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Meaning Relations and Modalities Bas C. van Fraassen Noûs, Vol. 3, No. 2. (May, 1969), pp. 155-167. Stable URL: http://links.jstor.org/sici?sici=0029-4624%28196905%293%3A2%3C155%3AMRAM%3E2.0.CO%3B2-R Noûs is currently published by Blackwell Publishing.

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Meaning Relations and Modalities1

The aim of this paper is to present a certain philosophical perspective on the basic concepts of modal logic. The essentials of our approach, both philosophical and formal, are found in a previous paper,2 but will be recounted briefly in sections 1 and 3. Section 2 contains an intuitive explanation of our interpretation of the modal operators, and section 4 its formal counterpart. Section 5 considers quantification and singular terms in modal contexts. In section 6 we return to philosophical issues with the question whether the interpretation of modal language involves metaphysical commitments. 1. Meaning relations and logical space. Until the present century, logicians seem to have paid little or no attention to the iteration of modal qualifiers. A consequence of (or reason for?) this, is that the interpretations of modalities given then seem to leave no room for the iteration of modal qualifiers. If a modal statement is one which "says how the predicate inheres in the subject7',3 or modality "concerns only the value of the ~ o p u l a " , ~ iteration must be either artificial or trivial. But I would like to 1 This work was partially supported by NSF grant GS-1566; the author also wishes to acknowledge his debt to many helpful criticisms and comments, especially by Professor K. Lambert (University of California, Irvine), Professor R. Montague ( U.C.L.A. ), and Professor D. Scott (Stanford University). 2 "Meaning Relations among Predicates" THIS JOURNAL, I ( 1967) : 161-179; henceforth MRP. 3 Cf. William of Sherwood's Introduction to Logic, tr. N. Kretzmann ( Minneapolis, 1966), p. 40. 4 Immanuel Kant's Critique of Pure Reason, tr. N.K. Smith (London, 1929), A74.

suggest that this is no mere oversight, that indeed in a large area of modal discourse iteration is de trop. As the above quotations suggest, the modal locutions there signal certain relations among the predicates. For example, in "Necessarily, whatever is scarlet is r e d , the modal qualifier signifies that the relation between red and scarlet is not merely extensional inclusion but meaning inclusion. Two kinds of meaning relations were distinguished in MRP: relations of intent and relations of content. The former was explicated, and given a formal representation, in terms of Wittgenstein's notion of logical space. Our example above is provided with the counterpart The region of logical space assigned to scarlet is part of the region assigned to red in the formal mode, and the counterpart Any possible object which is scarlet, is red in the material mode. In general, when "Necessarily, . . ." can be restated in the form "Any possible object - - -", iteration of modal qualifiers makes little or no sense, and the explication is in terms of the structure of logical space. For more detailed arguments and exposition we must refer to MRP, but we make the following claims. First, "possible entity" discourse is given an ametaphysical explication on our approach. Secondly, the use of logical space to build meaning relations into the language helps to make precise the notion that our conceptual scheme has its embodiment in the language we use. And thirdly, this approach gives a formal representation to a certain view of analyticity and the synthetic a priori which we consider philosophically acceptable (essentially that of W. Sellars, Science, Perception and Reality (New York, 1963), Ch. 10).

2. Natural and strict modalities While modal qualifiers sometimes have the function of expressing relations of intent (truths ex vi terminorurn), there are definitely cases which do not fit this pattern. The simplest kind of example still does not involve iteration: 1. It is not possible for a lion to become a cow. One might try to force this into the mold by arguing that if some-

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thing had been a Lion, we would not call it "a cow," but this seems highly implausible. An example explicitly constructed to avoid this move is: 2. Some possible object is an invisible man, but it is not (physically) possible for there to be invisible men.

It may seem now that what is at issue is only the distinction between physical and logical possibility. If that is so, we could eliminate the dBculty by singling out some subclass of logical space as representing the physical possibilities. But there are two problems which show that this solution will not do. The first is that there are non-trivial iterations of physical modality, and the second is that logical possibility and necessity cannot be explicated solely with reference to logical space. First consider the following statement by a (relatively pessimistic) social scientist: 3. We may no longer be in a position to achieve a stable population through education and economic aid, but this need not have been the case: greater effort should have been made ten to twenty years ago.

The first part of this statement has the form "possibly A but not necessarily (possibly A)," which cannot be explicated as true by the means of the previous section. With respect to logical modality, we face the second problem, namely, that on some positions such a statement as "Whatever is scarlet, is r e d is true, and true ex \:i terminorurn, but not logically true. This point is made very clear by Sellars (op. cit., 318-319), who draws a distinction between 'intra-conceptual' and 'extra-conceptual' certainty; in the case of the latter one considers alternatives to one's own conceptual scheme. An example, which involves a reference to a logical space (the color manifold) is found in Becker's famous essay Drawing on a distinction made by Husserl, on iterated m~dalities.~ he argues that "Necessarily, P" may be true without itself being necessary, and gives the example: Die apriorische Struktur der Farbmannigfaltigkeit ist 2.B. zwar notwendig, aber der Grund dieser Wesenstruktur ist uneinsich5 0.Becker, 'Zur Logik der Modalitaten" jahrbuch fiir Philosophie und ~hanomenologischeForschung 11 (1930), 497-548. For its historical role, see C.I. Lewis & C.H. Langford, Symbolic Logic (2d ed. New York, 1959), p. 497.

tig, sie scheint ,,zufalligerweisel' dieser ,,hyletischen Region" anzuhaften. (op. cit., 518) Another example is the familiar trick-question "How many legs does an ass have if we call its tail a leg?" Calling the tail a leg does not make it a leg, but if the supposition is an agreement between speaker and respondent henceforth to call the tail a leg, then the respondent's correct answer is "five." In the extreme, the admission of alternatives must countenance even the possibility of a change in the logic of one's language, although no system of modal logic has allowed for the possibility that a tautology be false. The general features of physical and logical modalities to which we have drawn attention are not unique to them, and we need some wider terms, analogous to Sellars' 'intraconceptual' and 'extra-conceptual'. We shall call a modality strict if it allows for changes in the structure of logical space (or even of logic), and natural if it admits as 'possible worlds' only those which can be embedded in logical space. From the point of view of the natural modalities, logical space is the form of any possible world: this seems to be the view of Kant's Inaugural Dissertation6 and Wittgenstein's Tractatus Logico-Philosophicus. This distinction is clearly not a very precise one, but since logical modality is a strict and physical modality a natural modality, the intuitive basis is not entirely undifferentiated. In our formal explication, the corresponding distinction will of course be an exact one. Having distinguished these modes of modality, we must now ask how they can be represented. Suppose that X is some physical system with state-space H: for example, X is a switch, and H is a set with two elements, 1and 0, where 1represents the 'on' position and 0 the 'off position, or X is a computer, and the elements of H represent all the possible states S1,. . . , S, of that computer. By "It is possible that X is F we may mean only "Some element x of H is such that if X is in the state represented by x, then X is F." On the other hand, we may mean by it "X is in a state from which there is a physically possible transition to another state in which X is F." In the former case, we are within the domain of the previous section. In the latter case we have what is properly called a natural modality. To represent this latter case we clearly need to represent the 6 The explication of modality in the C~itZqueof Pure Reason ("That with the formal conditions of experience, is possible." A21S) which agrees seems to me to differ from this point of view mainly in its epistemological orientation.

...

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physically possible transitions: we may do this by means of a set U of transformations u of H. (That is, a member u of U is a function mapping H into H; and u(x) represents a state to which there is a physically possible transition from the state represented by x.) Then:

4. "Possibly, X is F is true in state x if and only if, for some u in U, "X is F" is true in state u(x). Finally, by "Possibly, X is F we might mean that given some radical change in the situation, or our idea of it, "X is F" would be true. In that sense "Possibly, the switch has three positions7' is true, in that "The switch has three positions7' would be true if the switch were rebuilt. And in that sense "Possibly, any electron has a definite position at all times" is true, in that it is (logically) possible that quantum theory should turn out to have been altogether wrong. What we are considering here is a set U' of transformations u' of the structure of the logical space:

5. "Possibly, X is F is true in state x, in the logical space H, if and only if, for some u' in U', "X is F" is true in state x in the logical space u' ( H ).

To sum up, then, physically necessary truth is truth invariant under all possible physical transitions, and logically necessary truth is truth invariant under all possible changes in meaning relations among terms. Each of these kinds of possible changes (physical, linguistic) can be represented by a set of transformations. Now, we can represent both modes of modality at once, by forming for each u in U and each u' in U' a new function t, which acts like u on the domain of u and like u' on the domain of u'. (The two domains need to be disjoint, i.e. no logical space is an element of some logical space, in the ~ o n t e x t . We ) ~ are going to simplify this somewhat further, because strictly speaking, a logical space is a rather complex kind of thing, but the details of that will appear in our formalization in section 4. Of course, the point of all this is not just to provide modal logics with a semantic analysis; that exists already. Rather we wish to show that such a semantic analysis may be given from a specific (ametaphysical) philosophical point of view: we are attempting a 7 It might be thought that some u' in U' could do the job of a given u in U, so that the representation of natural modalities can be a special case of the representation of strict modalities. But we wish to allow functions in U to be not one-to-one; in D. Kaplan's terminology, we allow the trans-world heir lines to come together and continue as one.

philosophic retrenchment for formal semantics. To this subject we shall return in the last section. 3. Semi-interpreted languages In MRP, we defined a class C(S) of semi-interpreted languages. The syntax S of such languages is that of quanti£icational logic, except that the quantifiers are written (/x) and (/Ex), and that we have an identity predicate = and a special monadic predicate E!("exists7'). The ordinary quantifier is contextually defined as (x)A = (/x)(E!x > A). We take N, &, (/x) as primitive, and define the other logical signs as usual. Attention to this specific syntax makes our treatment less general than it could be, but will not exclude any of the familiar logics. When L is a member of C(S) it has besides this syntax associated with it a non-empty set H and a mapping f of the n-ary predicates (other than E!, =) into subsets of Hn (the set of n-tuples of members of H). The function f is called the interpretation function. Strictly speaking, the logical space of L should be identified with the couple < H, { f(P): P a predicate) >, but we also use "logical space" loosely to refer to set H. To change the structure of the logical space clearly means to change the function f. A model for L is any couple M = < loc, D >, where D is a set and loc a one-one mapping of D into H. D is the domain of discourse, and each member of D thus has a location in logical space. The truth-definition for L is quite straight-forward; for the sake of later convenience, we shall here use the term satisfaction. We use d, with or without accents or subscripts, to range over Hw (the set of denumerable sequences of members of H), and write d(x) for the ith member of d when x is the ith variable of L. We write "d satisfies A in model M as M A [dl, and "iff7 for "if and only if'.

I=

iff d(x) = loc(b) for some b in D, I= (El x) [dl where M = < Zoc, D > (2) M I = (Pnxl . . . x,) [dl iff f(Pn) iff d(x) = d(y) (3) M /=(x = y) [dl iff not M I= A[d] (4) M I= (-A)[dI (5)M/=(A&B)[d] iffM/=A[d]andMI=B[d] iff M I= A [d'] for all d' like d except (6) M I= (/x) A[d] (1) M

E

perhaps at x (briefly, d' =, d).

We shall call h a ,satisfaction-function for L if h is defined exactly for the sentences of L and there is a model M such that

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for all sentences A to L. It will be convenient henceforth to discuss logical relations in terms of satisfaction-functions. (Note that H, f, and M determine h, and that conversely, h determines H, f, and M up to isomorphism.) We say that a sentence A is valid (11-A) in L if all sequences in Hw satisfy A in all models; equivalently, if h(A) = Hu for all satisfaction-functions h for L. And we say that Al, . . . , A, semantically entail B (Al, .. . ,A, 11- B) in L if B is satisfied whenever Al, . . . , A, are satisfied; that is, if h(Al) n .. . n h(A,) is included in h(B), for all satisfaction-functions h for L. If X is a set of sentences, we understand X 11- A in that way: A is satisfied whenever all member of X are satisfied. And if C is a class of languages, we say that X 11-A in C if this is the case in every language in the class C. I t can now be proved (MRP section 6) that A1,. . . ,A, 11- B in C(S) if and only if B is derivable from Al, . . . , A, in classical quantificational logic (with identity and (/x) as the universal quantifier). In addition the logic of the restricted quantifier (x) is free logic.* We shall now have to generalize this apparatus so as to accommodate modality. 4 . Modal operators in semi-interpreted languages We shall now define a class C(S,) of semi-interpreted languages. The syntax S, is as before except that we add the proposiAssociated with a language L in C(S,) are not tional operator 0. only a non-empty set H and interpretation function f, but also a superset K of H, and a set T of transformations satisfying certain conditions which we shall now explain. We denote as K" the set of functions f defined on the sentences of L, such that f(A) KWfor any such sentence A. Each member t of T is now a transformation on K U K", with the special condition that for any element b for which t is defined, t(b) c K if and only if b e K. (So t has the effect of two transformations, one on K and one on K".) For any function f in Kn we define the space of f, Sp(f),to be the least set J such that f(A) C

Jw for all sentences A.

Clehrly not all the members of K" are proper generalizations of the satisfaction-functions of the preceding section. We call a 8 Cf. R. Meyer and K. Lambert "Universally Free Logic and Standard Quantification Theory" Journal of Symbolic Logic XXXIII (1968): 8-26.

member h of K" a satisfaction-function for L (in C(Sm))iff the following conditions are satisfied: (1') There is a subset h(E!) of Sp(h) such that h(E! x) = {d e Sp(h)~:d(x) c h(E!)) (2') There is for each pedicate P of degree n a subset h(P) of SP(h)" such that h(Px1. .. x,) = {d e S p ( h ) ~ h(P)) (3') h(x = y) = {d e Sp(h)w:d(x)= d(y)) (4') h(-A) = Sp(h)w- h(A) (5') h ( &~B) = h ( ~ n) h ( ~ ) (6') h((/x) A) = {d e S ~ ( h ) ~ :e dh(A) ' for all d' = ,d in S p ( h ) ~ ) (7') h ( n A ) = {d c Sp(h)w:td th(A) for all t e T)

If h is furthermore such that Sp(h) = H and h(P) = f(P) for each predicate P, we shall call h a principal satisfaction function for L. We have two ways of generalizing validity in L: we may call A valid in L iff h(A) = S p ( h ) f~or all principal satisfaction functions, or for all satisfaction functions. In the former case we would expect that there are sentences A such that 11- A but not 1) - UA, even if all desirable conditions are fulfilled: A would be intra-conceptually, but not extra-conceptually, necessary. Henceforth we shall mean "valid" in the second sense; and similarly we shall say that Al, .. . ,A, ] I - B in L iff h(Al) n . . . n h (A,) 1 - h(B) for all satisfaction functions h for L. Finally, we shall say again that these relations hold in a class C of languages in C(Sm)iff they hold in all members of C. For greater definiteness, we shall henceforce identify L with the quintuple . From the point of view of logic the laws which hold in a specific such language are not usually of much interest; one would rather consider those which hold in a class of such languages, a class characterized by some interesting set of assumptions about their structure.

5. Laws of modal logic We shall now introduce a number of assumptions concerning the set T of transformations, and show how they (cumulatively) entail the validity of certain well-known principles of modal logics.

Assumption 1. If f is a satisfaction function, then t(f) is a satisfaction function, for all t in T, and all f in K". This assumption rules out, for example, that 0 (A & -A) may be

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true. If one admits the possibility of changes in logic, one would not make this assumption. (We write "tf' for "t(f)", and so on, when convenient.) Assumption 2. If b e Sp(h) then t(b) E Sp(th), for all b in K, all satisfaction-functions h, and all t in T. Now we can prove that if 11- A then 11- OA. (For suppose that ~ . there must be a sequence, d in Sp(h)" such h(UA) # S P ( ~ )Then that td e th(A), for some t in T. By assumption 2., td c Sp(th), so then th(A) # Sp(th)u.) We can also prove that UA, u ( A I, B) 1)- U B at this point. (For suppose that d belongs to both h ( a A ) and h(m(A > B)). Then for each t r T, td is in &(A) and also in th(A > B). So for each t E T, td r th(B); that is, d a h(uB).)

Assumption 3. T contains an identity element, that is, a member t such that t(x) =x for all x in K U K O . From this it follows that CIA 11- A. The three preceding results correspond to the basic modal laws of von Wright's system M. To obtain similar results for Sq and S5, we add assumptions which make T respectively a semi-group and a

group.g Assumption 4. For all t, t' in T the composition tt' is in T. It now follows that O A 11- U n A . (For if, for some t and t', t(t'd) 6 t(tfh)(A), so that d c h(mOA), then T contains a transformation t" = tt' such that Yd t"h(A), so that d r h(OA).) Assumption 5. For all t in T, T contains an inverse, that is, an element t" such that t"t = i. Now it follows that 0 A 11- 17 0 A, where 0 is defined as usual. (For if d E h( 0A), then t'd a t'h(A) for some t' in T. Now let t be any element of T; then t't"t = t'. So for each t in T, T contains a transformation f' = t' t" such that t" td E t"th(A). That is, for all t in T, td is in th( 0 A). Hence d E h ( n 0A).) In quantified modal logic, the best known principles are the Barcan Principle and its converse: (BPI (Cv BP) 9 The usage of "semi-group" varies somewhat, but we shall use it when assumptions 3 and 4 are both fulfilled.

These are now usually argued not to hold in general.1° For example, the second has the consequence (x)nE! x: every actual existent exists necessarily. It is easy to see that these principles do not follow, in our framework, unless special assumptions are made relating th(E! x) to h(E! x). But it is not so implausible to have the analogues of these principles for the unrestricted quantifier (/x) hold. Assumptions 1 and 2 suffice to prove that D(/x)A 11- (/x)UA. (For suppose that d in d 4 S p ( h ) ~does not belong to h((/x)OA). Then some d' S p ( h ) d~oes not belong to h ( n A ) . So for some t in T, td' # th(A). But by assumption 2, td and td' belong to Sp(th)w, and td' = ,td, so td g th((/x)A), but then d r h(O(/x)A).) To prove the analogue to the Barcan principle, however, a further assumption is needed.

=.

Assumption 6. For every element b in Sp(th) there is an element c in Sp(h) such that b = tc, for all t E T and all satisfaction-functions h. Now the principle (/x)OA 11- O(/x)A holds. (For suppose that d 6 S p ( h ) ~does not belong to h(n(/x)A). Then for some t in T, td c th((/x)A). There are now two possibilities: either td t Sp(th)" or there is a sequence d' = td in Sp(th)w which does not belong to th(A). If the former, td r th(A), and hence d # h(OA), but then d c h((/x)OA). If the latter, let be a sequence in S p ( h ) such ~ that d and t(dr'(x)) = dr(x). That d" exists is guaranteed by asdr' sumption 6. Now t P = d' which is not in th(A), so d" # h ( n A ) , and hence .d B h((/x) q A).) It may be noted finally that these assumptions are not the only ones that lead to the results noted. For example, assumption 3 could be replaced by the assumption that for each b in K and h in K", there is a meniber t of T such that t(b) = b and t(h) = h. But in that case an identity element could be added to T without changing the set of valid arguments, so that there is no loss in the more elegant assumption which we actually made. The above results amount to soundness proofs for modal logics under our interpretation, and it is also possible to give completeness proofs for the systems discussed; specifically, completeness proofs have been constructed for tho mason:^ systems S4Q1 and S4Q3 ( Thoniason, op. cit. ) .

.

=.

10 Cf. the discussion in R. H. Thomason, "Modal Logic and Metaphysics" in T h e Logical W a y of Doing Things, ed. K. Lambert (Yale University Press, forthcoming).

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6. The question of essentialism Quine has argued that the assertion of certain modal statements commit one to the curious doctrine of essentialism, or at least, that these statements are not intelligible except in terms of this doctrine. In an examination of this claim, Terence Parsons has shown that no statement for which this can be argued (non-trivially) is a consequence of any of the familiar modal systems. Hence, he concludes, the use of modal logic does not commit us to e~sentialism.~~ This is a welcome conclusion to the 'anti-essentialist', but it cannot provide final comfort. For if we accept both Quine's and Parson's claims, it follows that one not committed to essentialism can still not make sense of completeness proofs for modal logic. For such a proof treats as possibly true all modal statements consistent with the theorems of the system in question. Parsons allows that certain statements do not seem to admit interpretation as possibly true except in metaphysical terms, and adds that it remains for the staunch anti-essentialist to add axioms excluding them to his system (op. cit., 190-191). There is of course another alternative: to attempt a philosophically acceptable, aGetaphysical interpretation of extant modal logic. Parsons notes two "linguistic" interpretations, inspired by Ch. V of Carnap's Meaning and Necessity. These rely on substitution operations in their truth-definitions for modal statements, and lead to versions of S4 and S 5 (op. cit., 187-188). They are not satisfactory, for any system complete under either interpretation must have such atypical consequences as "If A is an atomic wff, then /I- -.uA holds". Can there be a "linguistic" interpretation of modality adequate for the familiar systems, that makes sense of the currently accepted semantics of modal logic? Our claim is that previous attempts have faltered because of an overly restricted theory of meaning. \$'hat is (logically) necessary is not always true just in virtue of the meanings of the words; it may be so because it remains true even under all (admitted) changes in the meanings of the words. And the meanings of the words derive from semantic features of the language which may not be explicable in terms of such syntax-oriented notions as axioms and substitution operations. Our claim is that the interpretation of modalities in terms of transformations of (the structure of) logical space shows that no modal statement can involve one in any metaphysical commitment. 11

181-191.

"Grades of Essentialism in Quantified Modal Logic" Nous I (1967):

Of course, this requires that we have a theory of meaning which is not only not overly restrictive, but also acceptable. For a philosopher, the 'explanation' that "Possibly, . . ." is the case if and only if ". . ." is the case in some possible world, cannot be a final answer to his question. At best, it can be imagery used to aid the mastery of such more accurate "explanations" as "(Possibly A ) is true in model M if and only if there is a model M', bearing R to M, in which A is true". But the imagery has the form of a purported answer to the question "Under what conditions is an assertion of possibility truer', whereas that for which it is imagery does not. Perhaps the latter can be given the form of such an answer by adding that, like the scientist, the logician constructs a modeP2 of certain facts? In that case, the question is: of what does he construct a model? And here there seem to be two answers in vogue: the modal logician constructs a model of intentional objects like a physicist of electrons; the modal logician constructs a model of modal discourse like a linguist of Swahili. If to the first answer we reply with a question about the analogue to experimental data, we are referred to modal discourse-which seems to bring us to the second answer. However, if the logician is so like a linguist, why his disregard for the syntactic form of natural language, why the regimentation of language, and why no theory of the exact relations between logical syntax and the statistical facts of usage? These general questions concerning the enterprise of formal logic and semantics must be raised if we wish to show that we have a philosophically acceptable interpretation of modal logic. Our interpretation is, briefly, that in a given language there exists a structure of meaning relations among terms, and that this structure determines truth and falsity ex ui terminorum. But in addition, there is associated with the language a set of transformations of this structure, and that these transformations determine necessity and possibility. Pressed for an explication, we can exhibit the formal representation of these semantic relations and transformations. But for philosophy of logic, we must say more; and here we point to meaning relations in natural language constituted by linguistic commitments (MRP, section 3 ) and the ability of natural language users to consider, and admit as relevant, alternative linguistic commitments (Sellars, v.cit., 318-319). But it does not look as if we are doing linguistics, and we must specify what we consider to be the l2In this context "model" has of course a different (though surely familiar) sense from that in previous contexts, at least pm'ma facie.

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relation between the semantic concepts (such as meaning inclusion) which we employ, and the pragmatic concepts (such as language use and linguistic commitment) to which we refer. Our answer here (MRP, 167) is that the use of a concept in formal semantics is at once warranted and of philosophical interest if and only if the concept has a clear pragmatic counterpart. And we call the second concept a pragmatic counterpart of the first if the first is arrived at from the second by abstraction from relations to user and to context of use.13 Thus, "the term W denotes the object 0 has as pragmatic counterpart "the person X uses the term W to refer to the object 0 ( at time t )". The former is clearly not definable in terms of the latter ( i t is not that simple a kind of abstraction). This is, in part, how logic differs from any empirical science of language: its notions are not definable in terms of pragmatic notions, and its theses do not explain pragmatic facts. But the existence of the pragmatic counterpart at once warrants and gives significance to thk semantic concept of reference. And as for reference, so for meaning and modality. 13 Formal or pure pragmatics belonging a t a level of abstraction between formal semantics and empirical pragmatics.