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COMPLETENESS. IN MODAL. LANGUAGE'. It is a task of philosophical logic to investigate the concepts central to our reasoning. This is a matter of conceptual ...
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It is a task of philosophical logic to investigate the concepts central to our reasoning. This is a matter of conceptual analysis, distinguished by the fact that the concepts analyzed are basic ones, occurring in reasoning about a wide range of topics. Since reasoning is generally expressible in speech or writing (this seems a necessary truth, though some modification of the reasoner’s usual language may be needed), the task amounts to giving a partial semantic analysis of our language. The philosophical logician need not, however, adopt whole the modus operandi of contemporary linguistics. Natural languages are hellishly complex, so the logician may, like many another scientist, choose to study simplified analytical models. For the logician these models will be artificial languages of simpler and more easily understood structure than the natural ones. As always, the study of phenomena (in this case the logical properties of our language) by means of analytical models becomes two-sided: on the one hand the formal properties of the models themselves are investigated, and on the other inquiry is made into how tiell they represent the observed phenomena. In the case of logic the latter sort of investigation takes such forms as attempting to formulate in the artificial language interesting claims and arguments that have been made in our natural language, and seeing whether amphibolies in the natural language can be explained in a regular way by postulating alternate translations in the artificial language for amphibolous sentences in the natural one. This paper is intended as a contribution - mainly on the former of the two sides mentioned - to the study of elementary modal logic: modal logic with quantification over individuals. I

The logic of the modal operators - of possibility and necessity - has strong analogies with that of the quantitiers - of existence and universality. So strong is this analogy that it has seemed that the only reasonable course is to consider modal language as involving a peculiar sort of quantification over

Journal of Philosophical Logic 5 (1976) 25-46. All Rights Reserved Copyright 0 1976 by D. Reidel Publishing Company, Dordrecht-Holland

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some more or lessarcane sort of entity. One prominent tradition has taken these entities to be ‘possible worlds’ or ‘cases’- things each of which, whatever its other properties, is able to serve as a model for the semantic interpretation of the non-modal portion of the language under investigation. Philosophical logicians have thus been linguistically creative in two distinct ways. On the one hand they have formulated simple languages for use as analytical models and on the other they have exercised their prerogative, as scientists, of enriching their natural languages with words for the new entities they have discovered or postulated. The English of the analytic philosopher, if not the English of the common man, now contains a highly sophisticated repertoire of locutions for discussing possible worlds and the individuals existing in them. The question immediately presents itself of what the relation is between this enriched natural language and the original natural language with its modal operators. After all, if the philosophical logicians are correct in their analysis of the original language, the same things are talked about and many of the same things are said about them in the two languages. It would seem that in these two languages we have an instance of two conceptual schemes, of different logical structure, being used to ‘handle’ the same subject-matter. The question is, however, too difficult for us at the present stage of the development of logic. It involves natural languages, and indeed natural languages used to discuss their own semantics. For whatever reason, whether to pursue such investigations or because the simplicity of the artificial languages encourages their use in the formulation of complex arguments, or both, David Lewis* has reproduced the important features of this ‘philosophers’ English’ within the best known and most intensively studied of the artificial languages, the classicalfirst order predicate calculus. With some variant of his formulations and various artificial languages containing modal operators, we have a tractable version of our question: what is the relation between these simplified representatives of the two conceptual schemes?3 II A review of the general features of the theory of possible worlds is in order. It is supposed, first of all, that there are possible worlds, one of which is the actual one. There are also individuals - persons, bodies and whatnot - existing

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in and having properties in the worlds. We shall presuppose a version of the theory on which the same individuals can exist in different worlds and have different properties in them. I am avoiding discussing one of the most controversial aspects of the theory: the problem of trans-world identification. Lewis has discussed the problem, and his version of the theory holds that different worlds have different individuals related by a counterpart relation, but for our purposes this would just add needless complication. The properties the individuals in a world have in it and the relations they bear one another in it endow the world with a relation structure4 which comprises all the intrinsic character the world has: that is, the only non-logical properties a world can be said to possesswithout regard to the character or existence of other worlds or of individuals in them are those properties definable in terms of its relation structure. We may go farther and say that the only special primitive notions of the theory involving worlds are those of existing in and having a property in a world, and thus that worlds have no individuating properties for the theory other than those definable in terms of the individuals in them and their properties.’ Such are the general outlines of the theory of possible worlds. If the conclusions of one tradition in philosophical logic are true, it is presupposed by and cryptically expressed in modal speech. It is easy enough to formulate a first-order Ianguage in which to describe the system of possible worlds and individuals. It will be convenient to use a double-sorted language.6 There will be quantifiable variables ranging over worlds (w, with or without primes and subscripts) and individuals (x, y and z, similarly clothed or naked), a name for the actual world (a), which may appear grammatically wherever world variables may, a two place (individual, world) predicate expressing the existence-in relation, (I), an identity predicate, which may only occur between terms of the same type, and whatever non-logical predicates are desired. Non-logical predicates will be n+l -adic (n>O), with their first n argument places taking individual variables and the last a world term. The idea is to have each n+l -adic non-logical predicate express the holding of some n-adic relation at some world of n individuals. Thus the property of being red would be expressed by a dyadic predicate, Rxw, “x is red at world w.” Given the grammar of our language, very few of our assumptions about the system of possible worlds and individuals need be codified as axioms. For example, the assumption that worlds and individuals form disjoint

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classescan neither be asserted nor denied in it.7 In what follows I shall assume an axiom,

to the effect that every individual is in at least one world. I shall also assume, for each n+l-adic non-logical predicate, an axiom of the form, (x,)...(x,)(w)(Rxr.

..x,w3(Ix,w&...8cIx,w)),

to the effect that if an individual is one of an n-tuple of individuals instantiating the relation expressed by the predicate at a certain world, then it is in that world. There are arguments against assuming the axioms (2). As it hap pens, I don’t accept these arguments, but I am not concerned to rebut them here. Readers who find the assumption embodied in the axioms (2) implausible may regard it as merely a simplifying assumption. It is not, I think, essential to the points I want to make.* III

We need a modal language for comparison. Let us start with a quite ordinary one: it will contain quantifiable variables ranging only over individuals, and only over the individuals of one world at a time. For each n+l-adic nonlogical predicate of the non-modal language, the modal will have an n-adic,’ but it will of course lack I. In places of this predicate and the variables (name) ranging over (denoting) possible worlds it will have the modal operators for necessity (L) and possibility (M). Many different semantic interpretations can be given of this language: the literature is full of them. For a start I shall consider one based on Lewis’s article. The basic concept defined will be that of truth at a world on an assignment. Truth at a world is truth at that world on all assignments, and truth simpliciter is truth at the actual world. An atomic formula is true at a world on an assignment if and only if (a) the assignment assignseach variable of the formula to some individual existing in that world, and (b) the n-tuple of individuals assigned to those variables is one of the n-tuples of which the n-adic predicate of the formula is true at that world.” In particular, an identity is true only if both of its variables are assigned the same individual. Truth values (at a world, on an assignment) of negations, conjunctions, disjunctions and conditionals depend in the usual way on those of

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their components. A universal (existential) quantification is true at a world and on an assignment if and only if its matrix is true at that world on every (at least one) assignment differing from the original at most with respect to the operator variable of the quantifier, and assigning that variable some individual in the world in question. A necessitate (a formula starting with an ‘L’ whose scope extends to the end of the formula) is true at a world and on an assignment if and only if its matrix is true on that assignment at every world in which all the individuals assigned to its free variables exist. Two consequences of this clause should be noted: for necessitates of closed formulassentences- it reduces to “true if and only if matrix true at every world,” and, for open formulas, a world may be ignored if it does not contain all the individuals assigned to the free variables of the formula. A possibilitate is true at a world and on an assignment if and only if there is some world, containing all the individuals assigned to the free variables of the formula, at which its matrix is true on that assignment. The corresponding consequences should be noted. This semantic interpretation of our modal language we shah call the weak interpretation.” There are a number of interesting facts about the modal language on its weak interpretation. One is that every sentence of it means the same as some sentence of the non-modal language, in the sensethat there is a translation procedure which, given a sentence of the modal language, produces a sentence of the non-modal language which has the same truth conditions.12 It is worthy of note that if the sentence of the modal language begins with a necessity (possibility) operator, its translation will begin with a universal (existential) quantification over possible worlds. Thus there is a very strong sense in which one can say that “p is necessarily true” means ‘p is true in all possible worlds” - a translation procedure which allows one to identify parts of the target sentence as translations of certain parts of the original maps modal operators onto quantifiers over possible worlds. We may say that ‘necessarily’ means ‘in all possible worlds’ in just the sense of ‘means’ in which we say that ‘et’ (in French) means ‘and’. On the other hand, not every sentence of the non-modal language can be translated by some sentence of the modal language.r3 We need not go far afield to find a statement expressible in the non-modal but not the modal language. Philosophers have sometimes been moved to say, in the course of an ontological argument, that there is something which exists necessarily. It is not altogether implausible to interpret them as meaning that there is

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something which exists in every possible world. This claim is made by the sentence (3)

(3x)(Ixa

& (w)(Ixw))

of the non-modal language. I think it is plausible to require that, just as (3) contains, in addition to variables and logical constants, only the special vocabulary of the theory of possible worlds, any proposed modal translation should only contain, in addition to variables and logical constants, the predicate of identity. After all, in non-modal logic existence is expressed by quantification and identity, and to use nonlogical predicates in formulating the claim that there is a necessary existent would be to make that claim dependent on facts about the non-logical properties of individuals. So my claim will be that no sentence of the modal language, couched entirely in logical terms, on the weak interpretation of that language, is a translation of (3). The obvious candidate, (4)

mmNx

=v)9

won’t do. Far from making the ontological claim of (3), (4) is a triviality: only those possible worlds in which an individual exists are considered in evaluating modal statements about it, so, on the weak interpretation, everything exists necessarily, or (5)

c4wJNx

=y).14

To show that no other logical sentence of the modal language will do either, suppose that (3) is true and consider the set of all true sentences of the modal language, couched entirely in logical terms. Now add to the system of possible worlds a world for each existing world, so that there is a one-to-one function mapping the old worlds onto the new. Choose individuals to inhabit the new worlds so that (i) no individual is in both a new and an old world, and (ii) there is a one-to-one function mapping the individuals in the old worlds onto those in the new such that an old individual is in an old world just in case its image is in the corresponding new world. (3) will obviously be false in this extended system, but all the sentences in the set mentioned above will still be true. Those containing no modal operators will be true because only the actuai world is relevant to them. Modal operators with closed matrices will not change truth values, for, insofar as it can be described in terms of identity, quantification and connectives, the structure

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of the old system of possible worlds is perfectly reproduced in the new. Anything that can be expressed in purely logical terms that is true in all the old worlds will be true in all the new worlds as well. Nor will modal operators whose matrices contain free variables affect truth values, for only worlds containing the individuals mentioned will be relevant: half of the extended system of possible worlds can be completely ignored.” Two comments are in order. First note that not only is no purely logical sentence of the modal language a translation of (3) but also that no set of purely logical sentences of the modal language entail the claim that there is an individual which exists in every world: i.e., were we to define a relation of semantic entailment in the natural way over the union of the sentences of the modal and non-modal languages, it would turn out that no set of purely logical sentences of the modal language would semantically entail (3). This will allow us to prove that certain other claims cannot be expressed in the modal language - if they could, there would be such a set. Second, if the assertion can’t be formulated, neither can the denial. Consider now the claim that there could be individual_s other than those there actually are - that there are unactualized possibilia, or (6)

(3w)@x)(Ixw

& -1x7).

No purely logical sentence of the modal language is a translation of this. For suppose one were. Then its negation, together with (7) “It is necessary that there be exactly one individual,” would entail (3). There are, however, purely logical sentences that entail (6). The negation of any instance of the Barcan Formula will do, including the logical instance,

(x)KY)(x =r>’ WKY)(x=Y)* Our modal language, on its weak interpretation, is beginning to seem quite limited in comparison to the non-modal language. We have seeen two different claims, each of which can apparently be made in English by using modal locutions, which cannot be made in it. As a final example, consider the relation of ‘ontological priority’ some metaphysicians have spoken of. The idea is that one object is ontologically prior to a second just in case it is not possible for the second to exist without the first one’s existing as well (apparently, then, it can be denned in informal modal speech). We may

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define it in the non-modal language as follows: (9)

XOPY =df (W)(IYW 1 Ixw).

On the other hand, the obvious definition in the modal language, (10)

XoPy =df L((%)(z ‘y)

1 (%)(z =x)),

will not do, for the simple reason that it would, contrary to the intended meaning, make everything ontologically prior to everything else: (11)

(X)(Y)WZ)(Z

=Y) ’ (W(z ‘4)

is valid on the weak interpretation, for much the same reason that (5) is. Ontological priority is not definable in the modal language because it concerns existence in various possible worlds. To say of one thing that it was ontologically prior to another, it would be necessary to have terms referring to the two both occurring free within the scope of a modal operator, but once free variables occur within the scope of such an operator, the rangel of the operator is automatically restricted to worlds in which all the individuals mentioned exist-r7 The weak interpretation of the modal language is by no means universally accepted by contemporary modal logicians. Let us now turn to the consideration of an interpretation - I shall refer to it as the ‘strong interpretation’ more like those usually adopted. l8 Again we define truth at a world and on an interpretation. The clauses of the truth definition for atomic formulas, truth-functional compounds, and quantifications are the same as for the weak interpretation. A necessitate (possibilitate) is true at a world on an assignment just in case its matrix is true at every (at least one) world on that assignment. The change is that the range of a modal operator is no longer restricted to worlds containing all the individuals assigned to the free variables of its matrix. Notice that a tautologous formula, and thus its necessitate, will be true at every world on every assignment, but a formula like (12)

x=x

(13)

(3Y)(Y ‘X)

or will be false at a world cn an assignment assigning to ‘x’ an individual not existing in that world. On the strong interpretation, (4) can serve as a translation of (3) and (10)

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as a definition of ontological priority. The essentialist claims that were the pride of the weak interpretation can still be expressed on the strong one: instead of saying that something has a property essentially, we can say that it necessarily has it if it exists. Indeed, we may translate any sentence on the weak interpretation into an equivalent sentence on the strong interpretation by, starting with the innermost, replacing any subformula of the form or~4~efomU...XI

(15)

,...>

%...I

M(. . .x,, . . . ,x, . . .),

having n (n >0) variables free within the scope of the modal operator, with one of the form (16)

WYlMYl

‘Xl)

‘(0 - . (3Yn)(Y,=x,)

( . . . Xl,. . . ,x,. Or (17)

M(@Y,)(YI ( . . . x ,,..

. *) . . .)

=x11 & (. - . @Y,&Y, .,x,...



=xd &

1 . . .I,

where the yi are variables new to the formula. We have thus characterized the relation in terms of expressive power between the weak and strong modal languages. Anything which can be expressed in the modal language on the weak interpretation can also be expressed on the strong one. On the other hand, some things, like (3) which can be expressed on the strong interpretation, cannot be expressed in the modal language on the weak one. It remains to compare the expressive power of the modal language, on its new interpretation, with that of the non-modal language. Anything that can be expressed in the modal language can be expressed in the non-modal: there is a method of translating sentences of the modal. language into sentences of the non-modal. For sentences containing no modalities de re - sentences no subformula of which is a necessitate or possibilitate of an open fomiula - the two interpretations come to the same thing, and the translation procedure for the weak interpretation can be used. The translations of the necessitates and possibilitates of open formulas will, as in the case of the weak interpretation, begin with universal and existential quantifiers over possible worlds. Instead of immediately following the world quantifiers, however, the provisos that the individuals referred to by the free

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variables exist in world referred to by the operator variable of the quantifier will be put off until the atomic formulas in which the free variables occur (if these atomic formulas are not within the scope of a modal operator itself within the scope of the first modal operator - if they are, don’t bother with the proviso at this stage of the translation procedure, but wait until you do the inner necessitate or possibilitate), and moreover, the provisos will always be conjoined with these atomic formulas.20 Finally there is the question of whether there are propositions expressible in the non-modal language which are not expressible in the modal. “There might have been things other than those there actually are,” or “There are unactualized possibies,” is such. (6), which is a fairly direct rendering into our non-modal language of the second of the English versions above (the ‘philosophers’ English’ version), has no translation in the modal language on either its weak or its strong interpretation, though, as the first of our two versions shows, it can be expressed in ordinary, modal, English. As for our artificial modal language on its strong interpretation, to assert the existence of an individual in a world other than the actual it is necessary to use a quantifier inside the scope of a modal operator, but to say that it is distinct from any individual existing in the actual world it would be necessary to use a second quantifier which, though inside the scope ofthe first, would not be governed by the modal operator. ‘r This given the way subformulas nest, is impossible, though one of the functions’of the word ‘actual’ in English seems to be to allow just this. Still, the strong interpretation gives us a certain flexibility - we can say a good many related things in it. There are purely logical sentences of the modal language entailing the truth of (6) such as (18)

(x) U3Y)(X ‘U) & M(3x) - L(3Y)(X =Yh

and entailing its falsity, such as (19)

L(X)L(3Y)(X ‘U)?

and for every finite number, n, there is a purely logical sentence which is equivalent to (6) on the hypothesis that there are exactly n individuals in the actual world: (20)

(3x,) . . . (3X,)(X, #x2 & . . . &x,-1 fx, (Y)(Y =x1 ” * ’ .v~=x,)&M(3y)(y#x,

& &. ..&y+x&

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On the other hand there is no purely logical sentence of the modal language which can be depended on to translate (6) in the absence of such auxiliary hypotheses. For suppose that (6) is false, that the actual world is the only one with infinitely many individuals, and that for every finite set of individuals in the actual world there is a world containing just those individuals, and consider the purely logical sentences true under those suppositions. Now suppose there is added to the system of possible worlds a new world for each old world, cbntaining all the same individuals plus one new individual (the same for each new world) not in any old world. (6) will have become true, but no purely logical sentence of the modal language will have changed its truth value. These results are significant in that they show that, quite apart from the occurrence of apparently higher-order notions in English, and from the possibility of using a natural language as its own semantic metalanguage, our artificial language, even on its strong interpretation, is incapable of expressing things which can be expressed in ordinary English using modal notions. Our analytical model is oversimplified. IV

The quantifiers in out modal language, on both of its interpretations, ranged over the individuals in a single possible world, in the sense that, in evaluating the truth value at a world of a quantified formula, only those assignments assigning to the variable of quantification an individual in that world were relevant. Let us, in an attempt to develope a more adequate modal language, add a new style of quantifiers, which we may write 7x>’ and ‘(Ex)’ and call outer quantifiers, ranging ovec individuals in general - ranging, that is, over possibilia. We will add these quantifiers to our previous modal language, whose quantifiers (hereinafter referred to as inner qwntifiers) we will give their strong interpretation. The clauses of the truth definition for the enriched modal language concerning atomic formulas, truth functions, modal operators, and inner quantifications $11 be exactly like the corresponding clausesin the truth definition for the first modal language on its strong interpretation. An outer universal (existential) quantification is true at a world and on an assignment just in case its matrix is true at that world on every (at least one) assignment like the original assignment except perhaps with respect to the variable of quantification.

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One may wonder whether ordinary English contains anything identifiable as outer quantification. I have harbored doubts, and am still worried by the fact that the examples I have found are not marked syntactically - if they do involve outer quantification, and not, say, some sort of higher-order quantification (over descriptions or proposals or what not), then ordinary English uses the same locutions for both inner and outer quantification. A typical example is the sentence “All the buildings the zoning board has prevented from being built would have been monstrosities,” which, assuming buildings are the only individuals, and that unbuilt buildings don’t exist, letting ‘Px’ mean ‘the zoning board prevented x from being built,ln and ‘MY mean ‘X is a monstrosity,’ and for the moment ignoring the difference between a counterfactual and a strict implication, we might try to express by: (21)

(x)(Px 2 L@y)(y

=x) 1 Mx)).

But let us, with only that perfunctory nod to the question of its realism, go on to study the relation between our new modal language and its predecessor, and the relation between it and the non-modal language. It should be obvious that the sentences of the extended modal language are all translatable into the non-modal language. Further, the translation function from the sentences of the first modal language into the second is simply the identity map - the first modal language is a proper part of the second. In the other direction, we may note that outer quantification gives a real increase of expressive power to the modal language. (6), which has no translation in the first modal language, may be translated into the second as (22)

(3x)(-(3y)(y

=x) & M(3y)(y =x)),

In addition, the extended modal language is strong enough to provide translations of certain sentences of the non-modal language which, though not contradictory in themselves, contradict our assumptions about the system of possible worlds. Thus (23)

@x)(4(-1x4,

the denial of our axiom (I), may be translated by (24)

(3x)L-(Ey)(y

=x).

@x)@w)(Pxw

& -Ixw),

&ah (25)

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which contradicts the axiom of the form (2) for the predicate ‘P’, can be translated by (26)

&M(Px

& -@y)(y

=x)).=

For the ordinary purposes of life this may not be too important. The position could perhaps be defended that we should count as successful any translation that has the same truth conditions as the original, given the general principles of the theory of possible worlds. Then sentences like (23) and (25) which contradict the axioms of the theory could all be translated by any arbitrary contradiction in the modal language. On the other hand, the axioms and their denials are significant statements about the systems of possible worlds and not tautologies. It is surely of interest, at least to metaphysicians and Iogicians, that they can be expressed in a version of the modal language. The increase in expressive power given to the modal language by outer quantifiers is not, however, enough to make it the equal of the non-modal. One thing we can’t express in the modal language is the metaphysical thesis, mentioned earlier, that samenessof relation structure is a sufficient condition for identity of possible worlds. Given that there are finitely many non-logical predicates, we can go some way toward expressing this thesis by a sentence of the form (27)

(w)(w’)((x)(Ixw ((x)(Ixw’

3 Ixw’) 3

3 Ixw) 1

((x1) . . . (x,)(R,x,

. . . x,w 1 R,xr . . .xnw’) 2

((x1). . . (x,)(R,xr

. . .x,w’ 1 Rrx, . . . x,w) 1

((x1). . . (x,)(R,x,

. . .xnw’ 1 Rmx, . . .x,w) 1

w = w’) . . .),

where Rr, . . . , R, are all the non-logical predicates of the language, n is the largest number for which there is an n+l-adic non-logical predicate, and where, if Rj is a ktl-adic predicate, k & -(~Y)(v

=x)N,

=x’)) &

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but not by the use of the actuality operator without outer quantifiers. If, similarly, we replace the constant ‘a’ with an existentially quantified variable in (28) we get (34)

(3w’)(w)(3x)(Ixw

& Ixw’),

which resists even the combined powers of outer quantification and the actuality operator. The actuality operator can only exempt clauses about the actual world from the influence of modal operators within whose scopes they lie; it is thus a far weaker and more specialized device than the operator variables of explicit quantifiers. It is not at all clear to me that (32) or (34) can be expressed in ordinary, as opposed to ‘philosophers’ English’, English. CONCLUSION The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special classof entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, and I think that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worlds Considered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages. A final note. Qume has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.29 He has also dropped%nts about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by

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the concept of identity. He has dubbed the resulting concept a ‘preindividuative’ concept of existence, or a concept of ‘entity without identity.’ What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: “is that the same coffee that was in the cup last night?” I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit qu+ifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Lelbniz or a David Lewis, conceives of possible worlds as entities without identity. University of Pittsburgh NOTES 1 As should become obvious on reading it, this paper is inspired by the work of David Lewis, particularly his classic ‘Counterpart Theory and Quantified Modal Logic,’ Journal ofPhi~osophy 7, Maxch 1968. I would like to thank Lewis and the referee for this journal for encouragement and advice. An earlier version of the paper was formulated in terms of Lewis’s counterpart theory rather than in terms of individuals being in more than one world, but, since I considered only the case in which the counterpart relation was an equivalence, I felt that the added complexity was not justified. Doing things in a counterpart-theoretic framework does produce two new classes of sentences of the non-modal language which lack translations in the modal: speaking of the properties individuals have simpliciter rather than of those they have at a world allows us to discuss relations obtaining between individuals in different worlds (e.g. the longer than relation obtaining between actual yachts and their counterparts), and the assumption that no individual is in more than one world allows a tricky way of asserting that there are at most n worlds without using the identity predicate between terms for worlds. Otherwise, given the assumption that the counterpart relation was an equivalence with at most one member of each equivalence class being in each world, the transition from the counterpart theoretic framework to the current one was perfectly straightforward. 2 David K. Lewis, op. cit. 3 A similar question with respect non-modal containing quantified

to higher-order modal and non-modal variables ranging over possible worlds,

languages, has been

the

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studied by Aldo Bressan in his A General Interpreted Modal Calculus, Yale University Press, 1973. Bressan’s results contrast with ours - the sentences of his non-modal language are translatable into his modal one. This reflects the incredible strength of higher-order logic: given high enough types, virtually anything is definable! 4 For this concept, see Carnap’s The Logical Construction of the World. 5 To be sure, relations such as accessibility, defined over the class of possible worlds have been suggested in order to account for deviant modal logics, but we may dismiss these as excrescences on the theory, motivated solely by a desire to provide each proof procedure with a semantics with respect to which it will be sound and complete. More importantly, I would hope that much of what I have to say would carry over to temporal logic, with momentary stages in the history of one world taking the place of different possible worlds, and there the relation of temporal order is essential. 6 Double-sortedness is convenient but, of course, theoretically eliminable. Lewis uses a singlesorted language with a monadic predicate expressing worldhood. Since most of the quantified variables in most of the formulas he wants to write down are restricted either to worlds or to individuals, this becomes cumbersome in practise. ’ For comparison: in the notation of Principia Mathemutica, the assumption that a set has no members not of the immediately lower type can neither be made nor denied. a For those who found our reference to the relation structures of possible worlds disturbingly indefinite, let us suppose that the non-logical predicates come indexed by some initial segment of the positive integers. We define the associated structure of a world, w, as a sequence one longer than the sequence of non-logical predicates. Its first member is the set of individuals existing in w. If the k-th non-logical predicate is n+ ladic, its k+l-th member is the set of n-tuples of individuals, (x,, . . . , x,j, such that the k-th non-logical predicate is true of the n+l-tuple (x,. . . . , x,, w). Two worlds are said to have the same relation structure if their associated structures are isomorphic. 9 The order of the n argument places is the same as that of the first n of the corresponding predicate of the non-modal language. Thus, if Bxyw means that x is bigger than y in world w in the non-modal language, Bxy means that x is bigger than y in the modal. lo An n-adic non-logical predicate.of the modal language being true of the n-tuple of individuals (x,, . _ . , x,) at a worid, w, just in case the corresponding n+ I-adic predicate of the non-modal language is true of the n+L-tuple (x,, . . . , x,, w). l1 We are primarily interested in sentences. We wouId have to complicate our account if we wanted to avoid such odd results as the failure of the inference to a formula from its necessitate for open formulas, but we may consider truth values given to open formulas as being purely auxiliary to the aim of giving them to sentences. As a semantic account of the sentences of the modal language, our account can be gotten from that of Lewis, op. cit., by (a) requiring that the counterpart relation be an equivalence, (b) taking the equivalence classes as new individuals, and (c) making such changes in the account of the predicates as are required by (b). (Lewis also has an axiom to the effect that there is at least one individual in the actual world.) t2 Making the necessary changes in Lewis’s translation procedure (Lewis, op. cit.) to tit our version of the theory of possible worlds is left as an exercise for the zader. l3 Lewis points this fact out and gives, without proof, an example similar to one of OUTS.

t4 One common definition of the essential is that an object has essentially those properties which it could not lack while still existing. Everything, on this definition,

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essentially existent, for nothing could lack existence and still exist. The modal operator ‘L’, when its matrix contains a single free variable, can be read ‘essentially’ in just this sense when the modal language has its weak interpretation. ” This proof could be extended to cover sentences in general as well as purely logical sentences. To do so, however, wotild be to suppose distinct posstble worlds - an old ’ world and the corresponding new world - with the same relation structure, and this might be thought dubious on metaphysical grounds. It would also conflict with the idea that the identification of individuals from one possible world to another is a matter of similarity, for an individual in a new world will have exactly the same relational and non-relational properties as the corresponding individual in the corresponding old world, but will be distinct from it, and this too might be though repugnant: cf. Wilson, N.L., ‘Substances without Substrata.’ Review ofbferuphysics 12 (1959) and Chilsolm, R., ‘Identity through possible worlds: some questions,‘Nous 1 (1967). Of course, for all 1 have said, a more detailed analysis of the notion of identity across possible worlds might disclose some metaphysical repugnancy in the construction I have given. What I have proven, then, is that there is no purely logical sentence of the modal language on its weak interpretation which can be depended upon, without a thorough analysis of bans-world identity, to be a translation of (3): I6 ‘Range’ is a term ordinarily used in the discussion of explicit quantification. One speaks of the domain over which the bound variables range as the ‘range’ of the bound variables. I have found it convenient to relocate the concept and speak, not of the range of the variables, but of the range of the quantifiers. The usage in the text is a further extension: modal operators are taken as disguised quantifiers ranging over possible worlds. All this will become clearer. l7 One may prove rigorously that there is no definition of ontological priority in purely logical terms by a modification of the proof that (3) has no translation: leave the ontologically prior individual out of the new worlds but put the individual to which it is prior in. Ontological priority is discussed by Strawson in Individuals; at this late date 1 don’t care to argue about how faithful (9) is to his intentions. I8 Properly speaking, this is a new language, despite the fact that the syntax is the same. The occurrence of equiform sentences in the two languages, possibly with different meanings, should be taken on a par with the fact that the sequence of letters deeayeess-see-you-essessaye-oh-en is a word, though with different meanings, in both French and English: it is a matter of what language teachers call ‘misleading cognates.’ It will, however, be convenient to take the displayed formulas as ambiguous between the two languages, and to speak, for example, of comparing the meanings (3) has on the two interpretations. The strong interpretation is closely related to those proposed by Kripke. It could be taken as the interpretation underlying his ‘Identity and Necessity’ (in Milton K. Munitz (ed.), Identity and Individuution, NYU Press, 1971) and ‘Naming and Necessity’ (in Donald Davidson and Gilbert Harman (eds.), Semantics of NaturaZ Language, Reidel, 1972). and differs only in detail from the interpretation proposed in his ‘Semantic Considerations on Modal Logics (in the Acta Philosophical Fennim volume on modal and many-valued logics, Helsinki, 1963). My comments on the strong interpretation should hold for the interpretation of the last-mentioned paper as well. l9 Lewis, op. cit., briefly considers two alternatives to the weak interpretation. Neither is equivalent to our strong interpretation. Like it, they allow (4) to be a translation of (3), and they correspond to very simple translation manuals between the modal and

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non-modal languages, but they have other, very undesirable, logical chracteristics. 2o The proof that such a translation procedure takes sentences of the modal language into sentences of the non-modal having the same truth conditions is somewhat involved and won’t be given here. It involves taking the strings produced in the course of the procedure as sentences of a composite language, defining a semantic interpretation of that language, and showing that no step in the procedure starting with a true sentence yields a false one. ” David Kaplan has characterized this sort of situation as one requiring, not quantifying in, but quantifying out of a modal context. 22 Note that the predicate ‘P’ is required to be true at a world of buildings that don’t exist in that world. Thus (21) is not really a translation of the English example into the extended modal language as I have defined it. My temptation would be to take this as evidence that the English has been misconstrued, and that it should be taken, not as containing an outer quantification over buildings, but as being elliptical for some claim about building proposals. We needn’t take this line, however. We could preserve the spirit, if not the letter, of (21) by replacing the atomic predicate ‘P’ with some defined expression with a counterfactual in the middle of it, and not have any basic predicates true of individual at worlds in which they aren’t. 23 I conjecture, though I have no proof, that (23) and (25) - and so (1) and the axioms (2) - have no translation in the modal language without outer quantifiers. I would even hazard the guess that no sentence of the non-modal language which is inconsistent with those axioms though logically consistent has a modal translation using only inner quantifiers. 24 Three comments: (a) The “we can go some way toward expressing” is becaule the antecedent of (27) does not exactly say that there is an isomorphism between the associated structures of w and w’. Rather, it says of an antecedently given function - the identity mapping -- that it is an isomorphism. The difference did not bother me when I wrote the first version of this paper because 1 took for granted a related metaphysical thesis to the effect that trans-world identification of individuals was a matter of similarity. Given that thesis, obviously, if them is an isomorphism between the associated structures of two worlds, the identifying mapping is an isomorphism between them. (b) This related thesis is apparently not expressible even in the non-modal language. Without knowing what degree of similarity is required for the idenrification of individuals in different worlds, the best we could do would be to lay it down that an individual in one world exactly similar to an individual in a second world should bc identified with the second individual. Without knowing which similarities - similarities in what respects - arc relevant, we would have to allow for the possibility that even being in the same world as some XYZ individual might be a relevant relational property, so we could only be sure of counting individuals in exactly similar worlds - worlds with the same relation structure - as being exactly similar. But whereas the antecedents of (27) say of a given mapping that it is an isomorphism. here we would have to say that if there is some isomorphism between the associated structures of two worlds, then some such isomorphism is the identity mapping. But the antecedent here is beyond the capability of a first-order language. Of course, given some specification of the degree and respects of similarity required for trans-world identification, it may well be possible to do better: it depends on the specification. Ic) It is also worth pointing out that this similarity thesis strongly suggests the identity of indiscernrbles within a single world. For if there were two exactly similar individuals in one world, it might be

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necessary, under the similarity thesis, to identify both with the same individual in a second world, which is absurd. The reader is referred to Fred Feldman’s ‘Counterparts,’ Journal of!‘htiosop~y 1, July 1971, pp. 406-409. I am now prepared to argue that trans-world identification, though perhaps related to similarity, is quite complicated. In saying what properties an individual or a group of individual may have, we must consider different choices or “stipulations” of how to identify individuals in one world with those in another, with similarity of the various pairs of individuals identified with one another at best a necessary condition for the acceptability of a stipulation. 25 Identity receives the attention it does here because the identity predicate is the only dyadic predicate of our non-modal language taking terms for possible worlds in both argument places. Modal operators correspond to quantifiers over possible worlds in a language in which no more than one possible world term, constant or variable, occurs in any atom - which accounts for the similarities between (propositional) modal logic and the monadic first-order predicate calculus. If we had some other dyadic predicate of possible worlds - accessibility, say - we would have a whole new class of sentences untranslatable in the modal language. There is no formula of S4, for example, equivalent to “p holds at some world not accessible from the actual world.” 26 Suggestion for a model-theoretic proof: suppose there are infinitely many individuals. By a ‘middle set’ of individuals let us mean an infinite set of individuals whose complement relative to the set of all individuals is also infinite. Let A be a middle set. For each middle set of individuals which is not C_A, let there be a world containing all and only its members, let there be no other worlds, and let the actual world contain all and only the members of the complement of A. (28) is true. Now add worlds for the remaining middle sets. (28) has become false, but I do not think that any purely logical sentence of the extended modal language has changed its truth value. 27 In this regard a possible world name - like ‘a’ - may be treated as if it were a variable bound to a quantifier at the extreme left hand side of the sentence. 28 Essential, that is, if we do not have other combinatory resources. Cf. Quine’s ‘Variables Explained Away,’ in his Selected Logic Papers, New York, Random House, 1966. 29 Cf. his ‘Speaking of Objects,’ in Ontological Rekztivity, New York, Columbia University Press, 1969.