Assertion, denial and some cancellation rules in modal logic

TIMOTHY WILLIAMSON

ASSERTION, DENIAL AND SOME CANCELLATION

RULES IN MODAL LOGIC

0. The technical parts of this paper concern the effect of some non- standard rules of proof on normal modal logics. There is also a philosophical motive for investigating this topic, which it is as well to explain first.

Consider two philosophical claims:

(A) Assertibility-conditions determine truth-conditions.

(W Assertibility-conditions are truth-conditions.

(A) and (B) arc deliberately schematic. In particular: the discussion will apply to a variety of epistemological concepts, each of which can bc used as a reading for ‘asscrtibility’. The assertibility of a sentence might be a dcscriptivc matter (Quine’s dispositions to assent), or it might have prescriptive overtones (Dewey and Dummett’s warranted assertions). It might require conclusive proof. or only defcasible cvi- dence. (A) and (B) could also be rephrased in a less linguistic mode. as concerning the relation between the truth-conditions of thoughts and the conditions under which they can bc verified. accepted or con- firmed. They clearly have something to do with philosophical disputes about ‘verificationism’ and ‘anti-realism’. On the face of it, (A) seems to bc the’natural destination of certain arguments summarized in the slogan ‘Meaning cannot transcend USC’ and (B) an absurdly strong form of verificationism: any dificulty in holding (A) without (B) would therefore constitute a prirnajbcie objection to those arguments. A better understanding of the logical relations between (A) and (B) might throw some light on this confusing area.

(B) obviously entails (A); identity is a special case of determination, provided that (as here) the latter is taken in a logical rather than a causal sense. (A) does not obviously entail (B); after all. someone who holds that falsity-conditions determine truth-conditions is not thereby

300 TIMOTHY WILLIAMSON

committed to holding that fdlsity-conditions are truth-conditions. If (A) does not entail (B), them is a philosophical position to be con- sidered on which (A) is asserted and (B) denied; it would offer a non- trivial explanation of how sentences get their truth-conditions, without doing so at the expense of an implausibly verificationist consequence.’ That might be a good way of reconciling realist and anti-realist insights. On the other hand, if (A) does in some non- obvious way entail (B), then anyone who wishes to deny the vcrifi- cationism of the latter faces the task of saying what beyond asscrtibility-conditions determines truth-conditions, if they arc deter- minate at all. There would be the prospect of non-trivial connections between semantics and metaphysics, for it looks as though semantic premises might entail (A), while (B) might entail metaphysical conclusions.

Section 1 shows that on a variety of assumptions (A) does entail (Hf. The assumptions have some prima ,fucie plausibility, but their detailed evaluation is not attempted here; it will be sensitive to the notion of assertibility at issue. Section 2 is technical; it investigates a rule of proof in modal logic corresponding to (A). Section 3 discusses the proposal that someone who wants to avoid (B) but thinks that epistemological facts do determine truth-conditions might replace (A) by:

(A’) Assertibility-conditions and dcniability-conditions together determine truth-conditions.

Section 4 is technical; it investigates a rule of proof in modal logic corresponding to (A’). Section 5 concludes that (A’) is not in a much stronger position philosophically than (A).

1. Consider a language L of whose (indicative) sentences (A) is true. L will be assumed to contain the usual truth-functional connectivcs, satisfying classical propositional logic; since some anti-realists reject classical logic, the scope of this assumption can itself be regarded as under test. It will also be assumed that a concept of assertibility is rcprcsented in L by an operator (not a predicate) L with the sense ‘it is assertiblc that’. For present purposes, asscrtibility-conditions and truth-conditions can be thought of as classes of possible situations in

ASSERTION AND DENIAL IN hIOD.41. LOGIC 301

which a sentence may be uttered; such situations will be called contexts.’ The usc of an assertibility operator in the object language will be defended at the end of Section I.

The gist of (A) is that two scntenccs with the same asscrtibility- conditions have the same truth-conditions. A quick argument from (A) to (B) then goes as follows: it is asscrtible that A if and only if it is asscrtiblo that it is assertiblc that 4: therefore A and LA have the same assertibility-conditions; therefore, by (A), they have the same truth-conditions: since the truth-condition of LA is the assertibility- condition of A, the asscrtibility-condition of A is its truth-condition, which gives (B). More formally. (A) corresponds to the L-cancellation pru~err~., that for any sentences A and LJ of L. if LA ++ LB is true in all contexts then so is 4 H B. (B) corresponds to the L-colhpse pwyerz~~: that for any sentence A of L, A t, 15.4 is true in all contexts. The assumption used above in moving from the former to the latter is a kind of S4 biconditional, that for any sentence 4 of L, LA ++ Ll,A is true in all contexts; it has considerable intuitive plausibility, and by itself does not require asscrtibility to be either neccs- sary or sufficient for truth. The idea is that warrants for assertion must be recognizable to speakers, for otherwise such warrants could not play the role required of them in governing the speakers’ USC of the language; but then speakers can reflect that a warrant to assert that it is assertible (in their context) that 4 gives them a warrant to assert that ‘4: conversely. they can reflect that a warrant to assert that A gives them a warrant to assert that it is assertibie (in their context) that 4. Nevertheless, this equivalence can itself be regarded as under test. ?;otc that the argument did not appeal to any laws of truth- functional logic for L al alL3

Different assumptions about L can be used to construct another argument from (A) to (B). It may seem reasonable to assume that the assertibility of a conjunction entails the asscrtibility of its conjuncts and that if a sentence is assertible, it is not also assertiblc (in the same context) that the sentence is trot asscrtiblc, nor is the negation of the sentence assertiblc. Thus for any sentences A and LI of L. L(A & B) -+ (LA & I&), LA + -L -- LA and LA -+ -L - A (that is, LA + ,WL4 and LA + MA) are true in all contexts. In


contrast to the previous case, the S4 principle LA --f LLA is not assumed. By the first assumption we have f&4 & -LA) --f (LA & L - LA) and L(A 8~ -A) + (LA & L - A), whose con- sequcnts the second and third assumptions rcspectivcly make inconsistent. Thus I,(,4 8~ -LA) and L(,4 & -A) are true in no contexts, and so L(A & - LA) ++ L(A & -,4) in all. By L-cancellation, (A & -LA) c, (A & -A) and so A -+ LA arc true in all contexts. As a special CBSC we have -A --) L - A; combining this with the third assumption gives LA -+ A. Thus A c1 LA is true in all contexts.4 (A) again yields (B).

Although the arguments above require L and the truth-functional operators to be present in L, this is less of a restriction than it might appear. All t.hat is really required is that these operators can coherently be added to any language, for then - given the extra assumptions -- it follows that if (A) is true of all languages, so is (B). Of course, it has not been ruled out that the sentences of a restricted language (one not containing L, for instance) should obey (A) but not (B), even if the extra assumptions hold in all languages: but the ambitions of the litcraturc on realism and anti-realism call for rcad- ings of (A) and (B) on which they aspire to characterize all languages, rather than merely distinguishing some from others. For instance, the claim ‘Meaning cannot transcend USC’ is not intended to state a contingent fact about English.

This is not the place to decide whether one should USC the conditional ‘If (A) then (B)’ for a Modus Ponrns or for a Modus To1ien.s (the latter looks more promising); that may partly depend on which concept of assertibility is at issue. The present point is just that it is much harder than one might imagine to combine (A) with the negation of(B).

2. The discussion above can be set in the context of normal modal logics.’ The syntax is standard: sentential variables p, 9, r, . . . , the usual truth-functions, including the contradiction I and the tautology T, L, and M abbreviating -.-. L - . For present purposes, a normal modal logic can be defined simply as a set of formulae that contains all truth-functional tautologies and the formula L(p + y) + (Lp -+ Lq) and is closed under the rules of Modus Ponens,

ASSERTION AND DENIAL 1K MODAL LOGIC 303

Ncccssitation and Uniform Substitution; thus the logic is idcntilicd with the set of its theorems.” This formal language L’ should not be confused with the language L of Section I. L may contain all sorts of compositional devices not definable from the resources of L’ (L and the truth-functions). Rather L’ can be thought of as a meta-language for L. with the atomic sentential variables of the former ranging over all scntcnces of the latter. On one such interpretation, a formula of L’ will be valid if and only if the result of uniform substitution of sentences of L for its variables always yields a sentence of L true in all contexts (note that the truth-functional operators and L are thcrcfore thought of as in common between L and L’). Thus the rule of Uni- form Substitution takes vahd formulae of L’ to valid formulae of L’. Upper-case italic letters are used as meta-linguistic variables ranging over sentences of L’. The standard notation for rules of proof is used, on which Xcccssitation becomes: A/LA.

That the valid sentences of L’ arc closed under Necessitation and that L(p + q) + (LI, + Lq) is a valid sentence. on this inter- prctation, are substantive assumptions. Their plausibility will, of course, depend on which concept of assertibility is at issue. There might indeed be a decisive objective to them if they amounted to the assumption that the set of sentences of L assertible in 0 given confext is closed under logical consequence, for then membership of such a set would characteristically not bc a dccidablc property and it may be essential to the philosophical motivation for (A) that assertibility in a given context is a decidable property (othenvise it might lack the right connection with the u.re of language). However, all that is really being assumed is that the set of valid sentences of L’ is closed under logical consequence, and membership of this set may well be a decidable property even if membership in the set of assertiblc scntcnccs of L is not. It will turn out that the most relevant modal Iogics for present purposes arc indeed decidable: trivially, where (B) is true of the sentences of L, so that p c) Lp is valid in L’ (modal collapse), validity in L’ reduces to truth-functional validity and is therefore decidable.’ Thus the USC of normal systems may not involve too damaging an idealization.

Just as p tf 1-p is valid on the above interpretation if(B) is true of the scntenccs of L, so the following rule of proof L c* C holds in L’


if (A) is true of the sentences of L: if LA t+ LB is valid, so is A tf B. For if LA w LB is valid, then LA’ H LB’ is true in all contexts for any sentences A’ and B’ of L resulting from n and B respectively by uniform substitution of sentences of L for their variables, so by (A) A’ t* R’ is always true in all contexts, so A c-f B is valid (note that the convcrsc argument from the validity-preservingncss of L t, C to the satisfaction of (A) by sentcnccs of L is not valid without extra assumptions, since a particular sentence LA’ t, LB’ of L might bc true in all contexts for reasons dependent on the content of A’ and H’, so that no scntcncc LA H LB of L’ of which it was a substitution-instance would be valid). Obviously, L tt C no more says that any sentence (LA tf L/3) --f (A +-+ B) is valid than the rule of Necessitation says that any scntcncc A --f LA is valid. Questions about the rule indepcndcnt of the original interpretation, but with consequences for it, now suggest themselves. What is the effect of L t) C on normal modal logics?

WC may begin by noting that I, ++ C has a number of equivalent forms:

L + c: LA + LB/A + B L t, C: LA ++ LB/A t, B

M -+ C: MA + MB/A -+ B M++C:MActMBjA++B

L&l -+ C: LA + LB, MA -+ MB/A -+ BR

It is obvious that L, M -+ C is derivable from L -+ C and from M + C, that L ++ C is derivable from I, + C and M H C from M -+ C, that L cf C is interderivable with M +-+ C and that I, -+ C is intcrdcrivable with M + C. L + C follows from L ++ C because in any normal system LA +-+ L(A & B) follows from LA + LB. To complete the chain of equivalences it is sufficient to show that L + C is derivable from L, M --f C. So suppose that LA + LB E S (a normal modal system). Evidently LA + L(A + B) E S. Now - M(A + B) + (LA & L - B) is in any normal system, so - M(A + B) + (LB&L- B)E!$so-M(A--+B)-, -MAES,SOMA-+ M(A + B) E S. Thus, given L, M + C, A + (A + B) E S, so A + B E S. Thus all five of the above rules are equivalent. To revert to the interpretation above: M H C says in effect that

i\SSEK’l’lON AND DENIPIL IN MODAL LOGIC 305

clerziuhilit!:-conditions determine truth-conditions; L -+ C might be compared with the intuitionist semantics for the conditional, which allows one to assert a conditional when one has seen the assertibility of its antecedent to guarantee the assertibility of its consequent.’

In Section 1, it was in effect observed (N. 4) that the addition of L ++ C to any normal system containing Lp --t 11 and L(p & 4) + (1,~ & Ly) causes modal collapse; since any normal system contains the latter formula. the addition of L +-+ C to any normal system that includes the system T (KT) causes modal collapse. Similarly. it was observed that the addition of L tt C to any normal system containing I217 tt LLp has the same effect. Sot all normal systems collapse under L tf C. however. For instance K. the smallest normal system. is closed under L ++ C. For suppose that A tf H $ K. Since K is complete with respect to the class of all models. there is a world x in some model at which A t, B dots not hold. The model can be extended by the addition of a world J’. accessible from no world and from which only .Y is accessible. so that LA and LB hold at J‘ if and only if A and B respectively hold at s. Thus I.4 tf LB does not hold at J, and so is not in K. Contraposing, if LA +-+ LB E K then A * H E K.“’ This argument also shows the closure under L +-+ C of the system D. which results from the addition of MT to K, for D is complete with respect to the class of serial models (those in which from every world at least one world is accessible) and the cxtended model will be in this class if the original one is. A quirkier system closed under L c* C results from the addition of p cf LLp to K: for if LA t* LB is a theorem. so is L(LA cf LB). therefore LLA H L1.H and therefore 4 * B. None of these systems contains p +-+ Lp (in the last case, consider models with two worlds. each acccssiblc from the other but not from itself).

Two somewhat bcttcr-known rules of proof can bc derived from I- ++ C: they are M.4,:4 and L,,f/A. For in any normal system S, if MA E S then L - A -+ LI E S. so if S is closed under L H C and therefore under L --f C then -A + i E S, and therefore A E S. Similarly. if LA E S then LT + LA, giving T + A E S and therefore A E S by L + C. Both of these rules are weaker than L -+ C. This is obvious for LA/A. under which all systems that include T arc closed,


whcrcas I, c+ C has been seen to cause modal collapse in them. The rule of proof MA/A must of course be distinguished from the schema MA -+ A; the former says that if MA is a thesis then so is A, not that MA entails A. The addition of MA/A also causes any nominal system that includes T to collapse, since it contains the thesis M(P + Lp), but the consistent system which results from the addition of Ll to K is vacuously closed under this rule, not having any theorem of the form MA, whereas the addition of I, ++ C would make it inconsistent, since LT ++ LI is a theorem.”

Finally, a gcncral connection can be made between L H C and the standard semantics, where a model (W, R, V) consists of a set of worlds W, an accessibility relation R and a valuation V, a formula A holds at a world iff V(A, x) = T and a characteristic model for a system is one such that the formulae holding at all worlds in the model are precisely the thcorcms of the system:

FACT. A normal modal Iogic S is closed under 1, H C iff it has a characteristic model (W, R, V) in which Vx E W 3y E W V’r E W O’RZOZ = x)

Proof: Suppose that S has such a characteristic model (W, R, V) and that LA tf L,B E S. Let x E W, and choose y E W such that Vz E W(yRz o z = x). Thus V(LA, y) = V(A, x) and V(LB, y) = l’(B, x), so V(A - B, x) = V(LA ti LB, JJ) = T. Thus A ++ B holds at each world in W, so A cf B E S.

Conversely, suppose that S is closed under L t, C. Let (W, R, V) be the canonical model (whose worlds are maximal S-consistent sets of formulae) for S, which is always charactcrislic.” Let x E W. Con- sider the set X = {MA: A E x) u {LB: B E x). Suppose that X is S-inconsistent. Thus there arc A,, . . . , A,,, H,, . . . , B, E x such that (LB, & . . . & LB,) + -(MA, & . . . & MA,,) E S; since S is normalL(U,&...&R,)~L(--A, v ___ v -A,)ES.H~L-+C, andsobyI,tiC,(B,&...&B,)-+-A, v . . . v -A,,)eS, contradicting the S-consistency of x. Thus X is S-consistent. Let )’ E W be a maximal S-consistent extension of X. By the definition of R for the canonical model, yRx because {MA: A E -Y} c I’; yRz requires z = x because if B E x, LB E y so B E z, so x G z, so z = x.

/\SSER-l’ION AND DENIAL IN MODAL LOGIC 307

3. The following principle is at lcast plausible:

((3 It is assertible that it is assertible that A if and only if it is assertiblc that A.

Howwer. (C) was seen in Section 1 to yield the equivalence of (A) with (13). There is thus a problem for those who feel that (A) has the right end of the stick and (D) the wrong one. An obvious move for them to make, and one that has been proposed in the literature, is to treat n’c~ziuf on a par with assertion, and to hold that truth-conditions arc dctcrmined only by assertibility-conditions and deniability- conditions together, as in (.4’).” This feature is already built into Quint‘s model of radical translation. on which the natiw speakers’ assent and dissent bchaviour have equal status.” In less linguistic terms: a version of (A’) would say that the content of a thought is determined by the conditions for its falsification (disconfirmation. rejection) as well as those for its verification (confirmation, accep- tancc). The truth-conditions of ‘It is assertible that A’ could then be distinguished from those of A, cvcn on the assumption (C) that they have the same asscrtibility-conditions, for when there is no evidence either for or against A. ‘It is asscrtible that A’ is deniable while A is not. Of course. this shows only that one quick argument from [A) and (C) to (B) cannot immediately be transformed into an argument from (A’) and (C) to (R); it dots not rule out the possibility of a different argument from (A’) and (C) to (B), for (A’) says that identity in assertibility- and deniability-conditions is .wfi~bzt for identity in truth-conditions, not that it is necessury. The acceptance of (A’) and (C).does turn out to be consistent with the rejection of (U). but more work will be nccdcd to show this.

The denial of a sentence is, of course, equivalent to the assertion of its negation. It has been objected that to put denial on a par with assertion in determining truth-conditions is to treat ncgatice sentences as semantically unstructured, and thereby to flout an obvious con- straint of compositionality: for the meaning of - /I should be exhibited as a function of the meaning of A.” This point might be well taken if the denial of a sentence could bc expressed onI4 as the assertion of its negation. However, Quine’s model of radical

308 TIMOTIIY WILLIAMSON

translation suggests otherwise: assent and dissent are conceived sym- metrically (saying ‘Yes’ or ‘No’, shaking or nodding one’s head). For the purposes of (B), assertion and denial should be treated as equally basic speech acts, with the cquivalcncc bctwccn the denial of a sentence and the assertion of its negation defining negation rather than denial.

4. Just as (A) corresponds to the rule of proof L ++ C on a certain interpretation of modal logic, so (A’) corresponds to the rule:

L. M t, C: LA +-+ LB, MA +-+ MB/A +-+ B

For ‘A is deniable if and only if B is deniable’ is equivalent to *.- A is assertiblc if and only if - B is asscrtiblc’, which is formalizable as I, ~~- A t-+ L -~ B. which is equivalent to MA t-+ MB. Now L, M t--) C would fit into the conspicuously empty corner in the table of rules in Section 2; it stands to L, M + C just as L tt C and M t) C stand to L - C and M + C, and to L tt C and M - C just as I,, M 4 C stands to L + C and M + C. Since those other live rules are all equivalent to each other, it is a natural guess that L, M + C is equivalent to them too (it is obviously a consequence of L 1--f C), in which case there would be an argument from (A’) and (C) to (B) after all, at least on the assumption that the logic of assertibility is a normal one. Surprisingly, this is not the case. Some modal logics which contain the principle Lp H LLp (corresponding to (C)) arc closed under L, M t) C but not under L t* C, and therefore do not contain p c--f Lp. Thus (A’) and (C) do not entail (B).

To establish this result, it suffices to consider the following frame, where arrows as usual represent the accessibility relation:

The effect of the L and M operators on propositions (qua sets of worlds) can bc shown in a table:

ASSF.RTION AI\ID DENIAL IN MODAL LOGIC 309

Note that Lp 1--t LLp is valid on the frame (since accessibility is tran- sitive) but that p t) Lp is not. Now L. M H C preserves validity on the frame. For suppose that L,4 c* LB and MA ++ MB arc valid on it; thus in any model LA holds at the same worlds at LB and MA as MB, in which case A holds at the same worlds as B, by inspection of the table (no two rows of which have the same entries in both the second and third columns); thus A ++ B is valid on the frame. In contrast. L c, C dots not preserve validity on the frame: for instance. L(p & -1,~) t* L-t is valid while (1~ & - Lp) tt I is not.

The reader may have noticed that the formula Z,p + p (indeed, the system S4) is valid on the above frame. This formula is not logically valid on the intended interpretation of L as the ordinary, dcfcasiblc notion of assertibility. However, it can be shown to be a consequence of the assumptions made so far about asscrtibility together with the uncontroversial assumption that a blatant contradiction is not assertihle (- LI, or equivalently MT). That is, any normal system S closed under L, M t) C which contains I-p ++ LZ,p and MT also contains Lp --t p. For one can shown in order that S contains the following formulae:

(1) up -+ L(p & Lp)

(2) Z>(I, & Zap) + z,zJl

(3) LO - UP & -0)

(4) :W( p & Lp) -+ MZ’

(5) Lp + L( p & Lp)

(6) LP + M(p & Lp)

(7) MLp + MM(p & Z-p)

(8) MLp + M(p & Lp)

LLp + Lp E s

1, 2

Lp -+ LLp E s

5. MT E S

6

7. z-p + LLp E s

310 TIMOTHY WILI.IAMSON

(9 MLp t, M(p & Lp) 4, 8

(10) LP +-+ (P & 1-P) 3, 9, L, M t-) C

(11) LP -+ P IO

Thus in particular K + Lp tf LLp + MT + L, M t* C is simply s4 + L, M cf c.

Within the context of normal systems, the rccoursc to denial in determining truth-conditions fails to have its intcndcd effect; (A’) and (C) cannot bc satisfied together in a normal system without violating the logic of dcfcasible assertibility, either by not making a tautology assertiblc or by making truth a consequence of assertibility. However, there are other concepts of asscrtibility on which Lp + p is accept- able as a theorem; for instance, one might read L as ‘Thcrc is conclusive evidence that’, or (in the case of mathematical language) as ‘It is informally provable that’. Another possible reading would be ‘It is implicitly known that’, where implicit knowledge had the requisite closure properties. Thus the possibility remains open for someone to adopt a concept of assertibility obedient to S4, and then endorse (A’), and thus I,, M t* C. In order to investigate the prospects for this move, it is necessary to find out what system S4 + L, M tf C is.

It can be shown that S4 + L, M cf C is a system called K4 by Sobocinski, which results from the addition of LMp ++ MLp to S4.4, which in turn results from the addition of p -+ (MLp + Lp) to S4 (this K4 should not bc confused with the one better known under the same name, which results simply from the addition of Lp + LLp to K).‘”

The system has a rather simple canonical model. For if R is the accessibility relation on this model, it follows by a standard result that, since 1~ --f (MLp + Lp) is a theorem, R obeys the condition:

VxVyVz((xRy and .xRz and x # y) + zRy)

Similarly, since p + (LMp -+ Lp) (and therefore Mp + (p v MLp)) is a theorem, R obeys:

tlxV~>((xR~ and x # JJ) =$

* 3z(xRz and Vw(zRw + HI = y)))”

ASSERTION AND DENIAL. IN MODAl. I.OGIC 311

Together with the reflexivity of R (from Lp -+ p). these conditions entail:

V.uVyVz((.uRy and ~Rz) =z. (.y = 1’ or r = z))

It follows that every generated subframc of the canonical model is isomorphic either to the two world frame illustrated carlicr or to a single self-accessible world. Since every such subframc is isomorphic either to the illustrated frame or to a gcncratcd subframe of it. and every theorem of the system is valid on that frame, it follows that the formulae valid on the illustrated frame arc prcciscly the theorems of the system. This, of course, provides a straightforward decision procedure for S4 + L1 !I4 c+ C: in fulfilmcnt of an earlier promise.

The presence of the formula p H (1-p v (Mp & - MLp)) in S4 + L, ;M H C is of particular interest. For, on the earlier interpretation, it means that truth. although not equivalent to assertibility, is at least definable in terms of asscrtibility and dcniability: it says. in cffcct. that a formula is true if and only if cithcr it is asscrtiblc or its asscrtibility but not the formula itself is deniable. It remains to bc seen whether there are any concepts of assertion and denial for which this rather bizarre equivalence is plausible. Clearly. (A’) is a consc- qucncc of any such definition of truth in terms of asscrtibility and dcniability. More formally: any normal system is closed under L, A4 t+ C if it has a theorem of the form p ++ A, where .4 is a modal function of Lp and Mp (just as it will be closed under L c, C and M tf C if A is a modal function cithcr of Lp or of Alp). On the other hand, a normal system can bc closed under f.. M ++ C even if it has no such theorem. The systems K and I> arc examples. For if they had such a theorem, so would S5, which would therefore bc closed under LI :%I tf C, which it is not - indeed S5 collapses under L, M q--t C1 since S4 + L, M t* C. and therefore S5 + I., IM ++ C, contains the thcorcm p + (LMp + Lp) and S5 contains p + LMp. Yet K and 1) arc closed under L. M ++ C. for they arc closed under 1, ++ C.

5. Is there any concept of asscrtibility for which S4 + L, A4 H C is a plausible system? One might attempt to model some sort of speculative

312 TIMOTIIY WII.I.IAMSON

interpretation on the characteristic two world frame. Suppose, for instance, that we could make sense of the phrases ‘the world of reality’ and ‘the world of appearance’ in such a way that scntenccs of the language L not containing L could be said to hold or to ladi1 to hold at these worlds. Then one might stipulate that LA holds at the world of appearance if and only if A hoids at the world of appearance, but that LA holds at the world of reality if and only if A holds at both the world of reality and the world of appearance, with standard clauses for the logical constants. One might read L as ‘it is evident that’, understood as a factivc. Perhaps there is some limited domain in which this intcrprctation would be useful one, but taken generally it seems both fanciful and crude.

Does any concept we actually employ validate S4 + L, M ++ C? If L is read as something like ‘it is informally provable that’, at least S4 is arguably validated: but many additional theorems of S4 + L, M t) C arc not. Consider the theorem L( -p + L(p 4 Lp)), for instance: on this reading, it says that it is informally provable that if I) is false then it is informally provable that p is informally provable if true; but no such informal proof seems to be generally forthcoming. Again, the theorem LMp v L - Lp is mad as making the not generally plausible assertion that cithcr it is informally provable that p is not informally disprovable or it is informally provable that p is not informally provable.

No concept of assertibility that I know of validates S4 + I,, A4 ++ C. The retreat from (A) to (A’) therefore does not by itself meet the objections to (A) in a philosophically satisfying manner. The discussion in this paper has of course been based on a number of simplifying assumptions, which may need to be called into question. Of these, the appeal to the concept of truth-conditions is perhaps the lcast significant, for although thcrc obviously are conceptions of truth which many tcmptcd by (A) or by (A’) would regard as suspect. they are not required by the preceding discussion. For just as the identity of A and B in assertibility-conditions amounted to the validity 01 LA ++ LB, so their identity in truth-conditions amounted to the validity of A tf B, where it was not essential that the notion of validity should itself be cxplaincd in terms of some controversial notion of truth. In effect, the operative notion of truth was that of a

ASSERTION AND DENIAL IN MODAL LOGIC 313

redundant sentential operator, representing the identity truth- function. More deserving of controversy are the closure assumptions about assertibility (the use of normal logics). the treatment of asscrtibility as though it did not come in degrees. even the appeal to classical logic. A claim often made at this point is that the truth- condition of a sentence is determined by its compositional structure as weli as by its asscrtibility- and deniability-conditions; that may be so: but as a bare claim it hardly comes to grips with the problem for it dots not explain what determines the effect of a compositional device other than the conditions for the assertion and denial of the sentences which serve as its arguments and values. In any case, if a more truc- to-life model does imply a relation between assertibility and truth essentially different from the results above, it should at least be illumi- nating to discover what the crucially misleading simplifications were. and why.”

APPENDIX

It will be proved that S4 + L, !+I ++ C is Sobocinski’s K4. In order to show that S4 + L, M t) C includes Sobociriski’s system. WC can start by showing the former to contain p + (:Mlq -+ Lp):

(1) A4 - p +-+ M(-p v (34 - p & MLp)) s4

(2) (-p v (M - p & MLp)) + A4 - p T

(3) L(-p v (M - p & MLp)) -+ LA4 - p 2

(4) q-p v (A4 - p & MLp)) -+ LLM - p 3, s4

(5) ((-p v (M - p & MLp)) &

& LiM - p) + -p

(6) (L(-p v (M - p & MLp)) &

(7)

(8)

& LLM - p) + L - p

L( -p v (M - p & ML/I)) + I, - p

L-pt*L-p v (M-p&MLp))

5

4, 6


(9) -P-(-P ” CM - P&MLPN

(10) p -+ (MZq -+ I-p)

We can now show S4 + L, M ++ also LO contain p

(1)

(2)

(3)

(4)

(9

(6)

(7)

(8)

(9)

(10)

(11)

(12)

-UP & Ml-p & MP))

-JFP&WP&M -PII

L(P & M(-P & MPH

- L(-P & M(P & M - PI)

-p -+ (ML -p + L - p)

t--p 8~ MP) -+ LMP

(-p & Mp) 4 LLMp

(-p & M(p & M - p)) + LLMp

(MCI, & M - PI & LLMP)

+ M(p & LMp & M - p)

(M(P ‘3~ M - P) ‘3~ LLMP)

-+ MP & M(-P ‘3~ MP>>

(-P&WP&M - ~1)

+ WP & Ml-p & MP>>

MC--P 8~ M(P & M -- ~1)

+ MM(P & MC-P & M/I)>

MC -P & f+f(~ & M - ~1)

+ M(P & MC-P & MP))

1, 8, L, M t, C

9

(LMp + Lp):

T

T

1, 2

Previous result

4

5, 54

6

8

7, 9

10

11, s4

(131

(14)

(15)

(16)

07)

(18)

ASSERTION AND DENIAL IN MODAL LOGIC: 315

M(p & M( -p & Mp))

-+ M(-p & M(p & M - p)) As 12

M(p & M( -p & Mp))

- M-p & M(p & M - p)) 12, 13

(P & *+f( - p 6% Mp))

+-+ (-p & M(p 6% M - p) 3, 14, L. M ++ C

-(p&M-p& Mp)) 1.5

P + Q/VP -+ P) 16

p --+ (LMp + Lp) 17

It is now straightforward to show that S4 + L, M t) C contains the remaining axiom of Sobociriski’s system, L:Wp ++ MLp:

(1)

(2) (3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(l-2)

(13)

p + (ML/l + Lp)

p + (MLp + LMp)

-p + (:ML - p --t I,

-p + (Mp + L:Mp)

--.p -+ (MLP + ZMp)

MLp 4 LMp

p + (LA4p ----t Lp)

p + ( zxp + :MZ2p)

-p ---) (LM - p --+ L

- p --* (A4p + MLp)

-p --$ (LMp + MLP)

LMP + MLp

LMp +> :WLp

PI

Previous result

1, T

Previous result

3

4. T

2, 5

Previous result

7. T

Previous result

9

10, T

8, 11

6. 12

Thus S4 + L, M ++ C includes Sobocifiski’s system. In order to prove the converse, it suffkes to show that his system, which


includes S4, is closed under L, M H C. We first show it to contain p tf (Lp v (Mp & - MLP)):

(1) P - (MLP -+ 0) Axiom

(2) p + (Lp v - MLp) I

(3) P + MP T

(4) P --f (r,l v (MP & - MLP)) 2, 3

(5) -p --t (ML -p -+ L - p) I, substitution

(6) (MP& -LMP) + P 5

(7) LMp 4 MLp Axiom

c-9 (MI, & - MLp) + p 6, 7

(9) LP + P T

(10) (LP v (MP & - M~>P)) + P 8, 9

(11) P * (LP v (Mp g - Ml,pN 4, IO

Soboci6ski’s system can now be shown to bc closed under L, M tf C, as follows:

(1) LA ++ LB Assumed theorem

(2) MA * MB Assumed theorem

(3) (LA v (MA & - MLA)) +-*

- (LB v (MB & -MI-B)) 1, 2

(4) A H (LA v (MA & -MLA)) Previous result

(5) B w (LB v (MB & - MLB)) Previous result

(6) A-B 3, 4, 5

This completes the proof that S4 + L, M H C (and therefore K + MT + Lp H LLp + L, M tf C) is equivalent to Sobocitiski’s system.

ASSERTIOY AND DENI.AL IN MDDAL LOGIC 317

Two alternative axiomatizations of this system may be noted. First, it is equivalent to K + p ++ (/-/I v (Ml> & -Ml-p)). For the latter formula has been seen to bc contained in it. and in turn to ensure closure under L, A+ H C. Thus it remains only to show that K + y ++ (Ly v (/MI, & -&IQ)) contains the characteristic S4 axioms. I-/, -+ p is obvious. For Lp 4 LLp, substitute p & -.. 1-p for p; the result simplifies to M(p & --- LIT) ---f (p & - I+), which yields L,o --, L(p -+ Lp). Second. the system is cquivalcnt to T + p -+ (MLy + 1,~) + p -+ (LMp + Lp); this follows without difficulty from what has been established so far.

NOTES

’ I:or a rccenl discussion bearing on this point SW Peacocke 181. .‘ In this scnsc. the trutbcondition of a sentence is the set of possible situations in

which it can be truly uttered; it is not the set of situations with respect to which thr sentence. as uttered in a fixed context, is true. The sophistications of t\%o-dimensional modal logic will bc avoided. The assertibility-condition of a sentence is. corrc- spondingly, something !ike the set of possible situations in which its assertive uttcrancc is warranted.

’ Arguments of this kind are propuscd in Brandom [?I, following Dummett [3] at pp. 350 -45 1. ’ The second and third assumptions are of course conscqurnccs of the validity of L.4

+ .-l. Given the first assumption, the argument therefore applies to any truth-entailing concept of asscrtibility, such as provability in mathematics. Its application is clearly more ycneral than that. however.

’ Hughes and Cresswcll [5] provide a suitable background for the technical discussion. ‘ The rewards of a finer-gained approach are illustrated in Humbcrstonc [6]. which

distingmshcs between the dcrivabllity of a rule in a system and the mere closure of the system under the rule Interesting questions could he raised in these terms about the rules discussed in this paper. hut they go hcyond its scope.

For some relevant results see I.emmon and Scott 171: pp. X1-85. ’ ‘c’ for ‘cancellation’; in other respcts the choice of labels should he self-explanatory.

liotc that L. :M + C differs from LM --t C which. by analogy, would bc the rule INA ---t L.!f WA - B. ” Note that L e, c‘ and L + C are the converses of the standard rules called RE and

R:W by Lemmon and .%wtt (ihid. p. I I ), “’ Ibid.. p. 47. ” I&f., p. 86: further wsults concerning the rule LA.‘,1 are at pp. 46 and 80, and it is discussed in Ilumherstonc [6j. ” The proof prcsuppuscs the mntcrial at pp. 22-X of Hughes and Cresswell [5], ” Dummett 141. p, 112. cp. Price [9]. ” l:or a discussion in the context of language learning see Quint [IO]. Part II. ” Appiah II]. pp. 135:‘6. I6 Sobocitiski [I I]. The proof is in the Appendix.


” Lemmon and Scott [7]. pp. 6668. lx The technical parts of this paper are largely the outcome of corrcspondcnce with Lloyd Humbcrstone. Itc did most of the work in Section 2. pointed out the equivalence of S4 + p c.+ (1,~ v (Mp & - MLp)) and T + p - (LMp + Lp) + p --t (MLp + l,p) and the facts about its canonical model, and made other helpful comments. The referee also made useful suggestions.

BlBI.IOGRAPIIY

[l] hppiah, Anthony: 1986, Fir Truth in Semanrics. Oxford, Blackwell. [2] Brandom, Robert: 1976, ‘Truth and Assertibility’, 7%~ Journrrl of Philosophy 83,

13749. [3] Dummctt, Michael: 1973, Fregc- Phihxwphy of Idnguage, London, Duckworth. 141 Dummett, Michael: 1976, ‘What Is a Theory of Meaning‘! (II)‘, in G. Evans and J.

McDowell (cds.), Truth und Meaning, Oxford, Oxford University Press, pp. 67.- 137.

[S] JIughes, G. E. and Cresswell, M. J.: 1984, A (bnr~cmion to 1%40&l Logic, London, Methuen.

[6] Humberstone, I. L.: ‘A More Discriminating Approach to Modal Logic’, unpublished.

[7] Lemmon. E. J. and Scott, D. S.: 1977, The “Lemmon No/es”: An Infruduction IO Modui Logic-, cd. K. Scgerbcrg, Oxford, Blackwell.

[8] Peacocke, Christopher: 1986, ‘What Determines Truth Conditions? in P. Pettit and J. McDowell (cds.), Subjerr, Thoughl. rind Conrrxf, Oxford, Clarendou Press, pp. 181-208.

[9] Price, H.: 1983, ‘Sense, hsscrtion, Dummett and Denial’, Mind 92, 161-73. [IO] Quine, W. V. 0.: 1973, The Roofs of Refwence, La Salle, Open Court. (1 I] Sobocitiski, B.: 1964, ‘Family X of the Non-Lewis Modal Systems’, Norre IJame

Journal of Formul Logic 5 313 -318.

Dcpurtment of’ Philosophy, Trinity College, University of Dublin Dublin 2, Ireland.

Assertion, denial and some cancellation rules in modal logic

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