Counterpart-Theoretic Semantics for Modal Logic

Counterpart-Theoretic Semantics for Modal Logic Allen Hazen The Journal of Philosophy, Vol. 76, No. 6. (Jun., 1979), pp. 319-338. Stable URL: http://links.jstor.org/sici?sici=0022-362X%28197906%2976%3A6%3C319%3ACSFML%3E2.0.CO%3B2-H The Journal of Philosophy is currently published by Journal of Philosophy, Inc..

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although the existence of particular theorems of logic may be explained in terms of rules that define individual constants, the notion of logical truth depends on the notion of truth for a language. If a nonstandard logic is possible, in a way that is not parasitic upon classical logic, then a nonclassical notion of truth and consequence is possible. But if a nonstandard logic must ultimately be explained using classical logic, then indeed we would have found something that "our thought can overflow, but never displace." IAN HACKING

Stanford University

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FOR MODAL LOGIC *

D

AVID LEWIS'S proposals for the analysis of de re modal locutions represent a major advance in thinking on the subject, but have not had the amount of critical study they deserve. In this paper I will inquire whether they constitute an adequate semantic theory of d e re modality. In doing so I will not discuss Lewis's notorious metaphysical realism with respect to possible worlds. T h e semantic theory embodied in his postulates for counterpart theory and his schema for translating a modal language into a counterpart-theoretic one are compatible with any of a wide range of metaphysical theories about possible worlds and the nature of possibility; arguments about which of these metaphysical theories is correct are therefore irrelevant to the semantic issue. In particular, adoption of a counterpart-theoretic semantics is compatible with taking possible worlds to be purely abstract structures, with the

* This paper is a counterpart of a portion of my dissertation (Unirersity of Pittsburgh, 1977), but has benefited from comments on an earlier version. I would like to thank my advisor, Joseph Camp, and also David Lewis, Robert Purdy, and Jay Hartman Hazen for their various kinds of help and encouragement. 1 In "Counterpart Theory and Quantified hlodal Logic," this JOURNAL, LXV, 5 (Xfarch 7, 1968): 113-126. Acquaintance with this paper is presupposed; referelices to Lewis, unless o t h e m i ~ enoted, are to it. Cf. also his "Counterparts of Persons and Their Bodies," this JOURSAL, L X ~ I I7I , (April 8, 1971): 203-211. 2 This paper considers only the first-order logic of necessity and possibility, treated by Lewis in the papers cited, with the addition of an actuality operator. Counterpart-theoretic semantics have also been proposed for other (more often employed?) intensional locutions, such as counterfactuals. Cf. Lewis, Cou?tterfactuals (Cambridge, hlass.: Harlaid, 1973), pp. 36-13. 0022-362X/i9/i606/0319$02.00

0 1979 ?'he Journal of Philobophy, Inc.

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same ontological status, whatever it may be, as the structures studied in pure mathematics. (There is, of course, no reason why all possible worlds have to have the same ontological status: one might, for example, hold that the possible worlds corresponding to merely logical possibility are merely abstract structures, but that those corresponding to some kind of physical possibility have a more concrete kind of existence. Certainly the radical treatments of the problem of future contingents don't seem to regard alternative futures as merely abstract structure^.^) There are arguments purporting to show that the central idea of counterpart-theoretic semantics-allowing an entity in the domain of one world to represent an entity distinct from itself in the domain of another world-is misguided. Thus Saul Kripke has written: Strictly speaking, Lewis's view is not a view of 'transworld identification.' Rather, he thinks that similarities across possible worlds determine a counterpart relation which need be neither symmetric nor transitive. T h e counterpart of something in another possible world is never identical with the thing itself. Thus if we say "Humphrey might have won the election (if only he had done such-and-such)." we are not talking about something that might have happened to Humphrey but to someone else, a 'counterpart'. Probably, however, Humphrey could not care less whether someone else, no matter how much resembling him, would have been victorious in another possible world.

This passage is an attempt to mobilize our intuitions (and, in the last sentence, Hubert Humphrey's putative intuitions) against Lewis's semantic theory. In judging it we must first be clear about the way in which linguistic intuition is relevant to semantic theory. T h e prime requirement on a semantic theory is that it assign truth conditions to sentences of our natural language (or to those of some language whose sentences are taken as translations of the sentences of our natural language) that are in accord with our intuitions. Our intuitive judgments, made "upon reflection," after we have assured ourselves of the nonlinguistic facts, of what is true and what implies what, are the appearances that a semantic theory must save. I t is, however, important to distinguish just what the subjects of these intuitive judgments are. T h e judgments are about sentences of 3 Cf. Richmond H. Thomason, "Indeterminist Time and Truth-value Gaps," Theoria, xxxv~,3 (1970): 264-281. 4 I n a footnote to "Naming and Necessity," in D. Davidson and G. Harman, eds, Semantics of Natural Language (Boston: Reidel, 1972), p. 344. Unless otherwise noted, references to Kripke will be to "Naming and Necessity."

our natural language. Kripke's argument confuses sentences of the technical language of Lewis's semantic theory, which are outside our natural language or at least constitute an extension of it, with sentences of our ordinary language, and so misapplies intuitive judgments about sentences of ordinary language to the technical ones. On Lewis's theory, to say that Humphrey might have won the election-a modal claim, made in our ordinary language-is to say something that is true just in case, to put the conditions in his technical language, Humphrey's counterpart in some world where Humphrey himself does not exist, did win. T o put Lewis's claim in the form "Not Humphrey himself, but someone resembling him, might have won the election," which is essentially what Kripke does, can only create confusion. It incorporates a modal locution ('might'), and so appears to be a sentence of ordinary language. As such, it is one our intuition rebels against, for it directly contradicts the intuitively acceptable claim that Humphrey might have won ("if only he had done such-and-such"). Intuitively, then, this sentence is false; but its falsity, as a sentence of our ordinary modal language, in no way counts against Lewis's theory. T o make our rejection of this sentence into a rejection of Lewis's theory, we must confuse it with the sentence "Humphrey himself does not exist outside the actual world, but a counterpart of Humphrey won in some world." This last is, on Lewis's view, true, but, mentioning as it does possible worlds and counterparts, is a sentence not of our ordinary modal language but of the technical language of counterpart theory. As such it is not a sentence that we, qua speakers of our particular natural language, are entitled to have intuitions about.5 Similarly, Humphrey's regrets about the (supposed) fact that (had he done such-and-such) he might have won the election would lead him to express regrets about the electoral successes of people in other possible worlds only if he was in the habit of expressing the propositional content of his regrets in the language of Lewis's analysis. Similar confusions lurk in Kripke's use of the phrase "we are not talking about something that might have happened to Humphrey." Clearly if we say "Humphrey might have won," we are 5 All of which is not to deny that Lewis and others who accept his framework can argue within it, and so can be said to have logical intuitions about its sentences. T h e arguments carried on within the framework of counterpart theory are ordinary, nonmodal, arguments; the logical intuitions they reflect are not intuitions about the particular content of counterpart theory, but only the ordinary intuitions about the validity of arguments involving quantifiers and connectives.

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talking about Humphrey, and so the fact that Humphrey might have won is a fact about Humphrey. What Lewis does is to offer an analysis of this fact: on his account it is a relational fact about Humphrey, consisting of his bearing certain relations to possible worlds and election winners in them. I t is agreed that winning the election is something that might have happened to Humphrey, but what is agreed upon is here expressed with the aid of an unanalyzed modal locution. Our intuitive acceptance of this modal claim gives us no reason to reject the claim, made in an artificial, nonmodal, language, that Humphrey is not identical with his counterpart in the domains of non-actual possible worlds. Alvin Plantinga 6 has offered a similar criticism of Lewis's position: A



. . . take any property Socrates has accidentally-wisdom, perhaps. According to Counterpart Theory, Socrates-the person who actually is Socrates, the Socrates of @, if you wish-exists in just one worltl: the actual world. In that world he is wise. Accordingly there is no world in which he is unwise. T h e r c is no possible state of affairs such that if it had been actual, this vcry person would have been unwise. Accordingly, it is impossible that Ile should have been unwise. But then he has the property of being wise essentially. T h e confusion is obvious. After stating certain consequences of Lewis's metaphysical theory, expressed in counterpart-theoretic language, Plantinga draws conclusions from them in our ordinary modal language-that Socrates could not have been unwise, and so that he was wise essentially. In drawing these conclusions lie is, of course, presupposing certain equivalences between sentences of a nonmodal language describing possible worlds and things in them and sentences of our ordinary modal language. T h e equivalences he presupposes, however, are not those of Lewis's schema for translating modal into counterpart-theoretic language, but are those suggested b y earlier versions of possible-worlds semantics for modal logic. Plantinga's argument against Lewis, then, shows only that Lctvis's theory must be taken ~vhole:you can't combiric Lcwis's metaphysical claim that an incliviclual exists in oilly one world with the pre-Lewis explication of 'x has I; essentially' as 'x has F in every world in which x exists' and get acceptable results. 0 T h e Aratirre o f Necessity (Scw York: Oxford, 1974), pp. 115/6. I l l a ~ csubstituted Lewis's symbol for the actual wo~ld,@, for Plantinga's.

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On the next page, Plantinga comes close to recognizing the confusion: We may imagine [the counterpart theorist] replying as follows. "MThen I say that Socrates could have been unwise I do not mean that there is a possible world in which Socrates-our Socrate-in the strict and literal sense is unwise; I mean only that there is a world in which in the new and looser sense he has that property. I so use the sentence 'Socrates could have been unwise' that what it expresses is entailcd by the truth that Socrates has foolish counterparts:" Thus perhaps he speaks with the vulgar and thinks with the learned. H e genially agrees that there is a world in which Socrates is unwise and concludes that Socrates could have been unwise. By adopting this course he preserves verbal agreement with the rest of us who do not look upon Socrates as a world-bound individual.

But the recognition is stillborn, for he continues: But of course the agreement is only verbal. For it is only in this loose and Pickwickian sense that he concedes the existence of a world in which Socrates is unwise; and his use of 'Socrates could have been unwise' is therefore similarly loose and Pickwickian. If in his use the sentence 'Socrates could have been unwise' expresses a proposition entailed by the fact that Socrates has unwise counterparts, then the Counterpart Theorist is using that sentence to express a proposition different from the one the rest of us express by it. While he assents to our sentence, he denies the proposition we take it to express.

But what Plantinga disparages as a merely verbal agreement about the truth value of the sentence 'Socrates could have been unwise' is the only agreement that can be demanded from the counterpart theorist: it is the only agreement that matters. Our logical intuitions about such sentences of our ordinary modal language are the evidence that both Plantinga and the counterpart theorist must appeal to and explain. What proposition is expressed by such a sentence, or, less tendentiously, how to state what is expressed by such a sentence in terms of possible worlds and objects in them, is a matter of theory, and the counterpart theorist has proposed a theory at variance with Plantinga's. T h e couriterpart theorist claims that the proposition he, and we, express by the sentence 'Socrates could have been unwise' has been mischaracterized by the semantic theory Plantinga uses, with its underlying ontology of objects that exist in more than one world. Plantinga has demanded agreement on the theoretical sentences of that theory, as if they were as binding on

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responsible semantic theory as the "observational" truth that the sentence 'Socrates could have been unwise' expresses a truth. Although the objections to the principle of interpreting de re modalities in terms of individuals being represented by other objects in the domains of worlds other than their own are misguided, there are problems with Lewis's account, to which I wish to devote the rest of this paper. T h e modal logic generated by Lewis's account (i.e., the set of object-language arguments it validates) is different from that generated by more conventional possible-worlds semantics and less plausible. In some cases it assigns truth values to garden-variety modal assertions that are at variance with clear intuitions. Some of these problems can be eliminated by changing the way in which truth conditions for modal statements are specified in terms of the counterpart relation; others necessitate modifications in the definition of the counterpart relation itself. In the end I shall argue that a simple counterpart relation is not a sufficiently discriminating way of choosing representatives in one world for objects in another. A methodological comment before we begin: perhaps because of his literal belief in possible worlds, Lewis does not present his theory as a model theory for modal language. Such a model theory is, however, easily extracted from his work. Let a Lewis model structure be an ordinary structure, in the sense of the conventional model theory of (nonmodal) predicate logic, satisfying Lewis's firstorder axiomatization of counterpart theory. A formula of modal predicate logic, then, may be said to be Lewis-satisfiable just in case its translation into the nonmodal counterpart-theoretic language (as given by Lewis's translation schema) is true on some assignment in some Lewis model structure. This much is fairly straightforward; claims about the truth or falsity of particular statements on Lewis's view present additional methodological problems, which I shall comment upon as occasion arises. The purely model-theoretic and logical points could be discussed simply in terms of the formal constraints placed on the counterpart relations of Lewis model structures by Lewis's axiomatization, without reference to his informal characterization of counterparthood in terms of similarity. However, the interest of Lewis's theory lies in its grounding of questions of "transworld identification" in less "mysterious" considerations of similarity,7 and it is reflection on the sort of similarity involved 7 T h e kind of similarity involved in making one object a counterpart of another need not be exactly what we would have in mind if we said they were very similar. Compare Lewis's theory of counterfactuals (in Counterfactuals),

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that motivates our choice of what formal constraints to place (and to refrain from placing) on the counterpart relation. Thus, though I hope it will be clear which of my claims are mathematical and model-theoretic and which are not, I shall intermingle my points about the nature of counterparthood and its relation to similarity with my more strictly model-theoretic points. I1

One of the simplest problems has to do with identity. Lewis's semantics allows it to be true that although an object x is identical with an object y, it is possible for x and y to exist and not be identical with each other. I n terms of the first-order modal predicate language (terms which are interesting insofar as some of our ordinary modal language can be translated into that formalism),

is Lewis-satisfiable. This, as Kripke has argued, is very implausible. T h e problem arises because one object may have more than one counterpart in some world, and because a formula beginning with a possibility operator is true [i.e., true at the actual world-for the purpose of informal exposition we may ignore the fact that similar problems arise when formulas like (1) occur within the scope of a modal operator] if the formula within the scope of the possibility sign is true at some world on some assignment assigning to each variable occurring free in it some counterpart in that world of the object in the actual world it is assigned by whatever assignment we are evaluating the whole formula on. Suppose we have an object in the actual world with two counterparts in some other world. where the model theory can be discussed without reference to similarity, b u t is motivated i n terms of it. Here again, what makes one world closer to another than to a third need not be what would make us say that the over-all history of events in the first resembled that in the second more than that in the third (cf. Lewis's "Counterfactual Dependence and Time's Arrow," forthcoming in Nods). Still, the use of the word 'similarity' is justified in both cases: a n object (world) is a counterpart of (close to) another in virtue of their similarities in certain respects. 8 In "Identity and Necessity," in Milton Munitz, ed., Identity and Zndividuation (New York: N Y U Press, 1971). 9 A formula of the modal language is true at a world and on a n assignment in a Lewis model structure if and only if (a) the assignment assigns objects in the domain of (i.e., bearing the relation I to) the world to all the variables free in the formula, and (b) the translation of the formula into the modal language in accordance with a translation schema like that given on p. 118 of "Counterpart Theory and Quantified Modal Logic" except for replacing '@' with a n arbitrary variable 'zu' in T 1 is true in the structure on a n assignment like the given assignment except for assigning the world in question to 'w'.

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Assign both the variable 'x' and the variable 'y' to that object, so that 'x = y' is true. is true at the world with the two counterparts of the object on an assignment assigning one counterpart to 'x' and the other to 'y'. T h e problem, then, could be eliminated either by somehow motivating a constraint on counterparthood to the effect that an object can have at most one counterpart in any world (I shall argue that this cannot plausibly be done) or by changing the satisfaction conditions of modal formulas.1° A more serious failing, because less easily mended, is that Lewis's account does not validate the inference l1 from

Rab 3x Rax (2) is true if, in every world containing counterparts of both a and b, every counterpart of a bears the relation R to every counterpart of b. This, however, is consistent with there being worlds in which a has a counterpart but b doesn't, and in which the counterpart of a doesn't bear R to anything. Such a world would falsify (3). Since theorems of first-order logic are necessary if anything is, the inference from (2) to (3) may be seen as a special case of the plausible inference from

Fa (4) and vx (Fx 2 Gx) (5) to Ga (6) Lewis's semantics validates some special cases of this pattern, such as that in which 'F' and 'G' are monadic predicate letters. With such special cases the failing is more subtle: Lewis's semantics may validate the inference, but it leaves in limbo the argument we would 10 T h e first way corresponds to a change in the class of model structures COIIsidered, or, in the context of Lewis's original exposition, a strengthening of the axioms of counterpart theory. T h e second remedy could be effected very simply, by adding a couple of clauses to Lewis's translation schema. I have omitted the details of this minor "fix" because other prohlerns require much more radical changes in Lewis's theory (and, incidentally, solve this problem). 11 To simplify the example I have used individual constants. Lewis uses no primitive constants in his examples, but his translation schema can accommodate formulas containing them.

naturally appeal to if we were asked to defend the inference. In our informal reasoning we make extensive use of what Van Fraassen has dubbed epitheoretic arguments, corresponding to the rules of a Fitclistyle natural-deduction system that involve subordinate proofs.12 One such pattern in modal reasoning consists of concluding that a conclusion, valiclly derived from premises that are themselves asserted to be necessarily true, is necessary. Thus, for example, since (4) and (5) tell us that

(7)

Fa

and 'dx (Fx2 Gx) (8) are both necessarily true, and since the argument from (7) and (8) to is valid, we may conclude that (9) is necessarily true: in other words, we may conclude that (6) is true. But if we accepted Lewis's semantics, we would have to reject this epitheoretic argument, for this mode of reasoning would also allow us to derive (3) from (2). In fact, when a natural-deduction system that is sound relative to Lewis's semantics is constructed, the derivation of (6) from (4) and (5), even where 'F' and 'G' are atomic predicates, turns out to involve a detour through the negation rules.13 I n order to see what has gone wrong, it will be illuminating to ask what the intuitive content of the square is on Lewis's semantics. One obvious candidate is necessity. When no singular terms occur in the formula governed by the square, it is quite plausible to take it as meaning 'it is necessary that'. Another plausible candidate is essentiality. Where necessity can be thought of as a property of propositions, essentiality is more of a relation between individuals and propositions: it is essential to an individual that a proposition be true if and only if it is necessary that that proposition be true if the individual exists. Suppose, as is not altogether implausible, 1 2 It is this correspondence between the method of subordinate proofs and our modes of informal reasoning that makes natural deduction natural. A natural-deduction system for quantified modal logic is developed in "7 dissertation. It is similar to and inspired by Frederic Fitch's system in his Symbolic Logic (New York: Ronald Press, 1952), which, however, is not based on S5 and does not contain an actuality operator. The propositional fragment is described in my "The Eliminability of the Actuality Operator in Propositional Modal Logic," hTotreDame Jourizal of Formal Logic, xrx, 4 (October 1978): 617622. 1 3 A detour corresponding to the informal mode of reasoning known as indirect proof (I have studied a natural-deduction system corresponding to Leiris's niodel theory in unpublished work).

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given Lewis's informal account of counterparthood, that all of my counterparts are human. Then, on Lewis's semantics (extended to cover personal pronouns in the obvious way), it would be correct for me to assert (10)

(I am human)

whatever it might mean. But surely it is more plausible to hold that I, a human being, am essentially human, than that the proposition saying of me that I am human is a necessary truth; for it might be false if I didn't exist. Since essentiality can be thought of as a relation, however, this leads to the question: essential to what? If one singular term occurs in the scope of the square, it is reasonable to take the square as expressing essentiality to the denotatum of that term. If there are two singular terms involved, however, Lewis's semantics makes the square represent essentiality t o the two together: it is necessary that if the denotata of both terms exist, then the proposition expressed by the sentence in the scope of the square is true. I n general we can say that, on Lewis's interpretation, prefixing a square to a sentence says that the proposition it expresses is essential to precisely those individuals mentioned i n the sentence (taken together). T h e counterintuitive results detailed above stem from the fact that this is an unnatural operator: there are various locutions of ordinary English that can be taken as necessity operators, and various ways of expressing essentiality to one or more objects, but nothing that can plausibly be taken as an operator of the sort Lewis's semantics makes the square out to be. If a formal language, interpreted counterpart-theoretically, is to be useful for representing the sorts of things we say in our ordinary language, we must give a counterpart-theoretic account of some better-behaved operator. We must, that is, try to make the square represent some less mercurial concept than that of essentiality-tothe-mentioned-objects: preferably necessity, since essentiality can be defined in terms of it. I have tried. Lewis also considers alternatives to his translation schema. I have found that a definition that meets one objection tends to have unpleasant consequences elsewhere.14 T h e next objection is of a different kind. I t is often held that, for at least some kinds of modality, some relations hold necessarily or essentially. For example, taking events as a special kind of individual, one might hold that the death of Caesar was essentially of Caesar: that it could not have occurred without being the death of 14 In the general case. Somewhat better results are obtainable if the assumption is made that an object has at most one counterpart in any world.

Caesar. If that were so, however, Lewis's semantics would have the consequence that Caesar and his death could have at most one counterpart apiece in any world. On Lewis's account, a sentence of the form 'U Rub' is true only if, in every world in which they both have counterparts, every counterpart of a bears R to every counterpart of b. Suppose in some possible world there were two counterparts of Caesar, living in opposite hemispheres of the globe. Each might be related appropriately-by dying it-to some counterpart of the death of Caesar, but neither could be related appropriately to the other's death. Thus neither counterpart of the death of Caesar is of all the counterparts of Caesar; so, if Lewis were right, the death of Caesar could not be essentially of Caesar. Note that this objection, unlike the earlier ones, does not concern the formal logic of the modal language, but is rather to the effect that a particular intuitively true (or, at least: intuitively not obviously false) statement of the modal language comes out false on Lewis's theory. The methodological status of the objection is perhaps worth commenting on. Lewis's semantic theory, embodied in his informal comments about similarity and counterparthood as well as in the more formal material, has consequences for modal logic broadly construed-for the entailment relations holding between de dicto and de re modal sentences-which go beyond what can be extracted from his model theory for first-order modal logic. In formulating the objection just given, we first note that according to Lewis's theory there must be possible worlds verifying any de dicto modal truth: in particular, there must be a world verifying the intuitively plausible de dicto claim that there might have been two soldier/politicians, living in widely separated countries, with careers and characters closely resembling that of Caesar. ?Ve then, guided not by any explicit definition of counterparthood in terms of similarity but by our judgment as to what kind of similarities it is plausible to think ought to contribute toward counterparthood, draw the consequence that Caesar would probably, if Lewis were right, have more than one counterpart in some possible world, from which, together with the truism that a death is the death of at most one person, follows the unacceptable de re conclusion that Caesar's death could have occurred without being the death of Caesar. The whole argument lacks the mathematical rigor of our model-theoretic objections to Lewis, but it is, in its reliance on auxiliary hypotheses that may not be fully explicit, typical of the way in which, in practice, "observational" consequences are drawn from theories in semantics and in the natural sciences generally.

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On a less elevated plane of methodological i-dotting, it must be admitted that whether or not there are internal relations is a question that has exercised the metaphysicians for years, and that we can hardly expect universal assent to our suggestions about the essences of such event-tokens as deaths. So be it. Readers with different opinions about internal relations are invited to construct their own examples. At least when we turn from the logical or metaphysical modalities to physical or causal ones, I think almost everyone will admit that some relations hold unavoidably. Thus, in the absence of a definition of counterparthood that will guarantee unique counterparts, Lewis's theory fails to explain the truth of some true modal assertions. Returning to model theory, Lewis considers a language whose only modal operators are the possibility and necessity operators. It is clear, however, that something like an actuality operator is an essential part of the conceptual mechanism of our ordinary modal 1anguage.lS The failure of the counterpart relation to associate unique representatives in other worlds with objects puts insuperable difficulties in the way of a counterpart-theoretic interpretation of an actuality operator. Suppose we tried to add a clause to Lewis's schema for translating modal into counterpart-theoretic language to cover formulas beginning with an actuality operator. If we imitate Lewis's clause for the necessity operator and stipulate that a possible object actually has a property just in case all its counterparts (or, all the things having it as a counterpart) in the actual world have that property, we will get a sort of failure of excluded middle-we will have to admit as satisfiable things like (using a circle for the actuality operator)

which we might read as 'There could have been an object which, though it actually exists, is neither actually F nor actually non-F'. If, on the other hand, we follow Lewis's clause for possibility, and require only that at least one counterpart (or converse counterpart) in the actual world have the property, we get something at least equally repellent:

or 'There could have been an object such that not only does it actually exist and is it actually F, but it is also actually non-F'. 16

Cf. my "Expressive Completeness in Modal Languages," Journal of Philo-

so~hicalLogic, V, 1 (Fa11 1976): 25-46.

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I11

What can (and what ought) to be done in the way of altering Lewis's informal characterization of the counterpart relation? Fred Feldman ls has called attention to the consequences for Lewis's position of what may be termed "there-but-for-the-grace-of-God" cases. It is often tempting to think of two people that they could (in some sense of 'could1) have "exchanged places" in the worldthat in some sense it was accidental that Harry was a prince and Tom a pauper rather than the other way around. But for the "accident of birth" each would have had the breaks the other actually had and lived pretty much the life the other actually did live, and in general had most of the properties the other actually did have. Rut if we are to give a counterpart-theoretic account of the sense of 'could' in which this is true, we must postulate a world in which one of Tom's counterparts is more like I-Iarry (more like what Harry actually is) than he is like Tom, and in which one of Harry's counterparts is more like Tom than he is like Harry. But this Lewis explicitly forbids; for he requires of your counterparts, not only that they must resemble you to at least some cut-off degree, but that they must resemble you more closely than any other objects in their worlds resemble you. But in the world required by the there-but-for-the-grace-of-God example, Tom's counterpart does not resemble Tom more closely than anything else in the world does-in particular, Tom's counterpart does not resemble Tom as closely as Harry's counterpart does. T o accommodate the there-but-for-the-grace-of-God examples, we simply drop Lewis's requirement that a counterpart resemble that of which it is a counterpart more closely than does anything else in its (the counterpart's) world. T h e new characterization of counterparthood in terms of similarity is just that there is some degree of similarity such that anything in another world resembling you to that degree is one of your counterparts.l7 This revision of our conception of counterparthood has two consequences for the formal nature of the counterpart relation: first, the new conception makes the supposition that the counterpart relation is symmetric In "Counterparts," this JOURNAL, LXVIII,1 3 (July 1, 1971): 406-409. If we also drop the stipulation that a n object is not a counterpart of anything other than itself in its own world-a restriction that demonstrably does no work in the model theory, and has semantic consequences only on certain theories of possible worlds-we can even say that what makes it true that T o m and Harry could have had each other's lives is that there is a world-the actual world-where T o m has a counterpart-Harry-with Harry's properties, and vice 16

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VCTSP.

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(a suppositioii Lewis considers and rejects) more plausible, and, second, it makes it much less plausible to suppose that an object will typically have a unique counterpart in most of the worlds in which it has counterparts at all. Both of these consequences affect formal modal logic (the effects of symmetry, however, will be somewhat ~echerche',involving iterated modalities), even though Feldman's objection, like that about internal relations, concerned truth conditions for particular statements rather than for formal logic. Interestingly, despite the notorious intransitivity of most similarity relations, there is even a characterization of counterparthood, for which thcre is some intuitive motivation, on which it comes out to be an equivalence relation.18 Suppose all objects-those in other possible worlds as well as those in the actual world-to be partitioned into kinds or sorts. Just how these sorts are to be defined, and how many of them there will be, will vary with different senses of possibility and necessity. Perhaps, for a very strict sense of logical necessity, all individuals will be construed as belonging to the same sort, and for certain kinds of causal modalities quite small differences in the physical structure of objects will be enough to put them in different sorts. T h e counterparts of an object, in the sense of counterparthood relevant to a given kind of modality, will be all and only the possible objects belonging to its sort. Alice couldn't reach the key on the glass table because she was too small, and she couldn't crawl under the door to enter the garden because she was too big. a counter- If the occurrences of 'could' here are to be given part-theoretic treatment as possibility operators, the appropriate counterpart relation will relate people to people of the same size. Notice, however, what these various changes in the characterization of counterparthood don't do. Many of the objections to Lewis's theory turned on an object's having more than one counterpart, and none of our suggested changes eliminate this possibility. Indeed, on Lewis's original version there was some hope that multiple counterparts would turn out to be the exception rather than the rule, in which case we could avoid some of the objections by adding the stipulation that nothing counts as a counterpart of an object if it shares a world with something having an equally good claim to being a counterpart of the same object. Our response to Feldman's example, however, and even more our suggestion of basing counterparthood on a classification of objects into sorts, make unique counterparts the exception. Nor is there a remedy readily available 1 8 Good news for those who like simple logics: this will get us all substitution instances of theorems of S5.

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-no plausible way of defining counterparthood in terms of some kind of similarity will guarantee that an object will have at most one counterpart in any given world. In particular, one method of defining transworld identity that has recently been popular will not work. It has been suggested, with variations of detail and with more or less hedging, that (to put it in terms of counterparts) an object has counterparts only in worlds whose histories, up to some time after the object has come into existence, are exactly like the history of the object's own world, and that the unique counterpart of an object in such a world is the object in that world that started its career in exactly the manner and circumstances in which the object started its own. There are two reasons for rejecting this kind of suggestion. For one thing, even if it succeeded as an account of some sense of "metaphysical necessity," l9 there are other kinds of modality it cannot handle. Any general theory of de re modal assertions will have to account for the likes of "He could have carried out his plans, had his ancestors not squandered the family fortune." Secondly, it won't always give unique counterparts. Imagine a (not quite deterministic) possible world which, u p to a certain moment of its history, is spatially symmetrical, with the regions on the two sides of the plane of symmetry developing differently after that moment. Now consider an object in such a world that comes into existence before the two sides stop mirroring each other: in general it will have two counterparts (or at least two objects beginning their careers in the same way it began its career) in any world "branching off" from its world between the time it came into existence and the time its world ceased to be symmetrical. IV

What is needed is some way of choosing one of an object's counterparts in a world, to serve as its representative and, to avoid the internal-relations problems, to make the choice of representative for one object depend on the choice of representatives for other objects. Eut how do you make the choice? There is no ground for choosing one of an object's counterparts over another-this is what Kripke means when he says he sees no reason to think that similarity between objects will provide a suficient condition for transworld identification. Well, set theory (the set-theoretic reification of a choice being called a function) lets us have our cake and eat it too. First choose representatives one way, and then another. Something 19 I have argued in my dissertation that this idea doesn't even capture Kripke's intentions in his talk of metapliysical necessity.

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i\ necessaril) true just in case it comes out necessarily true no matter which counterparts you choose as representatives. Oversimplifying a bit, we define a set of representative functio~zs from the domain of one world into that of another. A representative function maps objects into counterparts of themselves, but, to allo~v for internal relations, not every function mapping counterparts into counterparts will be a representative function. A representative function that maps Caesar (an inhabitant of the actual world) into Seezer rather than Kizer (Seezer and Kizer being counterparts of Caesar in some world) must map the deatli of Caesar into the deatli of Seezer rather than that of Kizer, and vice versa. This aspect of the theory can perhaps be visualized more readily in terms of a .te?irsof counterpart relations. I11 addition to the single-cozr?7fe~pn1i relation (the counterpart relation in the sense of Lewis, relating objects to similar objects), imagine a pair-countetpa~t relation, relating ordered pairs of objects to similar ordered pairs, and so on. Then a representative function f must not only meet the condition that f(x), for any x for which f is defined, be a single-counterpart of X, but also the condition that the ordered pair (f(x),f(y)) be a paircounterpart of the pair (x,y). With the class of representative functions doing the work of the counterpart relation in Lewis's semantics, we define dc ? e modal locutions along the following lines: a formula is necessarily true just in case, for every world and every representative function into the domain of that world, the formula is true in that world when each singular term in it is interpreted as denoting the image under the function of the object it actually denotes. In a way, this semantic theory conibines Lewis's approach with some of Kripke's insights. In common with counterpart-theoretic semantics, it defines de re modal notions in terms of some notion which, though vague, is perhaps somewhat more amenable to analysis than a primitive notion of transworld identification would be. On the other hand, Kripke has said that possible worlds are "stipulated" rather than discovered. T h e current theory allows us to explain, partially, what he meant. What takes the place, on this theory, of the possible worlds of conve~~tioilal possible-worlds semantics, several of which could have the same inhabitants, are combinntions (ordered pairs, in a set-theoretic regimentation of the theory) of worlds conceived of as fully specified "qualitatively," with stipzl-

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latiolzs or choices (reified as functions) of rcpicscntatires lor thc

objects in other worlds. ALLLK H.\ZEN

Trinity College, Dublin

APPENDIX: TECHXICALITIES AND SIDE-EFFECI'S Consider a fairly conventional possible-worlds model theory. A model structure comprises a set of worlds, with domains that may overlap, one being singled out as the actual world. An n-adic predicate letter designates, at any world, some set of n-tuples of objects from the domain of that world; an individual constant denotes either something present in the domain of the actual world or nothing at all; an assignment assigns to each variable some object in the union of the domains of the worlds. 4 n atomic formula is true at a world and on an assignment just in case the sequence of objects assigned to and denoted by its terms is one of the tuples designated at the world by its predicate; any atomic formula, even an identity, is false at a world if one or more of its terms fails to denote (or he assigned) an object in the domain of that world. Truth-functional composition has the usual results, and the truth values of quantifications are defined in the usual w-ay in ternis of alternate assignments assigning the ~ a r i a b l eof quantification some object in the domain of tile world in question. The result of prefixing a necessity (possibility, actuality) operator to a formula is true at a world and on an assigniment just in case the formula itself is true on that assignment at every (some, the actual) world. X sentence is true in the model just in case it is true on every assignment at the actual world. Consider first (here comes the oversimplification) that fragment-we call call it the first-degree fragment-of a first-order modal language consisting of those formulas in which no modal operator other than an actuality operator occurs within the scope of any other modal operator. This is actually a fairly natural fiagment, and perhaps the only part of first-order modal logic with direct application in conceptual analysis and the semantic representation of natural-language assertions. Certainly it is difficult to find decent English sentences that can be construed as involving one modal operator in the scope of another and do not involve tensed modalities or clifferelit kinds of modality (e.g., an alethic modality in the scope of ari epistemic one). (In philosophical usage one can find such claims as that it is logically possible that a certain proposition be physically necessary, etc., but then, philosophers do speak artificial dialects.) A first-degree fu?zctional model comprises a set of worlds, with disjoint domains, one being singled out as the actual world, and for each world oll~erthan the actual, a set of (oric-one, partial) flinctions from the clornail~ of the actual world into its domain. Interpretations of predicates and in-

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dividual constants, assignments to variables are as before. A stipulational world is either the actual world or an ordered pair of a world and a function from the domain of the actual world to its domain. Truth at a stipulational world on an assignment is defined in the usual way (for atomic formulas, truth-functional compounds, and quantified formulas) if the world is the actual world, and in the usual way, but relative to a reinterpretation of the individual constants assigning them the images, if any, under the function, of their usual denotata, and an assignment assigning to each free variable not already assigned an object in the domain of the world the image, if any, under the function, of the object it is assigned by the original assignment. T h e result of prefixing an actuality operator to a formula is true on an assignment at the actual world just in case the original formula is, and at any other stipulational world just in case the original formula is true at the actual world with its constants and free variables appropriately reinterpreted in accordance with the inverse of the function. T h e result of prefixing a necessity (possibility) operator to a formula is true at the actual world on an assignment if and only if the original formula is true at every (at least one) stipulational world on that assignment. (It's not as complicated as it sounds.) The theorem: as far as the first-degree fragment of the first-order modal language is concerned, the conventional and first-degree functional model theories generate the same logic: the same set of valid sentences, and the same sentences semantically entailed by any given set of sentences. Proof: given a model of one kind it is easy to construct a model of the other kind satisfying exactly the same sentences. If we consider the full first-order modal language, the functional model theory becomes much more complicated, to deal with sentences like:

We must now consider not only functions from the domain of the actual world into the domains of other worlds, but functions from the unions of the domains of arbitrary finite sets of worlds into the domain of some world, and the analogue of a stipulational world is now a finite sequence of worlds with a cornpallion sequence of functions uniting the objects in their domains. T h e functions of such a sequence must then fulfill certain conditions (intuitively, that they be consistent in what they "say" about what objects in the domain of one world are "the same" as what objects in the domains of others), and the whole set of functions in the model structure must fulfill certain closure conditions. Dotting the i's and crossing the t's is time-consuming, but in the end it can be proved that the logic generated for the whole first-order modal language is the same as that generated by the conventional model theory. T h e proof is tedious, but involves no conceptual novelties; details may be found in my dissertation.

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Why, if it's all so complicated, bother? Well, there are at least two areas where a functional model theory can, I think, provide illumination. I t is always possible to provide a functional model satisfying the same sentences as a given converltional structure [very easily: replace the objects in the domains of the worlds with ordered pairs (original object, world) to ensure that the worlds have disjoint domains, and allow as functions precisely those which would, in terms of the original objects, have been identity mappings]. I n many cases, however, the functions of the new model structure will not be based on similarity in the way representative functions are supposed to be - a t least, not similarity with respect to the predicates true of the objects in the different worlds. Suppose, now, that we have formulated some condition of similarity-basedness on functional model structures that partially embodies the idea that representative functions ought to reflect similarities between objects in different worlds. (For example, we might require that in a similarity-based structure any isomorphism, with regard to the extensions of the predicates, between the domains of two worlds should be a representative function.) Then we can define a class of theories whic11, thougll consistent-satisfiable in some functional model structure-are incoherent-they are not satisfiable in a similarity-based model structure interpreting just the predicates appearing in the theory. T h e sort of incoherence involved can perhaps be best illustrated with an example from nonmodal logic. Consider a definition of identity in terms of the descriptive predicates of a language with a finite set of descriptive predicates. T h e negation of any such definition is a consistent formula of predicate logic; but one might well hold that if the stock of descriptive predicates involved included all those used in your total theory of the world, it would be incoherent of you to reject the definition of identity in terms of tllem.20 T h e notion that representative functions should be based on similarities, where the relevant similarities are those defined in terms of the structures defined over the domains of the different worlds by the predicates of our language, allows us to formulate broadly logical constraints on acceptable theories of the essences of things, much as the definition of identity in terms of descriptive predicates allows us to formulate a constraint on an acceptable theory of identity. T h e otlier area where the fur~ctionalapproach may be more illumi. nating than the conventional possible-worlds semantics concerns nondenoting singular terms. Notice that, as I have specified the model theories, no name not denoting an object in the domain of the actual world denotes anything at all:

is valid. I t would be very easy to add constants denoting unactualized possibilia to the language, if we thought solely in terms of the conventional 20

Cf. TY. V. Quine, Word and Object (Cambridge, Mass.:

sec. 47.

MIT

Press, ISGO),

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model theory. I t is much more difficult to formulate satisfaction conditions for formulas containing such terms within the functional model theory. I regard this not as a conceptual weakness of my approach but as a positive strength, for there are grave difficulties with the notion of naming the nonexistent. T h e arguments ICripke gives in the addendum to "Naming and Necessity" against construing 'Sherlock Holmes' as the name of an unactualized possibile correspond exactly to the difficulties that must be overcome if constants denoting possible beings are to be accommodated within the framework of the functional model theory. I take this as an indication that my formal approach reflects the conceptual situation morc accurately than does the conventional possible-worlds approach. AH

BOOK REVIEWS Alethodological Pragnzatism. Press, 1977. xv, 3 15 p. $18.00.

NICHOLAS RESCHER.

New York:

NYU

Systematic pl~ilosophyis no longer dead. In a series of books, most especially a trilogy which includes the subject of this review as a member, Nicholas Rescher makes a self-confessed attempt to revitalize philosophical system building. T h e two earlier essays introduced the metaphysical elements of his system and sketched in the leading elements of a corresponding epistemology and its justification. Methodological Pragmatism continues (and perhaps roughly finishes) this latter task. Although Rescher's project might sound old-fashioned in style, its content on the other hand is completely modern. Rescher calls his system "pragmatic idealism." In its epistemological elements, his position "recognizes the shaping of our knowledge as subject to mind-external constraints, but takes these constraints to manifest themselves wholly or predominantly on the side of praxis" (xiv). T h e role of methodological pragmatism is to reveal how praxis works against the metaphysical constraints. Although this review is obviously neither the time nor the place to go into Rescher's conceptualistic metaphysics, a metaphysics of fundamentally mind-involved objectivity, let me at least mention that the present book is consistent with-indeed necessitated by-the meta~hysicalviewpoint developed in Conceptual Idealism. 1 Conceptual Idealism (Oxford: Basil Blackwell, 1973); Primacy of Practice (Oxford: Basil Blackwell, 1973); and the book under re~ielr.

0022-362X/79/7GOG/0338$00.50

0 1979 The Journal of Philosophy, Inc.

http://www.jstor.org

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[Footnotes] 1

Counterpart Theory and Quantified Modal Logic David K. Lewis The Journal of Philosophy, Vol. 65, No. 5. (Mar. 7, 1968), pp. 113-126. Stable URL: http://links.jstor.org/sici?sici=0022-362X%2819680307%2965%3A5%3C113%3ACTAQML%3E2.0.CO%3B2-G 1

Counterparts of Persons and Their Bodies David Lewis The Journal of Philosophy, Vol. 68, No. 7. (Apr. 8, 1971), pp. 203-211. Stable URL: http://links.jstor.org/sici?sici=0022-362X%2819710408%2968%3A7%3C203%3ACOPATB%3E2.0.CO%3B2-O 16

Counterparts Fred Feldman The Journal of Philosophy, Vol. 68, No. 13. (Jul. 1, 1971), pp. 406-409. Stable URL: http://links.jstor.org/sici?sici=0022-362X%2819710701%2968%3A13%3C406%3AC%3E2.0.CO%3B2-L

NOTE: The reference numbering from the original has been maintained in this citation list.