlap, this pictures (the state of affairs) that it is indeterminate whether a and b are the same:

A B C

A is identical to A It is indeterminate whether B = C

A is distinct from B

With this convention, Leibniz' s definition of identity is nicely captured. It is easy to see that if the images picture two objects as distinct, then the images do not overlap, and a circle can be drawn to represent the extension of a property that one of them wholly has and that the other lacks. So objects represented as distinct will definitely differ in their properties. If the images coincide then no circle representing the extension of a property can distinguish the object(s) in any way.


So objects represented as identical must agree completely in which properties they have, which they lack, and which they are indeterminate with respect to. Suppose finally that the objects are pictured by means of overlapping images. Then neither of the above cases hold; no property can wholly distinguish one from the other (since any circle that encloses one image must enclose at least part of the other), but there are properties that one has or lacks and it is indeterminate whether the other has or lacks it (just draw a circle that totally encloses one image but not all of the other image).

So if having a property means being in the extension of that property, these diagrams picture the kind of indeterminacy of states of affairs (including states of affairs involving identity) that were described at the outset of this essay. Further, the diagrams validate Leibniz's account of identity. At least they do so if any region in the diagram is the extension of a property. And that assumption meshes with everything we have said so far. (We'll make a weaker assumption below, but it will not change the way in which the diagrams validate Leibniz's definition.)

Nothing about these diagrams excludes the possibility that one object image might fall totally inside another, or inside a group of others. By fiat, we prohibit this from happening for the applications discussed in this paper:

No objects are completely inside others

We prohibit these cases because in all of the applications of the hypothesis of indeterminate identity to solve puzzles from the literature such (onto)logical con- tainments of objects in others never arises as a possibility. Such overlaps generate radically different conceptual schemes; these are worth exploring, but we will not do so here.2

With diagrams of this sort we can picture the three types of problem situa- tions from the literature as follows:


Person a M person b

Ship b f ship c & ship a M ship b & ship a M ship c

pcat1 #pcat2 & pcat1 #pca & b

pcat, fpcat2 & pcat, fpcat3 &..& cat=?pcatj & cat=?pcat2 &..


2.2 A more rigorous formulation

If we are to nail down the logic of the situation, we need a more rigorous formulation of the picturing described above. Here is one, within a classical bi- valent theory.

Suppose that the image space of the picture consists of a set D of points, called ontons. The term 'onton' is purely suggestive; the points form the ontological basis of the space. Ontons are purely artificial, much as possible worlds are artificial in modal model theory.

We suppose that in any total picturing of reality there are a number of images of objects; each image is a set of ontons. The only constraint we impose on such a modelling is that each object image contain some ontons (at least two) that are not in any other object. The ontons in an image that are not in any other image constitute the core of that image.

Object images represent identical objects if they consist of the same ontons, they represent distinct objects if they share no ontons, and they represent indeterminately identical objects if they share some ontons but do not share some others.

Each property on D has a figurative extension, which is a set of ontons, and each relation on D has a figurative extension which is a pair of sets of ontons. An object image i pictures its object as having a property p iff every onton in i is in the figurative extension of p. An object image i pictures its object as lacking a property p iff no onton in i is in the figurative extension of p. And an object image i pictures its object as neither having nor lacking a property p iff some of its ontons are in the figurative extension of p and some are not. (Similar conditions apply to pairs of objects and relations.)

Assume for the moment that every set of ontons is the figurative extension of some property. Then the following versions of Leibniz' s account of identity turn out to be true on this account:

a is (definitely) identical to b iff a and b both have and lack the same properties

a (definitely) differs from b iff a has some property that b lacks, or vice versa

It is indeterminate whether a is identical to b iff there is no property such that a has it and b lacks it, and no property such that b has it and a lacks it, and there is some property that one of them has or lacks and such that the other is indeterminate with respect to having it.

Finally, we suppose that there is such a picture that accurately pictures reality. That is, we assume that there are indeed objects and properties, and that any object image pictures a unique object, and that every property has a figurative extension. An object has (or lacks) a property iff it is pictured as having (or lacking) it.


2.3 Semantics

In discussing a metaphysical thesis about identity it is important to distinguish issues about language from issues about the world. There is no way around discussing both. And so here is a sample semantics that takes account of the worldly indeterminacy of states of affairs, including states of affairs concerning identity. Most of the semantics is stated in terms that do not appeal to our modelling at all; so much of it makes sense on any account of indeterminate identity.

1. We assume that each atomic predicate is true of certain objects, and is false of certain other objects, and is thus neither true nor false of the remainder, if any. An atomic predicate may or may not express a property. If it expresses a property, then it must be true of exactly those objects that have the property and false of exactly those objects that lack the property. (Similarly for relations.)

2. Satisfaction:

(i) An object o satisfies 'Px' if 'P' is true of o; o dissatisfies 'Px' if 'P' is false of o, and otherwise o neither satisfies nor dissatisfies 'Px'. (ii) 'x=y' is satisfied by a pair of objects if they both have and lack the same properties, dissatisfied if one of them has a property that the other lacks; otherwise it is neither satisfied nor dissatisfied by that pair of objects.

3. We have to make some arbitrary choices about how our connectives and quantifiers work, in order to have any logical terminology at all. We shall use the commonest ones, which are these:

--A is true if A is false, false if A is true, and truthvalueless if A lacks truth value.

A&B is true if A and B are both true, false if either A or B is false, and otherwise lacks truth value.

(3x)A is true if A is satisfied by at least one object, false if A is dissatisfied by every object, and otherwise neither true nor false. VA is true if A lacks truth value, and is otherwise false.

As usual, we define AvB as (-A&-B). It is also convenient to define a "truth" operator, '>', such that '>A' is true if 'A' itself is true, and is false if A is either false or lacking in truth value. Its definition is:

>A =df A & --VA.

In the Appendix we state a set of rules for the logic of indeterminate identity, based on the above explanations.

3 Applications

Any theory that allows for indeterminacy in the world places constraints on how language might relate to the world. Suppose that we wish to add to our lan-


guage some lambda abstracts to form complex predicates, such as 'Xx[Ax & Bx]', i.e. the predicate of 'being both A and B.' If the language is sufficiently rich then we cannot assume that any such abstract refers to a property whose application to objects is perfectly characterized in the usual way by lambda abstraction, the principle that '?(a)' is interchangeable with 'Xx[1(x)](a)' in all extensional contexts. Such a powerful assumption leads to paradoxes (like the Russell paradox) whenever the language is sufficiently rich. Constraints of this sort are well-known, and people are used to restricting either the abstraction axiom or quantification over properties to avoid paradoxes.

Here is a less familiar constraint: one cannot take for granted that lambda abstracts that bind variables in contexts governed by the indeterminacy operator '7' stand for properties and also fully satisfy lambda abstraction. This may seem to be a dry technicality, but it is a crucial one. This is because Evans' 1978 argument and all arguments similar to it (that is, virtually all arguments on this matter in the literature) beg the question by ignoring this point-by assuming without argument that such abstracts automatically stand for properties and that they simultaneously satisfy the abstraction principle. We are not in a position here to find fallacies in arguments for these joint assumptions, for such arguments are never given.3 But we can say with some confidence that the assumption that arbitrary abstracts both stand for properties and satisfy lambda abstraction cannot be proved from the general view of the world stated in section 1. It is an additional assumption, and one that must be rejected by any proponent of indeterminate identity. Indeed, Evans' argument proves this.

3.1 Evans'Argument and Leibniz's Law

To get clear on this, consider the Evans argument. It is a proof that if there were a case of indeterminacy of identity, the supposed indeterminate identity could be shown to be a determinate non-identity. That is, from the claim that there are objects a and b such that it is indeterminate whether a is b one can prove that a is not b:

1. V(a=b) Hypothesis. 2. Xx[V(x=b)](a) From 1 by abstraction. 3. V-7(b-b) (Definite) truth of self-identity. 4. -Axx[7(x=b)](b) From 3 by abstraction. 5. (3P)[P(a) & -1P(b)] Conjoin 2 and 4 and existentially generalize. 6. --(a=b) From 5 together with the definition of identity.

Thus from the assumption that it is not determinate whether a is b, we prove that a is not b.

Several authors point out that the argument as stated is too strong; surely identity statements can lack truth values if there is linguistic vagueness in the singular terms.4 We agree with most commentators that Evans intended his argument to prove that if the singular terms are not themselves vague, then there can


be no truth-valueless identity statements. If the only escape from the argument is to hold that the terms are linguistically vague, then the argument successfully disproves the possibility of indeterminate identity in the world.

Since one can coherently maintain that it is indeterminate whether a is b, we regard this proof as establishing instead that the fault may be taken to lie else- where, in particular, either in the abstraction steps or in the existential generalization step. We cannot assume that there is a genuine property ('not being determinately identical to b') that behaves in accordance with the full abstraction principle. To avoid assuming this, we have two options. One is to assume that the principle of abstraction always holds for predicates, but to leave it open whether the resulting predicates stand for properties. If we do this, the above argument begs the question at step 5. The other option is to take for granted that abstracts stand for properties, but reject the principle of abstraction as always providing the conditions under which such properties hold of objects. Then the abstraction steps 2 and 4 may be unjustified.

Recall what is at issue. It is not whether property abstracts are meaningful; that is a conceptual point, and says nothing about what there is. The issue is whether for every such abstract there is a property in the world for which the abstract stands. In the case of identity, the issue of how identity behaves in the world is not a conceptual matter, it is an ontological one. It is characterized in terms of properties and relations, not in terms of concepts or meanings. And so assuming that a property abstract is meaningful does not mean that it stands for a property. The abstract used in the argument does not stand for a property, or it stands for a property that is not fully equivalent with the subformula in the abstract.

Evans' own argument appears to avoid these considerations, since that argument does not quantify over properties at all. Evans apparently bypasses any appeal to properties by moving directly from lines 2 and 4 to line 6, citing Leib- niz's Law:

From a=b and ...a... infer ...b...

This is a good argument form. If one of your premises is 'a=b', then you are assuming that 'a=b' is true. But if it is true, it is determinately true, and a determinately true identity should sanction interchangeability of its terms (assuming that there are no non-extensional contexts at issue). We agree completely, and we note that the metaphysical account of identity sketched above sanctions this version of Leibniz's Law. But Evans does not appeal to Leibniz's Law; he appeals to its contrapositive. It is essential to distinguish Leibniz's Law, which is valid for anyone who is discussing real identity, from contrapositive versions of it, which we dispute. This is because most attacks on indeterminate identity in the literature employ principles that are contrapositive versions of Leibniz's Law, such as the following:


From ... a...

and -1(...b...) infer aVb.

Our framework does not endorse this principle, even though it does endorse Leib- niz' s Law.

It is hard to shake the idea that if an inference is valid, its contrapositive should be too. But contraposition does not hold for systems that allow indeterminacy, even if indeterminate identity is ruled out. In fact, it does not even hold for systems that presume that the world is completely determinate but that language is sometimes so flawed as to allow the formation of sentences that do not have truth value. Suppose there is some sentence S in some languge that lacks truth value, and suppose that this language contains the connective '7' that forms true sentences from truth-valueless ones and false sentences from truth valued ones. Then the following is certainly valid:

S/... -,7S

But its contrapositive is not even remotely plausible:

VS /.*. -is.

(When we discuss validity we mean that if the premise is true, the conclusion has to be true too. This notion of validity is the right one to focus on because it is what is at issue in all of the philosophical arguments on the topic. ) Leibniz's Law is a principle of our system, but its contrapositive holds only for certain "well- behaved" cases.5

3.2 Abstracts

None of the above discussion entails that we cannot have abstracts in our language.6 There are two options for handling abstracts. One is to insure that abstracts stand for properties, thereby abandoning full abstraction principles (as proved above); the other is to hold onto the full abstraction principles but without assuming that abstracts always stand for properties. Call the first 'ontological abstraction,' since such abstracts are guaranteed by the semantics to stand for properties, and call the second 'conceptual abstraction,' since these abstracts are guaranteed to reproduce the conceptual content of the formulas from which they are generated. Both can be added to the language.

Ontological abstraction is symbolized by X*x[FDx], where the asterisk on the A indicates the ontological loading. Ontological abstraction is interpreted as follows:

X*x[(Fx] stands for a property whose figurative extension contains all ontons that are in objects that (definitely) satisfy 'Fx plus some but not all ontons in the core of each object that neither satisfies nor dissatisfies


Conceptual abstraction (using a 'c' for 'conceptual') produces complex predicates that work as follows:

Xcx[(ix] is a predicate that is true of an object o iff o satisfies IDx, and is

false of o iff o dissatisfies 1'x.

What logical principles are validated by these accounts? To answer this, we

need to make some distinctions that are not needed in a classical setting. First,

there is the question of the validity of the ordinary abstraction inferences:

ABSTRACT INTRODUCTION:

(Da

X[. x](a)

ABSTRACT ELIMINATION:

Xx[1'x](a) (Da

These principles are valid without restriction for either sort of abstract even if (D

contains instances of the indeterminacy operator. That is, in any such case, if the

premise is true, so is the conclusion. Thus all of our abstracts are minimally

well-behaved.

But in a non-classical setting, we can ask another question: must the conclu-

sions of the above arguments be indeterminate (or true) if the premises are inde-

terminate? The answer is yes for conceptual abstracts for both inferences, since

the conclusion must be indeterminate if the premise is. For ontological abstracts,

this also holds for abstract introduction: if the original is indeterminate then the

conclusion will be indeterminate too. But abstract elimination for ontological

abstracts can take you from indeterminate to false. This means that the contra-

positive of ontological abstract elimination:

--a 8,X*x[(Dx](a)

can have a true premise and an indeterminate conclusion, and it is thus not valid.

This is the form of the inference in step 4 in the argument above.

In summary, the Evans argument can either be taken to appeal to conceptual

abstracts or to ontological ones. If the abstracts are conceptual, they do not nec-

essarily stand for properties, and the fallacy lies in assuming that they do (step 5). If the abstracts are ontological, the fallacy lies in assuming that the contrapositive of abstract elimination is valid (step 4).

4 Sets of Objects

Sets are important to discuss because there are arguments based on set theory

both for and against the claim that there is indeterminate identity. The point of this

section is to extend our picturing to include sets, and to assess these arguments.


Sets of objects are the extensions of properties of objects, and thus we already have them pictured in our model: a set image is any figurative extension. An object is pictured as (definitely) being a member of a set if the object image lies totally within the figurative extension that pictures the set, and it is pictured as (definitely) not being a member if its image is totally outside that figurative extension. If a set image lies partly within and partly outside of a figurative extension, that portrays a situation in which it is indeterminate whether the object is or is not a member of the set.8

The point of this section is to develop a precise account of sets of objects and explore their indeterminate identities.

4.1 Principles of set theory

It is unnatural to conceive of every region (every set of ontons) in the diagrams described above as representing a distinct possible extension, since some regions differ in completely artificial ways. Suppose, for example, part of our picture looks like this:

If there are no other objects in the logical vicinitiy, the following different picture should not picture anything different:


Let us say that two regions are ontologically equivalent if each wholly encloses the same object images as the other and each wholly excludes the same object images as the other (and thus each partially includes and partially excludes the same object images as the other). Ontologically equivalent regions should represent the same extension. The conceptually simplest way to accomplish this is to select from each class of ontologically equivalent regions one member of the class to represent all of them. Call this the canonical extension for that class. The fact that there is more than one region in a given equivalence class is a purely artificial product of our picturing conventions, and has no ontological signifi- cance, so we can just speak in terms of the canonical ones.

Canonical extensions picture sets of objects. Suppose that we call any canonical extension a set image, and we characterize membership of an object in a set as follows:

o is definitely a member of set S iff o's image is totally included in S

o is definitely not a member of set S iff o's image is totally disjoint from S

o is neither definitely a member of set S or definitely not a member of S iff o' s image is partly included in S and partly outside of S

We say that two set images represent the same set if they are in fact the same canonical extension; they represent distinct sets if there is an object image totally within one that is totally outside the other; otherwise it is indeterminate whether the sets they represent are the same.

What principles of set theory do these assumptions yield? To express the strongest principles, we need a biconditional stronger than the material one. We would like a biconditional A=B to be true when A and B have the same truth value or both lack truth value, false when they have opposite truth values, and lacking in truth value when either of A or B has a truth value and the other does not. Such a biconditional can be defined by:

A?B =df (A&B) v (-,A&-,B) v (VA&VB).

Using this connective, the following principle of set extensionality is always true in our modelings:

A=B = Vx[xEA = xEB]

To say that a biconditional of this sort is true is to make the strongest possible claim about the equivalence of its two sides; it means that they are true together, false together, and also lack truth value together. Suppose we write A-B for

>[A=B]-


Then saying that the former principle is validated by our semantics as true is equivalent to saying that this is true:

STRONG EXTENSIONALITY: A=B - Vx[xEA = xEB]

Thus our principle of set extensionality has the strongest formulation that one could hope for. The formulation allows that membership in a set might be either determinate or indeterminate, and the right hand side of the major biconditional says of sets A and B that any object is either determinately in both A and B or determinately not in either A or B or is such that it is indeterminate whether it is a member of either. And the whole sentence says that whether or not this coincidence in membership holds for all objects is definitive of whether A and B are the same; if the coincidence always holds, then A and B are the same, if it does not always hold, then A and B are distinct, and if it is indeterminate whether it always holds then it is indeterminate whether A and B are the same. The identity conditions for sets are thus determined exactly in terms of the membership of objects in them.

What about set comprehension? Is there a set corresponding exactly to every formula? That is, is this principle true:

STRONG COMPREHENSION: 3SVx[xES - (Dx].

This principle holds for a wide range of cases, but not all. Suppose that (F is the sort of formula typically employed in set theory; in particular, suppose that every atomic predicate in (F has an extension, and that (F is built up out of names and atomic predicates and identities by means of the Boolean connectives &, v, and the quantifiers 3 and V. Then the above formula holds for (D. But if (D contains some predicates that do not have extensions, or contains non-classical connectives, such as 'V', then it may not hold.

Here is another way to view things. Let us introduce set abstracts, which mimic ontological property abstracts, i.e. with the following semantics:

{x: ( } stands for the smallest canonical extension that contains every onton in every object that satisfies (D and contains some but not all ontons in the core of every object that neither satisfies nor dissatisfies (D.

(Recall that the core of an object is the set of ontons in the object that are not in another other object, and we assumed above that each object has a core with at least two ontons.) In our modeling there is always a unique canonical extension that satisfies the above condition, so such abstracts always pick out unique sets. Further, no matter how (D is formulated, the abstracts always satisfy these classical conditions:


CLASSICAL COMPREHENSION:

(Da aE {x:iFx}

aE{x:iFxl .-. (Da

However, the stronger three-valued condition

STRONG COMPREHENSION: Vy[yE{xAf x} _ (Dy]

does not always hold. Again, it holds for the typical formulas of classical set theory, but not necessarily for formulas containing non-classical terminology.9

We have confined our discussion here to sets of objects, because these can be pictured. We speculate that it is possible to extend the account to a full extended set theory of the usual sort, and to recover from this theory of indeterminate sets a subfragment that mirrors the classical Zermelo-Frankel universe.'0

4.2 An argumentfrom set theory against indeterminate identity

Salmon 1981 gives the following argument against the possiblity of indeterminate identity for objects:11

1. Suppose Vx=y 2. But --Vx=x 3. So {x,y)f{x,x) 4. So xOy.

There is nothing wrong with 1, which is a supposition, or 2, which is validated by our modelling. But how do we get to 3? Since there are no intermediate steps given, one can either speculate about the fill-in reasoning, or just evaluate things directly. In our modeling, hypothesis 1 entails:

3'. V({x,yj}{x,x}).

It does not validate 3. Since many people have endorsed Salmon's argument, it is tempting to try to

see what is behind their intuitions. And there is a natural idea in the wings: Leib- niz's Law, which is often cited in this connection. But the argument does not appeal to Leibniz's Law, it appeals to its contrapositive, which is not valid.

4.3 An argument from set theory in favor of indeterminate identity

The implementation of set theory in this framework is important for a second reason. Suppose that there is no indeterminacy of identity among objects at all, but suppose that there is some property indeterminacy; e.g. it is indeterminate whether a is P, where 'P' stands for some property. Then consider the sets:


A= {x:Px} B ={x:Px v x=a}

It is an immediate consequence of our modeling that it is indeterminate whether A=B. So any kind of indeterminacy for objects leads inevitably to indeterminacy of identity for sets. (This is then a kind of indeterminacy that differs from the three sorts of examples given at the beginning of this paper.) Indeterminacy of identity is much harder to avoid that most of its opponents suppose.

Is this just an artefact of our model? We think not, for arguments can be given for this conclusion that do not exploit the model at all. Consider first the following two proofs that jointly show that 'Pa' cannot be indeterminate:

1. SupposeA=B. 2. Then Vx[Px = Px v x=a] Comprehension and Extensionality 3. Then Pa = Pa v a=a Instantiation 4. Pa Tautologically implied by 3 5. -,VPa 4, Meaning of 'V'

1. Suppose A*B. 2. Then -,Vx[Px = Px v x=a] Comprehension and Extensionality 3. Then [-,IPb = Pb v b=a]

for some object b Quantifier Equiv. 4. Either Pb & -1(Pb v b=a)

or -1Pb & (Pb v b=a) Taut Impl by 3 5. The left disjunct of 4

is inconsistent. Taut 6. -1Pb&b=a 4,5,TautImpl 7. -1Pa 6, Leibniz's Law (not its contrapositive!) 8. -,VPa 7, Meaning of 'V'

These arguments assume nothing about indeterminacy of identity or about our models. But suppose that 'Pa' lacks truth value. Then each proof becomes a reductio of its hypothesis, and so the proofs jointly refute both A=B and A*B. This is not an inconsistency, but only because there is a third option: the refuted sentences lack truth value. Thus, the only way to maintain the indeterminacy of 'Pa' in the presence of normal principles of set theory is to admit the indeterminacy of 'A=B'.2

So the assumption that Pa is indeterminate leads to the conclusion that there are sets whose identity is indeterminate. The obvious way out of this argument is to abandon the possiblity of doing set theory on the assumption that there is genuine indeterminacy in the world (that is, giving up ordinary instances of set comprehension or extensionality), or to insist that all indeterminacy (not just that involving identity) be disavowed as a matter of theoretical simplicity. Quine 1981 seems to offer this dilemma, and to favor the latter choice. But the dilemma is a


false one. The argument given above shows that indeterminacy of set identity is a consequence of worldly indeterminacy regarding objects possessing properties; it does not show that a theory that accomodates these indeterminacies is less simple than one that tries to get around them. Indeed, as we have tried to show, the theory of sets of objects is simple and straightforward even if set identity can be indeterminate. There is no insuperable complexity here, there is only unfamiliarity.

Further, if simplicity is the issue, then one really needs to examine the simplicity of ways around admitting indeterminacy, other than just trying to look the other way. These usually don't bother people because these ways haven't been articulated and subjected to criticism. We suspect that if this were done, the indeterminate identity view would look at least as good.

5 A Determinate World

The object images discussed in the two preceding sections will look to some people like blurry parts of a picture of a world that may not itself be blurry. This is a theme that is found throughout the literature: that apparent indeterminacy of identity is a product of a deficiency in the conceptual apparatus (perhaps, but not necessarily, of our language) that we use to represent a world that is itself completely determinate. One version of this is the idea that identity puzzles are to be solved by "refining our concepts." (Cf. Stalnaker 1988 for a view something like this.) In this section we criticize this idea.

Some writers on identity have suggested that apparent indeterminacy of identity is due to unclarity in our concepts. If our concepts are clarified, we discover a world in which identity is completely determinate. When there is a genuine puzzle about identity, this is because the key concept(s) appealed to in the puzzle can be clarified in different ways. For example, if our concept of a ship is clarified in one way, then this identifies the original ship with the reassembled ship, and if it is clarified in another way, then this identifies the original ship with the repaired ship. This is not quite accurate, since in terms of unclarified concepts the phrase 'the original ship' does not pick out a ship at all. The more accurate thing to say is that if our clarified concepts are ship, and ship2, then 'the original ship' does not denote anything at all, and instead, we have that

the original ship, = the reassembled ship, the original ship2 the repaired ship2

These are equally true in the actual world, and there is no indeterminacy of identity to confound us.

This view is not compatible with the indeterminate identity view. For one thing, the refined-concept view requires a larger ontology than the indeterminate identify view. The total number of ships in the indeterminate identity model is more than one and less than three. (It is indeterminate whether there are two or three.) The refined concepts view requires more objects. Let place-o be the early


location of the original ship, place-a the later location of the repaired ship, and place-s the later location of the reassembled ship. The notion of ship can be refined so that the following are all simultaneously true:

the original ship, = the reassembled ship1, and the original ship1 0 the repaired ship,. So a ship, is located at both place-o and place-s, and a (different) ship, is located at place-a.

the original ship2 = the repaired ship2, and the original ship2 0 the reassembled ship2. So a ship2 is located at both place-o and place-a, and a (different) ship2 is located at place-s.

the original ship3 0 the reassembled ship3, and the repaired ship3 * the reassembled ship3. So a ship3 is located at place-o, and a different ship3 is located at place-s, and a (still different) ship3 is located at place-a.

This requires five "ships" in the actual world (using the unrefined notion of ship), more than any of the refinements of the indeterminate identity view. Of course, we shouldn't talk that way, since ship is a notion in need of refinement. Where we thought there was one ship (i.e. where "the original ship" was) there are actually three ship-like things: a ship, (which later ends up where "the reassembled ship" is), a ship2 (which later ends up where "the repaired ship" is), and a ship3 (which ceases to exist during the repair/reassembly). All of these are distinct from one another. (There are also multiple ships later on, two of them in each location.) These multiplicities are either the strength or the weakness of this alternative.

This sort of burgeoning ontology may be untroublesome for ships, but it is more difficult in the case of people who undergo disruptive changes. We need to ask what are the conditions for refinement of the notion of person. Some natural answers that come to mind are that people are things that think, they are things that pay taxes, and they are things that we marry. If these sorts of paradigms do not hold then it is not plausible to claim that we are refining the notion of person, we are simply changing the subject. Now refinements in the notion of person will not end up with there being multiple persons in one place at a time, since our concept of person stands in need of refinement, and should be dispensed with. But (like the ship case) the refinements will require there to be in a single place both a person, and a person2. If these "refined people" have the characteristics that are essential to refinements of persons, then in a given place will be located two or more things that think and feel and desire, two or more things that pay taxes (do they both pay the same taxes, or have there been duplicate payments, or...?), and, in one case, two or more things that you married. This is disconcert- ing to say the least.

There are two problems here. The immediate problem is that the view under consideration is not a developed view at all; we don't really have an option to consider, but many options, depending on what refined things are supposed to be like. The options need to be spelled out. We need to know, for example, whether


we get two person-like things that share a body, or two body-like things too. The second problem is the suspicion that if the view is spelled out it will be incoherent. But it is premature to address that before seeing the options.

A third issue lurks. We still need to make sense out of our ordinary talk of identity, in which we speak as if there is a single person in a given place at a given time. Even if this is an illusion, based on a sloppy use of unclear concepts, the sloppy illusory talk seems to be systematic, and one would like an account of it in terms of the unsloppy realistic view. But when this is done, perhaps the precise reconstruction of the sloppy talk about inexact identity will be what we all mean now when we apparently talk about identity. And perhaps this reconstructed identity will be indeterminate. And then perhaps the indeterminate identity view will turn out to be true. Derivatively true, perhaps, but true nonetheless.

Appendix: Modellings and Rules

In this appendix we make precise the three-valued logic we are dealing with, by giving two (equivalent) semantic modellings and an axiomatic formulation. The syntax of the languages has already been indicated: a first-order language whose primitives are negation (-), conjunction (&) and the existential quantifier (3), as well as identity (=) and the additional operator < (non-falsity).13 From the latter we may define V (indeterminacy), A (determinacy), and > (definite truth) as follows:

Vb =df

AO+ = df-__V

>D =dfV<--

1. Two Semantic Modellings

We will begin by formalizing the semantics given in the paper. Then we will give an apparently more general three-valued semantics, but will show that the two are equivalent in the sense that they recognize the same inferences as valid.

a. Ontic models i. Basic notions An ontic model for our language will be a triple M = (0, D, v) such

that 0 is a set (called the set of ontons of M), D a set of subsets of 0 (the objects of M) and v an interpretation which assigns to each individual constant an object in D and to each n-ary predicate a function from Dn to {t, u, f}. An assignment in D is a mapping from the variables of our language to objects in D; if a is an assignment and dED, a(d/x) is like a save for assigning d to x. An assignment determines for each singular term a denotation in D and for each, formula a truth-value in { t, u, f}; we write vae for the value of the expres- sion e and define this recursively as follows: 14

voa[t] = Pt if t is a constant, otherwise at.

va[P(tl, .. t)] t = P(Pa[tl], .*-, val[tn])


v-a[s=t] = t if vza[s] = va[t], = f if voa[s] n voa[t] = 0, and = u otherwise.

= t if v'a[b] = f, = f if va[b] = t, and = u otherwise

var[O&q] = t if vza[b] = va[g] = t, = f if voa[b] = f or voa[q] = f, and = u otherwise.

voa[3xb] = t if for some dED, voa(d/x)[b] = t, = f if for all d, voa(d/x)[(b] = f, and = u otherwise.

We say that a formula 4 is a consequence of formulas E (E k b) iff 4 is true whenever (i.e. on every model and assignment on which) all of E is.

ii. Nuclear models In sections 2.1 and 2.2 we made the assumption that every object has a core subset of at least two ontons which belong to no other object. Let us call ontic models which satisfy this assumption nuclear models. It might appear that limiting ourselves to nuclear models is a strong assumption, but we shall see that they are in fact as general as those above, and lead to the same theory.

iii. Figurative extensions Up to this point we have not mentioned figurative extensions (Section 2.2), nor the properties which have them. Indeed, as noted in section 2.3, the modelling given above is independent of these notions. But we can introduce them in the following way. Let p be a three-valued function on D (call such functions "concepts" for the time being). Then we say that a set S of ontons is a figurative extension of p iff for all dED, p(d)=t iff dCS and p(d) =f iff dnS =0. A concept is a property iff it has a figurative extension.

It is worth noting that we can give a more direct characterization of properties. To this end, let the center of a concept p be the set of objects which satisfy it; the penumbra be the set of objects which neither satisfy nor dissatisfy it; and the fringe be the set of objects which overlap some member of the center but do not themselves belong to the center.15 Then

Proposition 1: If p is a property, the fringe of p is a subset of the penumbra of p. In nuclear domains, the converse holds as well: a concept whose penumbra contains its fringe is a property.

Proof: If p has a figurative extension, say S, then every member of the center is a subset of S, hence every member of the fringe overlaps S, hence does not dissatisfy p and so belongs to the penumbra. Conversely, suppose p satisfies the condition and D is nuclear. Take S to be the union of the center of p, together with exactly one element from the core of each penumbral object.


Then if p(d) = t, d is in the center and hence a subset of S. If p(d) = u, then by construction d overlaps S; but also, since D is nuclear there will be some core element of d which is not in S. Finally, if p(d) = f, d is neither penumbral nor central; since (by assumption) the fringe of p is contained in the penumbra, d cannot be in the fringe, hence cannot overlap any central object. The only way, then, that it could overlap S is to contain a core element of some penumbral object, but then it would have to be that object, which is impossible.

The preceding discussion has been restricted (for simplicity) to singulary properties as opposed to relations, but the generalization is straightforward. Let one n-tuple of objects Overlap (Include) another iff it overlaps (includes) at each coordinate. Then all of the above definitions may be repeated with capitalized concepts, and the new version of Prop- osition 1 remains true.

b. General models

Ontic models give considerable structure to the objects of their domain. An apparently more general approach is to leave these objects unanalyzed, taking a general model to be simply a pair (D, v), with v as before. The one thing which must be modified is the recursive clause for identity, the only one which uses the ontic structure. We must now provide more directly for the interpretation of =. Thus = is treated like any binary predicate, except that we require

=1) v'=(d,e)=tiffd=e =2) v=(d,e) = f iff v =(e,d) = f

Clause =1) is as before; clause =2), on the other hand, is not a definition (as such it would be circular) but only a constraint on v. Given = 1), =2) is equivalent to postulating the symmetry of v= as a function: v=(d,e) = v=(e,d).

Because of the symmetry of disjointness, it is evident that every ontic model (0, D, P) contains an equivalent general model (D, v). But what may be surprising is that ontic models, even nuclear ontic models, are fully as general as (so-called) general models.'6

Proposition 2: For every general model (D, P) there is a nuclear model (0, D*, P*) and a bijection * from D onto D* such that for all terms t, (Pat)* =

P*a*t, and for all formulas b, Pao = P*a*4o.l7

Proof: Given (D, P), let a and b be distinct objects not in D, and let 0 consist of the set of (unordered) pairs of objects in DU { a, b}.'8 For dED, let d* have as members {a, d}, {b, d}, and all {e, d} such that eED and P=(d,e) 0 f. Then d* is a subset of 0, and we can choose D* to be the set of all d* for dED. Note that if d* = e* then { a, d } Ee*; since there is exactly one set in e* which contains a, we must have { a ,d } = { a, e }; since a must be distinct from both d and e, we must then have d = e.19 Thus * maps D 1-1 onto D*. We will need the following

Lemma: v=(d, e) = f iff d*ne* = 0.


Proof of Lemma: Suppose v=(d, e) = f. Then dOe and {d, e1Xd*. Now suppose {g, h} Ed*nle*. Then (since every member of d* contains d) either d = g or d = h, and similarly, either e = g or e = h. Since d and e are distinct, it follows that we have d = g and e = h or v.v., and hence { d, e } = { g, h I E d* after all. By reductio, the condition of the lemma follows. Conversely, if v=(d, e) 0 f, then by construction {d, elEd*; furthermore, by =2) and construction we have {d, e}Ee*. Hence d*ne* 0 0, q.e.d. for the lemma.

We may now prove the conclusion of the proposition by induction on terms and formulas. The only thing which really needs to be checked is that truth-values of identities are preserved. To this end, let d be vas and e be vat. We may assume by inductive hypothesis that v*a*s = d* and ,A*a*t = e*. It will suffice to show that v=(d, e) = v*a*s=t. Note first that v=(d, e) = t iff d = e iff (by the I-lness of *) d* = e* iff (by the above assumptions and the ontic semantics) v*a*s=t = t. By the lemma, v=(d, e) = f iff d*ne* = 0 iff (by assumption) v*a*snlr*a*t = 0 iff v*a*s=t = f, q.e.d.

It is a corollary of Prop. 2 that ontic and general models determine the same notion of consequence. For suppose, say, that X k 4k but that there is some general model in which E

is true and b not. Then that model would determine by Prop. 2 an ontic model which satisfied E but not 4, contrary to our supposition.

c. Artifacts of the model

It is clear from our results that very little use is made of the fact that the objects of an ontic model are sets of ontons. In fact, the only thing that matters is the ways in which objects overlap (something they could not do if they were not sets); overlaps are used to determine whether objects are distinct or merely indeterminately identical. Everything else about objects is an "artifact of the model" in the same way as is the fuzziness of a string model of a triangle: it is not part of what is being represented by the model.

To make this point precise, let an onticframe be a pair (0, D), D part of the power set of 0. A model (0, D, v) is said to be over (0, D). The theory of a frame is the set of all formulas valid in the frame, i.e. true on every assignment in every model over the frame. Two frames may be called overlap equivalent if there is a 1-1 map h from D onto D' such that for all d, e in D, doe iff h(d)oh(e). Then we have

Proposition 3: Overlap equivalent frames have the same theory.

The proof of this is very similar to that of proposition 2, consisting in setting up a correspondence between interpretations over the two frames, such that corresponding interpretations assign the same truth values; overlap equivalence is used to handle identity formulas. Reasoning similar to that in the proof of Proposition 2 also lets us establish:

Proposition 4: Every ontic frame is overlap equivalent to a nuclear frame.

This is why the restriction to nuclear frames is inessential. (Enriching our language with conceptual or ontic abstracts adds no essential expressive power; Propositions 3 and 4 remain true for the expanded language.)20


2. Natural Deduction for Indeterminate Identity

We now give a syntactical presentation of the set of valid inferences captured (in common) by the two modellings just given. These are natural deduction rules of two sorts, direct and indirect. The direct rules are typified by &elim, written

b&O kF

and read "4 is derivable from +&Xf". Indirect rules are typified by reductio, e.g.

E F <-,+

read "if if and <-1 / are both derivable from E and 4), then <-4) is derivable from E alone." The exact implementation of natural deduction (Fitch, Quine, Kalish/Montague, etc.) will be left to the reader's imagination.

The presence of < (non-falsity) in the above rules is characteristic for three-valued natural deduction, being required to mesh with the fact that asserting a line in a proof always means asserting it as true.

In the following rules, 4)(t/x) means the result of substituting t for x wherever x is free in 4); it is assumed that bound variables are rewritten as necessary to avoid clashes in carrying out such substitution.

a. Basic three-valued logic

Rules for Conjunction

&intro: b, o F ?b&/

&efim: O&O k ?> ? & qj k

<&intro: <sb, <q F <(?b&0)

<&elim: <(b&0) F <b <(b&q/) F <b

Rules for Negation21

RAA:

Rules for Non-Falsity22

<intro: b F <b


Rules for Quantification

3Eintro: b(t/x) F 3xX

Eelim:

<3intro: <b(t/x) F <3xb

< 3 elim: <3 x q

(provided y is not free in the conclusions of the elimination rules).

b. Theories of identity

The following two rules characterize determinately true identity.

Reflexivity: F t = t

Leibniz' Law: b(s/x), s = t F b(t/x)

To these we add the symmetry of weak (non-false) identity:

Weak Symmetry: <(s = t) F <(t = s).

Proof of the following proposition can be obtained along the usual Henkin lines, making 4 true in the canonical model if it belongs to the maximal set, false if <10 does not so belong, and u otherwise. For details see Woodruff 1969 and (in preparation).

Proposition 3. The above system of rules is sound and complete with respect to the notion of consequence defined in 1.1.2 above.

These rules make obvious what was implicit in our semantics: the theory of true identity is just classical identity theory, while the theory of weak identity is simply the theory of a symmetrical relation.23 This means that most of the interest of three-valued identity theory lies in the properties of weak identity, and it is here that one might look for interesting new concepts and axioms.24

Let us just mention two possible additions. The first is the "contraposed" version of LL which we call the principle of Definite

Difference: 25

Definite Difference: b(s/x), --,b(t/x) F -i(s = t)


This is equivalent to the following replacement principle for weak identity:

LL*: ?b(s/x), <(s=t) F <?b(t/x)

and to Evans' principle:

Evans: V(s = t) F (s = t),

the last formulation being the conclusion of Evans' famous argument. As we have already observed, DDiff does not follow from LL, but only from LL*, and the failure to note this is the Achilles' heel of Evans' argument. In fact, that DDiff is valid in a particular model for a context b(x) amounts to saying that the concept expressed by this context is a property (see 1.1.2 above). Those whose intuitions agree with Evans' will want to add one of these principles to those we have espoused. Any one of them together with the other rules is equivalent to the determinacy of identity: F -,V(s = t).

The second possible addition is a genuine development of the theory of weak identity, one which is especially relevant to ontic models. We call it the principle of Definite Iden- tity of Weak Indiscernibles (DWI):

DWI: Vx(<(s = x) X4 <(t = x)) k s =t

The adoption of this axiom leads to a rich theory of the ontological structure of indefinite objects (see the note after Proposition 4 above).

Notes

1. This paper recapitulates parts of Parsons & Woodruff 1995 in slightly more technical form, and extends the discussion to a consideration of indeterminate identity for sets.

2. These are touched on in Woodruff (in progress). 3. E.g. Pelletier 1989 claims that rejection of these principles is based on incorrectly

construing the indeterminacy operator as a modal operator. (This may be true of some people's rejection of them, but it is not true of us.) He does not go on to show that the principles are true.

4. See Broome 1984, Cook 1986, Burgess 1989, Garrett 1988, Noonan 1982, 1984, Ras- mussen 1986, Thomason 1982.

5. Its contrapositive holds for any formula whose predicates all stand for properties, and that contains nothing else except identities, the connectives and &, and quantifiers. "Gap-filling" connnectives such as 'V' or '>'can spoil the validity of the principle.

6. We thus disagree with Parsons 1980:14 if he is interpreted as saying that we cannot have abstracts in a language that contains the indeterminacy operator. (That interpretation may be stronger than he intended.)

7. Such a figurative extension will always exist; the "some but not all" condition is sat- isfiable because of our earlier assumption that every object has a core with at least two ontons. It is possible for more than one figurative extension to meet the condition given in this clause for ontological abstracts, but when this happens the distinct figurative extensions completely include and completely exclude exactly the same objects.

8. Do not confuse the fact that a figurative extension is literally a set of ontons with the fact that that figurative extension is an image of a set of objects. This is no different


from Venn diagrams, in which sets of spatial points on the page represent sets of objects that are not themselves spatial points.

9. The condition holds for some of these as well. It is a necessary and sufficient condition for this condition to hold that there are no objects x and y such that it is indeterminate whether x=y and such that x satisfies CF and y dissatisfies (D.

10. This is described in Parsons & Woodruff "Indeterminate Set Theory" (in progress). 11. Salmon uses ordered pairs instead of pair-sets, but this is not an essential change. 12. The argument as stated makes use of set abstracts, but this is not essential. The rea-

soning can be duplicated so as to derive ']X 3YVX=Y' from 'VPa' without using set abstracts in any proofs.

13. We choose non-falsity, rather than indeterminacy as primitive because of its role in the natural deduction system to be given later.

The propositional connectives we use are not functionally complete for three- valued logic; they suffice to express all those functions which in the terminology of Langholm 1988 are determinable and persistent. We could make them complete by adding a constant * with the value u, but there appears to be little motivation in the present context for such a constant. Indeed, on the principles we espouse it appears likely that all cases of indeterminacy will be contingent.

14. The semantics for identity follows the discussion of the paper. The clauses for & and are those of the so-called Kleene strong connectives (Kleene 1952, Section 64).

15. Van Inwagen 1988:261-62 uses the term 'frontier' in a manner similar to our 'penumbra', and uses 'fringe-referent' differently from but suggestive of our 'fringe'.

16. The construction at the heart of the following proposition is similar to one used, for slightly different purposes, in van Inwagen 1988.

17. Here P* and a* are the maps on interpretations and assignments induced by * in the usual way. For instance, a*x =df (ax)*, v*P(d*) =df vP(d).

18. There is an assumption involved here that there will always be distinct objects a, b not belonging to D. If we take our background set theory to be ZFC, and recall that D must be a set, there will always be such objects (for instance, D and ID}).

19. This argument shows incidentally that I a, d } (and, for similar reasons, I b, d }, will not belong to e* for any e-d, so that D* satisfies the nuclearity condition.

20. Things would change radically if we were able to express more of the set-theoretic structure of objects in our language. Suppose, for instance, that we require enough objects in D to define inclusion in terms of overlap: d c e if for all gED, god only if goe. Then the following axiom will be verified:

Vxy. Vz(<x=z <-4 <y=z) -* x=y

and we can define "inclusion" of one object in another by

XEY =df VZ- <X=Z -* <y=Z.

Then the existence or non-existence of cores can be stated in the language, and nuclearity is no longer an artifact. We intend to investigate these richer models in a future paper.

21. To save space, we make the following convention: 4 and A- o are opposites, as are --b and AO. -4 stands for any opposite of b.

22. In view of the ubiquitous role of < in this system it would perhaps be more perspic- uous to treat these rules as structural rules in the sense of Curry 1963, p.186.


23. The first of these claims is true in a strong sense; if we restrict ourselves to formulas constructed out of atomic formulas of the form >s=t, then all other occurrences of <, V, etc. can be eliminated and (for these formulas) classical predicate logic holds. Thus the pure theory of true identity (in this sense) is just isomorphic to classical pure identity theory.

On the other hand, the theory of weak identity is wholly unconstrained except for symmetry, with the consequence that, in contrast to classical pure identity theory, three-valued pure identity theory is undecidable.

24. Essentially the only axioms which can be added to pure classical identity theory are statements about the size of the domain, for instance: 'VxVyx=y'.

25. Incidentally, though from the proof theoretic point of view it is preferable to concen- trate on weak identity, from a philosophical standpoint there is considerable evidence that true difference (the contradictory of weak identity) is the more fundamental notion.

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