GRADES OF MODALITY L. F. COBLE

GRADES OF MODALITY L. F. COBLE I. Despite its title, this paper does not deal with what Quine discussed in his "Three Grades o f Modal Involvement" [3]. I t may, however, apply somewhat to the sort of 'gradualism wit h regard to modal matters wh ich Goodman suggested ([2], p. 7) and which Quine further recommended ([5], p. 67). I shall develop here a notion o f modality according to wh ich propositions can be distinguished b y degrees o r grades o f necessity or possibility. Thus it will be possible to say of two necessarily true propositions that one is more necessary than the other, o r to say, simply, that one proposition is more o r less necessary. At this time I shall o n ly be concerned wit h the formal behavior o f these modalities; I shall n o t n o w discuss possible applications o f these notions o f necessity. Accordingly, after specifying the grammar I expect these modalities to follow, I will examine them from both syntactical and semantical points of view. The calculuses proposed will be shown to be consistent and complete with respect to their suggested semantics. I will o n ly be concerned here wit h the degrees o f modality in propositional logic; I expect no n e w problems to arise when the systems a re extended t o include quantifiers th a t d o n o t arise with any quantified modal logic. 2. Since we are investigating propositional logics, it will be presumed that they are cast in a language that meets the usual conditions fo r such systems; that it contains atomic formulas, p, q, r... etc., and the truth-functional connectives, conjunction (&) and negation (-7 ), obeying the usual grammatical rules. (Other truth-functional connectives, e.g. disjunction (v) a n d material implication (D) are to be defined in terms of conjunction and negation in the familiar ways.) I n addition, it is supposed that the language contains monadic modal operators, N

324 L. F. GOBLE is a we ll formed formula if and o n ly if A is. These operators represent th e different degrees o f necessity; a formula, N A, should be read as saying that A is necessary to, a t least, the degree j. This should be interpreted in such a wa y that the higher the degree, the stricter the necessity. Thus, fo r example, the necessity o f A is stronger when 1 \1 just N one can define possibility operators M is equivalent to 7 N With in this basic framework one can construct various languages, each containing a diferent number of modal operators N operator. I t might be, however, that one would prefer to distinguish t wo grades o f necessity, say, lo g ica l necessity and physical necessity (assuming that if a proposition is lo g ica lly necessary then it must also be physically necessary), wit h the former being the stricter of the two. Possibly one would want to introduce a th ird and weaker so rt o f 'practical necessity, for when one says, ordinarily, o f an event that it must occur. For o th e r purposes, i t mig h t b e appropriate t o h a ve st ill a greater number of distinct degrees of necessity. It is, of course, possible t o have a language wit h denumerably ma n y modal operators; indeed, the systems I describe below will be formulated in a way to allow for this case. A language o f the kin d ju st described w i l l be called 2 2 where k is the number o f operators fo r (different degrees of) necessity in the language. 3. Given a language 2 ' interpreted as degrees o f mo d a lity in as ascending order o f strictness, i t is f a irly clear what axioms and principles o f in - ference should govern these concepts. Accordingly, we define a calculus in 2 ' A.0 a set o f axioms wh ich wit h modus ponens generates the classical propositional calculus. A.1 N A.2 1\1,(AD B) N A.3 N

GRADES OF MODALITY 3 2 5 R.1 mo d u s ponens: from A and A D B, to infer B. R.2 necessitation: if 1 A, then 1 N,A, for every i, A.I is introduced to express the ranking of the modalities according to degree, the higher the degree, the stricter the necessity. The other postulates and rules are a ll fa milia r fro m the standard modal logics. Obviously, each individual modality, N by itself behave like the necessity defined b y the system T o r M o f G6del-Feys-von Wrig h t. Hence, I w i l l ca ll th is system Tk, k being the number of modal operators in the language Y Additional postulates ma y be added to define systems co r- responding to S.4 and S.5. Thus, S.4 axiom schemata: A.4 N to Tk and S.5 A.5 --7 N to Tk. Other systems, corresponding to other standard modal logics, ma y be defined in corresponding ways, but I shall not be concerned with them here. It might be suggested that a stronger axiom than A.2 should be postulated f o r these systems. On e mig h t propose i n it s place: A.2' N This corresponds t o th e wid e ly accepted p rin cip le th a t th e conclusion of an inference (B) is at least as sure (necessary) as the weakest o f its premisses ( A and A DB). A.2 wo u ld then follow as a special case of A.2' when i = A.2 i s, however, already derivable in Tk, g ive n A.2 and A.I. Thus, suppose that j <i, so that A.2' appears as N D N A DNB. By A.1 I NJADB) D N ( A ( a princ iple t o the effect that if a formula, A, is an ax iom, then so is N an ax iom f or ev ery i, I I n proofs of lat er theorems it w i l l often be convenient t o t hing o f t he systems formulated i n t his way, w i t h modus ponens as the sole rule of inference.

326 L. F. GOBLE by transitivity. Similarly, if i< j, so that A.2' is N DNB ; N is equivalent to N A D N N But, b y the same token, i f A.2 b e postulated as an axiom, not only is A.2 derivable as a special case, but A.1 is redundant, given R.2. Thus suppose that j ; N then an instance o f A.2'. Since 1 ADA, N Hence, N obtained from Tk (SAk, S.5k) b y replacing A.I and A.2 b y A.2' is equivalent to Tk (SA0 S - 4. I n another paper [ l a general semantical theory fo r the standard modal logics in the manner o f Kripke's semantics, but in such a wa y that it was n o t necessary t o assume among th e p rimitive s o f th e semantics a relation, R, of relative possibility between possible worlds. Those methods w i l l n o w be extended to provide an interpretation f o r th e different modalities N ' In [I] I suggested that we conceive a possible world, w defined by the ordered pair < P,, 13,>, where 13, is a complete and consistent set of formulas (in tu itive ly, the atomic formulas in B truths of w plete, set of formulas included in B 'fundamental postulates' governing w in P Since we n o w wish to distinguish grades o f necessity, we must distinguish grades amongst the fundamental postulates. Thus, for a world, w we should have different sets of formulas, P is 'fundamental to, at least, the degree 2'. We should then define evaluation rules f o r formulas in such a wa y that i f an atomic formula, p, belongs to P least, the degree i (i.e., N Let us put this more precisely. A normal world, w ordered set, a

GRADES OF MODALITY 3 2 7 (1 s a t i s f y i n g these conditions: (i) th a t as there a re k many distinct modal operators N many sets Pi nin is a consistent set o f formulas (i.e. there is no formula in.r,, such that both it and its denial belong to Pi,,,); (iii) mo re - over, that P,,, is a complete set (i.e. f o r every formula in..rk either it o r its denial belongs to ro t); r, thus plays the role played b y 13, in the account o f [ l and pi, in :T included in 1 ) Given this notion of a 'possible wo rld i t is n o w possible to define models b y which formulas can be evaluated. A s in i l l let a model, t, be a pair <wo, W >, where W is a set o f worlds as defined above and wo e W. For each model, (t. = <WO, W >, let the evaluation function, (N, wh ich maps pairs o f formulas, A, a n d possible worlds, w t o tru th values, b e defined a s follows: (a) I f A is an atomic formula, cp, (A, w A ei)% Suppose than qi i s defined f o r formulas B and C and a ll worlds w, in W. (b) ((AB & C, w,) = T i f a n d o n ly i f %(B, w,) = T a n d cp, (C, w (c)cp,(---713, w,) = T if and only if cp,(b, w To determine truth values fo r necessitative formulas, NB, we define relations between the possible wo rld s in W, different relations being used in the evaluation o f different modalities. Following the pattern o f [1], l e t u s define relations W follows: w, Ri We ca n then g ive a comprehensive definition o f (p formulas of the sort N (d) cp,(n in W such that w, Ri

328 L. F. GOBLE The clause (d) above defines a T-ish necessity, as w i l l be shown; notice that a ll the relations W necessarily transitive or symmetric. If an S.4 necessity is to be obtained, relations Ri which are transitive as we ll as reflexive must be defined. For an S.5 necessity relations which are symmetric, transitive and reflexive are required. These conditions a r e m e t b y R W, m r if and only if Pi, P i, for every j such that i j 1 k. w i- P (Notice that the Ri in the R i should be modified b y replacing 'Ri and 'Ri cases as one, letting 'cp' and correlatively ' N except as otherwise mentioned. Th e definitions o f ' Ri 103' m a y seem u n d u ly complex. Ho we ve r, i t i s necessary to introduce talk of the sets Pi to insure the va lid ity of A.1 as well as A.4 and A.5. Notice that the analogous condition fo r RI every i s already met since Pi cr,(a, w go on to say that A is true on 1.t(=<w since wo is supposed to be the actual world (for 14 Finally let us say that A is va lid if and only if A is true on every model pt. These formulations a re ambiguous, depending o n wh ich re lations RI are used in clause (d) o f the definition o f (p. To be accurate, we should speak of a formula's being T valid, o r S. 5 R 5. We are n o w in a position t o establish consistency and completeness theorems f o r the calculuses defined in section 3 relative to the semantics just given. Theorem 1. I f A i s provable i n Tk (SAk, S 5k), th e n A i s Tk (S.4 I leave the proof of this to the reader; it is easily shown that

GRADES OF MODALITY 3 2 9 all the axioms are valid and that the rules preserve this property. Theorem 2. If A is Tk (S.4 T k ( S A " S. 5 This ma y be established along the lines given in [1] utilizing a Henkin-type argument. W e observe f irst t h a t some quite general results hold for these systems: Lemma Ï. I f X is any system containing the classical propositional calculus, and i f 7A is n o t provable in X, then the system X obtained from X by the addition of A as an axiom is consistent (i.e there is no formula B such that both B and 7B are provable in X'). Lemma 2. If X is any consistent system containing Tk (S.4k, S.5 consistent extension of X in which A is not provable. We can now show that any non-theorem of these systems is falsifiable, which gives us Theorem 2. Let K be a n y consistent extension of Tk (S.4k, S 5k) in which both R.1 and R.2 are admissible, and le t L, M, N, e t c. be complete and consistent extensions o f K (lemma 2). Fo r a n y such syste m M, le t th e wo rld determined b y M, w < I n provable in M, and each P i such that 1 m M is consistent, and To will be one set P i moreover, if i j, then P each w of section 4. Let us speak of a model determined by K,! I pair < w complete and consistent extensions o f K, and w determined by L, any complete and consistent extension o f K. Given IL

330 L. F. GOBLE Lemma 3. For all w Proof is by induction on B. (a) If B is an atomic formula, then m tion that the lemma holds for formulas C and D. (b) I f B = C&D, M T (inductive hypothesis) i f f cp, sistency of M), so c r hence, cr, wm) = T, then (r,,, (c, wm) T, so n o t H m then H 3 1 Let N be any complete and consistent extension of K such that w SO Crp,K W N ) = T inductive hypothesis). Hence, t r T. Suppose that (NI( (NC, w the system all of whose axioms are the formulas, A, such that H m N a consistent extension of K, for if H KA, H b ility of R.2 in K, so H and consistent extension, NI*, in which C is not provable (lemma 2). w then P i Ri A 1 ) i H H so, th e n H m Now, given that q),,, wo rld w T. But b y the inductive hypothesis, this implies that 1 contrary t o th e specification o f N. Consequently, w e mu st conclude that H Theorem 2 n o w follows immediately. For, i f a formula, A, is not provable in Tk (S.4k, So5k), there is a system L such that L is a complete and consistent extension o f Tk (So4k, S.5k) and

GRADES OF MODALITY 3 3 1 A is not provable in L (lemma 2). Let the model la be < w where W is the set of all worlds determined b y complete and consistent extensions o f Tk (S.4k, S.5k) a n d w determined b y the aforementioned L. B y lemma 3, (pp, (A, wl) T if f 1 - falsified by this [t. Therefore, if A is Tk (S.4k, S. 5 be provable in T 6. Thus far we have constructed equivalent syntactical and semantical frameworks in which true propositions can be distinguished according t o th e degrees o f th e ir necessity. W e could say that A is more necessary than B if A's greatest degree of necessity is greater than the greatest necessity o f B. Th a t is to say, A is more necessary than B when N A and N true and i j and there is no m such that m i and N true. Nevertheless, n o t a ll (true) propositions belong t o th e spectrum of necessity. There remains a gulf between those propositions wh ich a necessary to some degree, and those tru e but contingent propositions wh ich are in no wise necessary. It wo u ld be desirable to arrange a ll (true) propositions along a single continuum of degrees of necessity. Along these lines it would seen natural to introduce into the language of these systems another operator, No, wh ich should satisfy the principle.a.3' A N o A. This postulate wo u ld replace A.3 in Tk (S Lik, S 5k), it s being understood that a ll the other postulates o f the system wo u ld now apply to No as we ll as the other operators fo r necessity. Since N0A would be equivalent to A, we could then say that any true proposition was necessary to, a t least, the degree O. This is not to say very much, to be sure; it does not commit one to saying that contingent propositions a re necessary in a n y orthodox sense, but it does allow one to locate them with in the spectrum of modality. From the semantical point o f vie w presented in section 4, one would like to introduce a relation, W Wm R

332 L. F. GOBLE (similarly fo r re form N This w i l l not do, however. The a xio m A.3 i s not va lid b y this account. Those instances o f A.3' in wh ich A contains no well formed parts of the form N N the fo rmu la N (wo, w Pl show that (f ( N T and w this 1.1), b u t cp,, ( N T as W A.3' falls in cases like this because distinct worlds w, and w differ i n some fundamental postulates, P 'truth operator' says, in effect, wh e n evaluating formulas N in w, look at A only in the world w,. IV as given above does not n a rro w one's visio n enough; i t allows one to shift one's view to some other w. Accordingly, le t us instead stipulate that w, w if and only if w A.3' is va lid on this account, as are a ll the other axioms fo r N. The proof o f semantic completeness is unaffected b y this addition to the semantics for TL. Thus in the proof of lemma 3, case (d) (2nd part) in wh ich B might have the fo rm NoC, we observe that for the systems M, N, and N* described, if C P i, (i?-0), then 1 m and 1 i?-0. Similarly, P It is unfortunate that in the semantics fo r Tk as enriched b y N be defined significantly d iffe re n tly than th e o th e r relations Perhaps one would expect this, however, as one notices that in Tk N And there's A.3' itself. R. must also be given special handling in the semantics for SA R

GRADES OF MODALITY 3 3 3 For S.5 the relations Ri other relations: 7. A t the opening of this paper I suggested that the account of modalities given here might have some application to the modal 'gradualism o f Goodman, Qu in e and others. I should now like to caution that suggestion somewhat. I believe some progress has been made here in developing the idea of a spectrum of modal degrees; we can explicate how one proposition, p, is more necessary than another, q, in terms o f p's possessing a higher degree of necessity than q. Moreover, one could say simp ly that a proposition, p, is mo re o r less necessary according as the degree of necessity on p is more o r less high. Nevertheless, I doubt that the critics of modal logic would find these notions much more congenial than they found the more fa milia r ideas o f (absolute) necessity and its correlative, analyticity. For arguments simila r to those wrought against strict necessity and analyticity might be turned against the present graded modalities. Thus, it might be argued against the absolute necessity as formalized b y the fa milia r modal logics, th a t no clear sense can be made of this concept since there are no useful criteria which unequivocally locate propositions some as 'fundamental postulates' and so as necessary, others as true but contigent (as described in [1]). This problem need not arise with in the present framework. A ll that is required is that one be able to say of a proposition, p, that it is more fundamental than another, q, or that p is more central to one's conceptual framework than q. p should then be located in a set Pi and q in Pi where i j. It does n o t matter if this grants some degree o f necessity to statements like 'there have been black dogs' as well as to those like '2 + 2 = 4', though one wo u ld expect i t t o be a lesser degree. Some sort of relative centrality of propositions in conceptual frameworks d o e s b e lo n g w i t h i n t h e p ict u re sketched b y Quine in, e.g. [41. Nevertheless, it is not clear that the behavioral crit e ria wh ich Qu in e wo u ld a llo w t o determine h o w

334 L. F. GOBLE central o r h o w fundamental a statement is, wo u ld define the linear order among classes o f statements which is required by the present semantical account. I f not, th e n th e th e o ry o f grades o f modality proposed here wo u ld not provide a true explication of that sort of gradualism. University of Wisconsin L. F. GOBLE REFERENCES [ ] GOBLE, L. F., " A Simplified Semantics f o r Modal Logic," forthc oming. [2] GOODMAN, N., "O n Likeness of Meaning," Analy s is, 10 (1949), pp. 1-7. [3] QUINE, W. V. O., "Three Grades o f Modal Inv olv ement," i n Th e Way s of Paradox, New York, 1966, pp. 156-174. [4] QUINE, W. V. O., " Two Dogmas of Empiric is m," in From a Logic al Point of View, New York, 1961, pp. 20-46. [5] QUINE, W. V. O., Word and Object, Cambridge, Mass., 1960.