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13 Counterpart Theory 13.2 Syntax of CT Counterpart Theory Does Not Make Use of Individual 13.1 Philosophical Preliminaries Constants

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13 Counterpart Theory 13.2 Syntax of CT Counterpart Theory Does Not Make Use of Individual 13.1 Philosophical Preliminaries Constants 13 Counterpart Theory 13.2 Syntax of CT Counterpart theory does not make use of individual 13.1 Philosophical Preliminaries constants. (This requires slightly adjusting the in- ference rules so that \arbitrary" constants are not As originally presented in 1968, David Lewis's coun- needed for (Q8In); instead free variables are used. terpart theory represents a way to translate the lan- Some of you may already be familiar with the use of gauge of quantified modal logic into a first-order log- free variables in proofs.) ical theory (one with classical quantifier rules) with- Counterpart theory (CT) makes use of the follow- out any intensional operators, but in which, instead, ing predicates along with the intended meaning listed: we quantify over worlds in the object language, and introduce additional predicates for such things as in- clusion within worlds, and the counterpart relation Wx x is a world for objects in different worlds. Ixy x is in (world) y Lewis regards possible worlds as genuine entities, Ax x is actual (exists in the actual world) and the things in them as well. He accepts a theory Cxy x is a counterpart of y known as modal realism according to which possible objects, although not actual, are taken to be gen- Lewis also adopts the following definition: uinely real things in the full sense. Considered as a formal theory, one need not accept Lewis's meta- @ abbreviates x 8y(Iyx $ Ay) physics of modality to accept CT; however, one must maintain that it is somehow intelligible to quantify Here, @ is the actual world, or the world contain- over worlds and non-actual objects. ing all and only actual things. Lewis defines this with a description; but it seems clear that he accepts (Of course, we do that in the metalanguage when the Russellian method for eliminating descriptions in we do Garson-style semantics for modal logic; how- context. (He should have given us conventions for re- ever, it is arguable that the commitment to possible solving scope ambiguities for @, but in virtue of (P7) worlds in accepting a theory like fS5 is at least much below, such ambiguities won't matter much. more oblique. Again considered as a formal system, Lewis might have instead taken @ as the theory's one might accept fS5 and think of modality as a sui own individual constant and defined \Ax" instead as generis notion. Garson's models are then perhaps \Ix@". heuristics or constructs that can be used to define a notion of validity for a logic with modal operators that is extensionally right, without showing the true 13.3 Axioms of CT nature of necessity or possibility.) System CT is built on pure first-order logic with iden- Lewis's counterpart relation C is paramount his tity (roughly, Garson's QL, modulo the lack of indi- way of thinking of de re modality. In standard (Krip- vidual constants), and adds the following particular kean) objectual semantics for modal logics, whether axioms: or not it is possible for me to have a property I don't have, or whether it is possible for me not to have (P1) 8x 8y(Ixy ! Wy) some property I do have, depends on what properties (P2) 8x 8y 8z(Ixy & Ixz ! y = z) I have in other possible worlds. This presupposes the (P3) 8x 8y(Cxy ! 9z Ixz) intelligibility of trans-world identity, the same thing (P4) 8x 8y(Cxy ! 9z Iyz) (me) existing at multiple possible worlds. Lewis in- (P5) 8x 8y 8z(Ixy & Izy & Cxz ! x = z) stead understands de re modal questions in terms (P6) 8x 8y(Ixy ! Cxx) of the properties had by my counterparts in other (P7) 9x(Wx & 8y(Iyx $ Ay)) worlds, people who closely resemble me in some rel- (P8) 9x Ax evant sense. It is possible for me to have worn a dif- ferent color shirt today because in some other world (P1) says that things are only in worlds. (P2) says there is someone qualitatively exactly like me, except that nothing is in more than one world. This is some- wearing a different color shirt (sitting in a room qual- times called the principle of world-bound individuals, itatively like this one, teaching students very similar and is in effect Lewis's rejection of transworld iden- to you, etc.) tity, strictly speaking. 72 (P3) and (P4) together say that all counterparts (It then follows from their definitions that if A are in worlds. (P5) and (P6) together say anything in takes the form (B & C), then jAjw is (jBjw & any world has itself, and only itself, as its counterpart jCjw), and so on for ∼, _ and $.) in that world. w (P7) says that there is a world that contains all and 3. If A takes the form 8x Bx, then jAj is w only actual things; it is this axiom that allows us to 8x(Ixw ! jBxj ). introduce \@" by description (without fear that the (It follows that if A takes the form 9x Bx then existential clause of the contextual definition will be jAjw is 9x(Ixw & jBxjw).) false). (P8) says that something is actual. Lewis considers but rejects certain additional 4. If A takes the form 2 B, and the free w principles. variables of B are x1; : : : ; xn, then jAj 0 0 0 is 8w 8y1 ::: 8yn(Ww &Iy1w & ::: & 0 Iynw &Cy1x1 & ::: &Cynxn ! 8x 8y(Cxy ! Cyx) 0 0CT j[::: [B]y1 =x :::]yn =x jw ). (The counterpart relation need not be 1 n symmetric.) (It follows that if A takes the form 3 B, CT 8x 8y 8y(Cxy & Cyz ! Cxz) and the free variables of B are x1; : : : ; xn, 0 w 0 0 0 (The counterpart relation need not be then jAj is 9w 9y1 ::: 9yn(Ww &Iy1w & 0 ::: &Iynw &Cy1x1 & ::: &Cynxn & transitive.) 0 j[::: [B]y1 =x :::]yn =x jw ).) 0CT 8x 8y 8z 8w(Cxy & Cxz & Iyw & Izw ! y = z) 1 n (Something needn't be counterpart • Hence, if B is a sentence without free vari- to only one thing in a world.) 0 ables, then j 2 Bjw is 8w0(Ww0 ! jBjw ). 0CT 8x 8y 8z(Cyx & Czx & Iyw & Izw ! y = z) (Something needn't have only one • If Bx has one free variable, x, then j 2 Bxjw 0 counterpart in a world.) is 8w0 8y(Ww0 &Iyw0 &Cyx ! jByjw ). 8x 8w 8w0(Ww & Ww0 & Ixw ! 0CT w 9y(Iyw0 & Cyx)) jAj is basically the translation of A relativized to (Something needn't have a world w. The complete translation of a sentence A counterpart in every other world.) of QML in Lewis's CT is A relativized to the actual @ w 0 0 world jAj , or more fully 9w(8x(Ixw $ Ax)& jAj ). 0CT 8x 8w 8w (Ww & Ww & Ixw ! 9y(Iyw0 & Cxy)) (Don't confuse the schematic \A", italics, with the (Something needn't be the counterpart predicate \A" of Lewis's langauge, no italics.) of something in every other world.) Examples: • The translation of \8x Fx" is \8x(Ix@ ! Fx)". 13.4 Translating from QML to CT • The translation of \3 9x Rxx" is \9w(Ww & Lewis provides a means for translating sentences 9x(Ixw & Rxx))". stated in the language of quantified modal logic (one The translation of \9x 3 Fx" is \9x(Ix@ & that adds only \2", or it with \3" to a standard • 9w 9y(Ww & Iyw & Cyx & Fy))." first-order predicate language), such as Garson's qK or fK or their extensions, into the language of CT. Homework: Translate the following from QML to At the core of a translation is a function, which CT: we may write jAjw, where A is formula and w is a variable for worlds. We can define this recursively as 1. 2 8x Fx follows: 2. 8x 2 Fx 1. If A is an atomic sentence or \?", then jAjw is 3. 8x 3 9y Ryx A itself. 4. 2 8x(Fx ! 3 Gx) 2. If A takes the form (B ! C), then jAjw is (jBjw ! jCjw). Translation in the other direction (from CT to QML), however, is not always possible. 73 13.5 Comparison with other systems • We translation of the Converse Barcan Formula: • While the rules for the quantifiers are classical, (CBF) 2 8x Ax ! 8x 2 Ax and have a univocal domain, ranging over all pos- sibilia, quantification is often restricted to actual is a theorem of CT, but not the translation of things, or all those in some world, so in practice, the Barcan Formula: the theory has some of the features of a free logic. However, it has no genuine terms, and a fortiori, (BF) 8x 2 Ax ! 2 8x Ax none that don't refer to an existent thing. The reason the latter fails is that there may be • The de re/de dicto distinction is captured by objects in other possible worlds that are not the treating all individually referring expressions ac- counterparts of any objects in this world. (As we cording to Russell's theory of descriptions and saw, Lewis rejects the idea that every object in a applying the contextual definitions accordingly. given world must be the counterpart of anything in a given other world.) • In some ways the modality espoused is close to that of S5, since it in effect defines necessity in Most difficulties with (CBF), moreover, are dif- terms of truth in all possible worlds whatever. fused by the nature of the system as a whole. In line with this, for every A which is a closed sentence, the translations of the appropriate in- 13.6 Possible Expansions/ stances of the following are theorems: Modifications (M) 2 A ! A (B) A ! 2 3 A As Lewis points out, counterpart theory could also (4) 2 A ! 2 2 A be made to proxy for logics with other kinds of inten- (5) 3 A ! 2 3 A sional or modal operators (deontic operators, epis- However, these are restricted to closed sentences.
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