Modal Semantics without Possible Worlds

Modal Semantics without Possible WorldsAuthor(s): John T. KearnsSource: The Journal of Symbolic Logic, Vol. 46, No. 1 (Mar., 1981), pp. 77-86Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273259 .

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THE JOURNAL OF SYMBOLIC LoGic Volume 46, Number 1, March 1981

MODAL SEMANTICS WITHOUT POSSIBLE WORLDS

JOHN T. KEARNS

?1. Four values. In this paper I will develop a semantic account for modal logic by considering only the values of sentences (and formulas). This account makes no use of possible worlds. To develop such an account, we must recognize four values. These are obtained by subdividing (plain) truth into necessary truth (T) and contingent truth (t); and by subdividing falsity into contingent falsity (f) and necessary falsity (impossibility: F). The semantic account results from reflecting on these concepts and on the meanings of the logical operators.

To begin with, we shall consider the propositional language Lo. The language Lo has (1) infinitely many atomic sentences, (2) the two truth-functional connectives

%, V, and the modal operator Cl. (Square brackets are used for punctuation.) The other logical expressions are defined as follows:

Dl [A&B] =(def)-[ -A V -B], D2 [A v B] (def)[-A V B], D3 0 A =(def)FIp-A. I shall use matrices to give partial characterizations of the significance of logical

expressions in Lo. For negation, this matrix is wholly adequate:

A ~A T F t f f t F T

Upon reflection, it should be clear that this matrix is the obviously correct matrix for negation. The matrix for disjunction is completely natural, but it does not completely characterize the significance of the wedge:

A B [A V B] T T T T t T T f T T F T

t T T t t T,t t f Tt t F t

Received February 13, 1979.

77 ? 1981, Association for Symbolic Logic

0022-4812/8 1/4601-0010/$03.50



78 JOHN T. KEARNS

A B [AVB] f T T f t T,t f f f f F f

F T T F t t F f f F F F

There are three cases where the values of the components of a disjunction do not (completely) determine the value of the disjunction. The wedge is not functional in the four-valued semantics. If A, B are contingently true, then [A V B] may be contingently true. Similarly, if one of A, B is contingently true and the other is contingently false, [A V B] may be contingently true. But even if A is contingently true, [A V - A] must be necessary. If A, B are contingently true, [A v [ ~ A V B]] is necessary.

When we try to construct a matrix for C], reflecting on the concept of necessity yields inconclusive results. There are different concepts of necessity, and these have different properties. To begin with, let us consider a "minimal" concept. If A has value T, then GA is true. But we cannot say which kind of truth it has. If A does not have value T, then FA is false-but there are two varieties of falsity. The matrix for the box is:

A GA

T T,t t fF f fF F fF

The defined symbols have these matrices:

A B [A&B] [ADB] A OA

T T T T T T,t T t t t t T,t T f f f f T,t T F F F F f, F

t T t T t t t T,t t f f, F f t F F f



MODAL SEMANTICS WITHOUT POSSIBLE WORLDS 79

A B [A & B] [A v B] f T f T f t f, F T. t

f f f,F T,t f F F t F T F T F t F T F f F T F F F T

The matrices give a partial characterization of the logical operators, but they must be supplemented. In supplementing the matrices, I will define valuations that I call T-valuations, because, as it turns out, these valuations exactly "fit" the system T. (I am using the terminology of [1].)

As a first stab at defining 'T-valuation', we might say that any sentence A which, according to the matrices, cannot come out false should be assigned the value T. So every classical tautology would get the value T (and every classical contradiction would receive value F). This proposal is inadequate because it would not assign T to enough sentences. The matrices prevent a sentence [A V A] from being false; so such a sentence would receive the value T. But the matrices allow a sentence [A V - A] to have value t. So the matrices allow the sentence LI [A V - A] to be false, when this should not be allowed.

A more satisfactory definition follows. A Oth-level T-valuation of Lo is a function which assigns one of T, t, f, F to each

sentence of Lo in a manner consistent with the matrices. Let *' be an mth-level T-valuation of Lo. f' is an m + 1st level T-valuation of Lo

iff V assigns T to every sentence A which is true (which has value T or t) for every mth-level T-valuation.

A valuation Y is a T-valuation of Lo iff V is an mth-level T-valuation of Lo for every m ? 0.

The system P is a natural-deduction system of propositional logic employing tree proofs. The rules for constructing these proofs are below. An elementary rule is one for which each premise is a sentence on the line of the inference figure which exemplifies the rule. A nonelementary rule is a rule for which at least one premise is a subproof whose conclusion is on the line of the inference figure exemplifying the rule. A nonelementary rule cancels (or discharges) occurrences of one of the hypotheses of a subproof-premise. In illustrating the rules of P, the hypotheses that are cancelled are enclosed in braces.

V Introduction (VI) V Elimination (V E) A B {A} {B}

[A V B] [A V B] [A V B] C C C

-.Elimination ( ~ E) { A} Contradiction Elimination (CE)

A A -A A B



80 JOHN T. KEARNS

The system P is used to prove inference sequences. These have zero or more premises and a single conclusion. An inference sequence is written: Al, ..., An / B. (The slant line separates premises from the conclusion.)

The rule of inference for P is this: A tree proof whose conclusion is B and whose uncancelled hypotheses are included among Al, ..., An establishes Al, ..., An / B as a theorem of P.

A proof of (an instance of) the law of excluded middle is given below to illustrate proofs in P. In this proof, numerals are written above hypotheses. The numerals of cancelled hypotheses have lines through them. To the right of each inference figure is an abbreviation for the name of the rule exemplified. For nonelementary rules, there are also the numerals of hypotheses cancelled by this application of the rule.

[A V A] AV -A] CE A ,E 1 A

[A V %A] VI

[A V -A] 2

The system T is obtained from P by adding these rules for constructing tree proofs:

LiElimination Li Introduction WA A A must be the conclusion of A DA a proof with no uncancelled

(T) hypotheses. [1A E[A D B]

FIB

Let the rank of a tree proof in T be the number of distinct occurrences of inference figures which it contains. Then it is easy to establish the following results.

LEMMA. Let F be a proof of Al, ..., AnIB. Let F have rank m. Let < be an mth- level T-valuation of LO for which each of Al, ..., An is true. Then B is true for V .

This is proved by induction on m. SOUNDNESS THEOREM FOR T. Let r be a proof of Al, ..., An/B. Let Y be a T-

valuation of Lofor which each of Al, ..., A,, is true. Then B is true for V . PROOF. Let F have rank m. Since V is a T-valuation, Y is also an mth-level

T-valuation. By the lemma, B is true for Y. Let X be a set of sentences of Lo. X is consistent with respect to T iff there is no

sentence A such that both A and -A are deducible in T from premises in X. Let X be a set of sentences of Lo. X is maximal consistent with respect to T iff X

is consistent with respect to T and for every sentence A, either A E X or X U {A} is not consistent with respect to T.

LEMMA 1. Let X be a set of sentences that is consistent with respect to T. Then X can be extended to a maximal consistent set with respect to T.

For the following lemmas, let X be a consistent set of sentences of Lo. Let X*




be a maximal consistent extension of X. Let 1" be the function defined on sentences of Lo such that

(i) <(A)= T iff L A E X*, (ii) Y(A) tiff n A 0 X*, A e X*, (iii) V(A) =f iff n , A X*, - A E= X*, (iv) V(A) = F iff - A EX*. LEMMA 2. Y is a Oth-level T-valuation of Lo. It is tedious to verify this lemma, but doing so is straightforward. LEMMA 3. Suppose Y (and every function generated like Y from a maximal con-

sistent set) is an mth-level T-valuation of Lo. Then V is an m + 1 st-level T-valuation of LO.

PROOF. Let A be a sentence that is true for every mth-level T-valuation. Suppose { A} is consistent. Then let Y be a maximal consistent extension of { - A}, and let Y'' be the valuation generated by Y. By hypothesis, Y' is an mth-level T-valuation of Lo. This is impossible, for A is false for V'. Hence, { -A} is not consistent. So /A is a theorem of T. By C] Introduction, / LIA is a theorem of T. Hence, LnAe X* and Y(A) = T.

LEMMA 4. Y' is a T-valuation of Lo. COMPLETENESS THEOREM FOR T. Let X be a set of sentences of Lo and let A be a

sentence of Lo such that A is satisfied by each T-valuation which satisfies each member of X (in symbols: X V T A). Then A is deducible in Tfrom premises in X.

If we change the matrix for modal operators to the following:

(S4) A GA 0$A T T T t f,F T,t f f,F T,t F F F

we can use the modified matrices to define 'S4-valuation'. The definitions are the same as for 'Oth-level T-valuation', 'mth-level T-valuation', and 'T-valuation', except that we replace 'T' by 'S4'. The system S4 is obtained from T by adding this rule:

(S4) EWA LI LIA

This system is easily shown to be sound and complete for S4-valuations of Lo. Another change in the modal matrix yields:

(S5) A EWA OA T T T t F T f F T F F F

We can use this to define 'S5-valuation', for which valuations the system S5 is sound and complete. S5 is obtained from T by adding:



82 JOHN T. KEARNS

(S5) OA C1 OA

?2. Quantification. To develop a semantics for propositional logic, there is no need to distinguish the different concepts of necessity and possibility from one another. The propositional semantics provides a general framework. Within this framework we can investigate the specific properties of this or that modal concept. (Here I am thinking of properties that can be specified with respect to the matrices.) We can similarly develop the semantics of a first-order language without considering one or another specific modal concept. But it is more difficult to do this for a first-order language, because there are more choices that must be allowed for. It is hard to keep them straight unless we can appeal to some heuristic considerations. In this section, I will provide a heuristic account, and develop a modal semantics to fit just one pair of modal concepts.

In thinking about concepts of necessity and possibility, I find it helpful to focus on possibility. There are different concepts of possibility for which we actually use the words 'possible' and 'possibility'. The corresponding concepts defined in terms of possibility and negation are not all (ordinarily) called concepts of necessity. I can distinguish four concepts of possibility (or four families of concepts). Two of these are epistemic. (1) Absolute epistemic possibility characterizes sentences (or propositions) that are not contradictory. Absolute epistemic possibility is logical possibility. (2) Relative epistemic possibility characterizes sentences that are not ruled out by what is (now) known.

I follow Kripke (in [2]) in calling the remaining concepts metaphysical. I distinguish relative metaphysical possibility from absolute metaphysical possibility. But I will not discuss these concepts further, for I am going to develop a semantic account to fit absolute epistemic possibility. This seems (to me) to be the simplest of the concepts of possibility, and the easiest to "capture". The concept of "necessity" which answers to absolute epistemic possibility is (ordinarily called) analyticity. In what follows we are to understand LA as "it is analytic that A", and O A as "it is logically possible that A".

L is a conventional first-order language in which the basic logical operators are a, V, C] and V. The symbols &, v and O are defined as before. The existential quantifier is defined:

D4 (3 a)A = (def)- (Va) A. In describing the semantics of L, I will consider only nonempty domains. I will

regard it as analytic that the domain is nonempty. It is not obvious that this is an accurate "representation" of the ordinary concept of analyticity. For I cannot decide whether 'something exists' is an analytic sentence of ordinary English.

I will regard the individual constants as names having no distinctive meanings that might contribute to the analytic character of a sentence. But it is analytic that an individual constant denotes a real individual-when this can be said. If p is a binary predicate that, by its very meaning, is reflexive, then ((a, a) will be analytic for every constant a. It is an essential feature of L that L contains no "ambiguous" name. If we wished to regard some individual constants as endowed with sense,




this would make the semantics more complicated. But even when this is allowed, it is technically convenient to have a recognizable class of constants without meanings.

The biggest problem with quantification is determining whether and how a formula can have the values T and F with respect to individuals as values of its free variables. The ordinary concept of analyticity only covers sentences (only sentences "fall under" it). We cannot reflect on this concept to determine what account of quantification is correct. The situation requires that we decide how the concept will be adapted. Although some treatments are unreasonable, different treatments seem equally reasonable. In this paper I will allow that a formula can be analytic or impossible with respect to individuals. But I insist that no distinctive feature of the individual be relevant to determining the "extreme" values. If a formula is analytic with respect to one individual, then it must be analytic with respect to every individual.

When a formula is analytic with respect to some individuals, we attribute its analytic character to the nonvariable expressions it contains. For example, suppose F(x) and G(x) both mean x is blue. The formula [F(x) V G(x)] will be analytic with respect to every individual. But it owes this feature to the meanings of F, G, V and a, not to any features of the affected individuals. Given this understanding, it is natural to hold that if a formula A(a) is analytic with respect to individuals in the domain, then (Va)A(a) must be analytic.

When we consider the (ordinary) concept of analyticity as it applies to sentences, the S5 matrix is clearly correct. So long as sentence A belongs to the language we are speaking, then if A is analytic, so is the sentence 'it is analytic that A'. If A is not analytic, the sentence 'it is analytic that A' is logically impossible (is contradictory). So for the quantificational semantics of L, I will not develop three ac- counts as I did for Lo. Only the S5-valuation will be defined.

I will use the same "ideas" for the first-order semantics as for the propositional semantics. Given a nonempty domain 9, an assignment function f will assign individuals in 9 to individual constants of L, butf will not assign sets to predicates. Instead, individuals in the domain will be treated as if they were constants. The result of replacing a variable by an individual in a formula is a pseudo-formula; these belong to a pseudo-language. If A is a formula (or pseudo-formula) containing free occurrences of distinct individual variables a,, ..., anx and j1, ..., n are individuals in the domain 9, then A[al, *-..-, an; j31 .., [3n] is the result of substituting 4 ...-, [8n for the free occurrence of a,, ..., cxn in A. (When real expressions r1, i rn are substituted, for the free occurrences, I use Church's notation S"1::,r- Al to denote the result.)

Let 9 be a nonempty domain, and letf be an assignment function from L to ?9. Then Lf is the pseudo-language containing the following pseudo-well-formed- formulas (pseudo-wffs):

(i) Every wif of L is a pseudo-wff of Lfz; (ii) If A is a pseudo-wif of Lf., A contains free occurrences of individual variable

a, and p is an individual in 9, then A[a; p] is a pseudo-wff; (iii) All pseudo-wffs are obtained by (i)-(ii). An S5 (analytic)-valuation will assign values to pseudo-sentences of Lf. in a



84 JOHN T. KEARNS

manner similar to that in which S5-valuations assign values to sentences in Lo. The matrices are the same as before. But the "idea" of the matrices must be extended to cover quantifiers.

For the universal quantifier, the "extended matrix conditions" are as follows: (i) A pseudo-sentence (Vcx)A has value T if A[a; p] has value T for every

individual p in the domain. (ii) A pseudo-sentence (Va)A has value t iff A[a; p] has value t for every in-

dividual p in the domain. (Remember, if an open formula has value T for one individual, it has value T for every individual.)

(iii) If pseudo-sentence (Va)A has value f, then A[a; p] has value f for some individual p in the domain, but A[a; p] does not have value F for any individual p in the domain.

(iv) A pseudo-sentence (Va)A has value F if A[a; p] has value F for some individual p in the domain.

Clauses (iii) and (iv) are not 'if and only if' clauses. The failure t provide such clauses is the analogue for 'V' of the rows in the matrix for '&' in which two values are possible.

Let 9 be a nonempty domain and letf be an assignment function from L to Q. Let Yl- be a function assigning one of T, t, f, F to each pseudo-sentence of Lf. r is a Oth-level S5 (analytic)-valuation of Lf. iff

(1) If A is a pseudo-wif containing free occurrences of a single individual variable a, j3 is an individual constant and p is an individual in 9 such that f(3) = p, then V(SAI) = (A [a; p]).

(2) If A is a pseudo-wif containing free occurrences of a single individual variable a, p is an individual in 9, and r(A[a; p]) = T, then r[(Va)A] = T.

(3) r satisfies the matrices for -, V and (the S5 matrix for) FI. (4) r satisfies the extended matrix conditions for V. Let r be an mth-level S5 (analytic)-valuation of Lf. r is an m + 1st-level

S5 (analytic)-valuation of Lf. iff r assigns T to each pseudo-sentence A which is true for every mth-level S5 (analytic)-valuation of Lf.

r is an S5 (analytic)-valuation of Lf iff r is an mth-level S5 (analytic)-valuation of Lf. for every m ? 0.

The system Q is obtained from P by adding these rules for tree proofs:

V Elimination V Introduction (Vat)A j3 is an individual SaAI a is an individual variable SaAI constant. (Vc)A that occurs free in A; 3 is

an individual constant that does not occur in A or in any uncancelled hypothesis.

The system S5 (analytic) is obtained from Q by adding the rules El Introduction, LI Elimination, (T), together with

(S5) (analytic) 0 A ElIAI j3 is an individual constant

LI O A LI(Va)A that does not occur in A.




It is easy to adapt the earlier soundness proof for T to establish the following result.

SOUNDNESS THEOREM. Let r be a proof in S5 (analytic) of A1, ..., An/B. Let 9 be a nonempty domain, f be an assignment function from L to A, and r be an S5 (analytic)-valuation of Lfg for which each of Al, ..., An is true. Then B is true for r.

The completeness proof for S5 (analytic) is quite ordinary. So I will not supply proofs for the lemmas and theorems.

For the completeness proof, L must be enlarged with new individual constants. L + is obtained from L by adding these: d, dl, d2, .... S5 (analytic) is extended to S5 (analytic) + which permits deductions with sentences of L +.

Let X be a set of sentences of L (of L + ). X is consistent with respect to S5 (analytic) (with respect to S5 (analytic) +) iff there is no sentence A such that both A and

A are deducible in S5 (analytic) (S5 (analytic) +) from premises in X. Let X be a set of sentences of L (of L +). X is maximal consistent with respect to

S5 (analytic) (S5 (analytic)+) iff X is consistent with respect to S5 (analytic) (S5 (analytic) + ) and for every sentence A of L (of L + ), either A E X or X U {A} is not consistent.

Let X be a set of sentences of L (of L +). X is instantially sufficient iff for every sentence (3a)A in X, there is a constant j3 such that SpAI E X.

For the following lemmas, let X be a set of sentences of L that is consistent with respect to S5 (analytic). Let X be extended to a maximal consistent set X* (of sentences of L).

Let the sentences of L + which have the form (33)B be enumerated: (3a,)Al, (3a2)A2. Let g be a one-one function from the positive integers to the constants d, dl, d2, ... such that g(m) does not occur in (3ca)Al, ..., (3am)Am. Let W

{[(3am)Am v S'1m)Aml] Im 2 1}. LEMMA 1. The set X* U W is consistent with respect to S5 (analytic) +. Let X* U W be extended to a set Y that is maximal consistent with respect to

S5 (analytic) +.

LEMMA 2. The set Y is instantially sufficient.

Let 9 = {a la is an individual constant of L+ }. Let f be a function defined on the individual constants of L + such that f(a) = a. Let r be a function defined on the pseudo-sentences of Lfg (these are just the sentences of L + ) such that

(i) r(A) = Tiff ]A E Y; (ii) r(A) = tiff LAX Y,AE Y; (iii) V (A) =f iff 1 MA ? Y. -MA E Y; (iv) r(A) = F iff - A E Y.

LEMMA 3. r is a Oth-level S5 (analytic)-valuation of Lfz.

LEMMA 4. Suppose r (and functions generated like r) is an mth-level S5 (analytic)-valuation of Lfz. Then r is an m + 1st-level S5 (analytic)-valuation of Lf.

LEMMA 5. -t is an S5 (analytic)-valuation of Lf.

COMPLETENESS THEOREM. Let X be a set of sentences of L and let A be a sentence of L such that every S5 (analytic)-valuation of an Lf which satisfies every sentence in X also satisfies A (i.e., X IHS5 (analytic) A). Then A is deducible in S5 (analytic) from premises in X.



86 JOHN T. KEARNS

?3. Concluding remarks. The present semantic account is equivalent to the standard account in providing soundness and completeness results for the customary modal systems. It is simpler than the standard account in virtue of having dis- pensed with possible worlds and their relations. I also think that my account is philosophically preferable to the standard account for having done this. For I do not think there are such things as possible worlds, or even that they constitute a useful fiction. My approach makes it possible to base modal logic on our ordinary modal concepts. However, for the present I will not enlarge further on the philo- sophical advantages of my semantics. The reader is left to her own speculations on these matters.

REFERENCES

[1] G. E. HUGHES and M. J. CRESSWELL, An introduction to modal logic, Methuen, London, 1968.

[2] SAUL A. KRIPKE, Naming and necessity, Semantics of natural language (Donald Davidson and Gilbert Harman, Editors), Reidel, Dordrecht, Holland, 1972, pp. 253-355.

DEPARTMENT OF PHILOSOPHY

STATE UNIVERSITY OF NEW YORK AT BUFFALO

BUFFALO, NEW YORK 14260



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Modal Semantics without Possible Worlds