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Propositional Quantifiers in Modal Logic1 Propositional quantifiers in modal logic' by KIT FINE (Oxford University) In this paper I shall present some of the results I have obtained on modal theories which contain quantifiers for propositions. The paper is in two parts: in the first part I consider theories whose non-quantificational part is S5; in the second part I consider theories whose non-quantificational part is weaker than or not contained in S5. Unless otherwise stated, each theory has the same language L. This consists of a countable set V of proposi- tional variables pl, pa, ... , the operators v (or), -(not) and 0 (necessarily),the universal quantifier (p), p a propositional variable, and brackets ( and ). The formulas of L are then defined in the usual way. 1.1. Semantics In Kripke's semantics for S5 [6],a proposition is, in effect, identi- fied with the set of possible worlds in which it is true. So on this view the propositional quantifier (p) should range over a set of subsets of possible worlds. Formally, a structure A is an ordered triple (W, P, u), where W (worlds) is a non-empty set, P (proposi- tions) is a non-empty set of subsets of Wand v (valuation) is a map from Vinto P. The truth-definition then goes as follows: for a structure A = (W, P, u), a world x in W,and formulas B and C, I am greatly indebted to the late Arthur Prior. Without his encouragement, this paper would not have been written. 2 The results of this part are contained in my Ph.D. thesis, submitted to the University of Warwick in the Summer term of 1969. PROPOSITIONAL QUANTIFIERS IN MODAL LOGIC 337 (v) (pi)B iff ILB for all structures A’ = (w,P, v’) X X such that v’(pJ= VIP3 for all j f i, i = 1, 2, . .. (Note: v’(pi) E P by the defini- tion of structure.) The formula A is valid in A, FEA, if A for all y in W. There are three natural conditions we can impose upon P: (i) P is Boolean i.e. closed under complementation and finite union. (The negation and disjunction of propositions are also propositions.) (ii) P is closedunder formulas i.e. for each structure A = (W, P,v) and for each formula A, IxEW : El- A EP. (Each formula expresses a proposition under any interpretation.) (iii) P is the power set of W, i.e. the set of all subsets of W. (For each set of worlds there is a proposition true in exactly those worlds.) These three conditions lead to three corresponding theories. Let S5x - (S5x; S5x +) be the set of formulas valid in each struc- ture A = (W, P, v) such that P is Boolean (closed under formulas, the power set of W). It should be clear that S5x - c S5x c S5x + . One can also show that the inclusions are proper. 338 KIT FINE 1.2. Axiomatizability Let Ax S5x- consist of the following axiom-schemes and rules of inference: 1. All tautologous formulas. 2. o(A+B)+(OA+OB). 3. UA+A. 4. -OA+O-OA. 5. (p)A(p)-+A(B), B a formula of PC, the non-modal propositional calculus, free for p in Alp). 6. (PICA 'B) -+llP)A +CP)B). 7. A+(p)A, p not free in A. MP. A, A+B/B. Nec. A/OA. Gen. A/(p)A. AxS5x is AxS5x -without scheme 5, Restricted Specification, but with Specification: 8. (p)A(p)-+A(B), B any formula free for p in A(p). AxS5x +is AxS5n with the axiom: 9. C3Pl)(Pl8Z CPaI(P2 -aPl 'Pa))) It should be clear that scheme 5 corresponds to the condition that the set of propositions P be Boolean and that scheme 8 corresponds to the condition that P be closed under formulas. Axiom 9 (with Nec.) corresponds to the condition that P be atomic over the set of worlds W, i.e. that for each x E W there is an atom in P (a minimal non-empty a in P) such that x E a. If one identifies indistinguishable worlds, i.e. those which belong to the same propositions in P, then axiom 9 says that each world is describable, i.e. that for each x E W, {x} E P. PROPOSITION l.3AxS5x - , AxS5x and AxS5x + are axiornatizations of S5x - , S5x and S5x + respectively. 8 R. Bull independently established this result for S5x and S5x+ [2]. His proof is by semantic tableaux, and his semantics is slightly different from mine. D. Kaplan independently established this result for S5x+ [5].His proof is by quantifier elimination, as in Proposition 2 below. PROPOSITIONAL QUANTIFIERS IN MODAL LOGIC 339 The simplest direct proof of these results is by the Henkin- Scott-Makinson method of maximally consistent (mc) theories (see [7]). One uses a technique of Cresswell's [3] to guarantee that each mc theory has the same language, and for S5x+ one also uses a construction which for each mc theory provides a variable which belongs to that theory and that theory alone. 1.3. Decidability PROPOSITION 2. The theories S5x + and S5x are decidable. The proof proceeds by the elimination of quantifiers. Let L, be the language L with the unary operators MI, k = 1,2, . ., and let L, be L1 with the propositional constant g. For the new symbols we add the following clauses to the truth-definition: (vi) EMSiff at least k distinct atoms of P are inclu- ded in { y : €81. (vii) g iff x belongs to an atom of P. Given that indistinguishable worlds are identified, MkA says that A is true in at least k describable worlds, and g says that the actual world x is describable. One proves the decidability of S5x + (S5x) by showing that: A. There is an effective way of finding for each formula A of L a quantifier-free formula B of L1 (LJ such that A-eB E S5x + (S5x); and B. The set of quantifier-free formulas of L, n S5x + (L, n S5x) is effective. It is possible to define Mk and g within the language L. Write: QA for OA & ($)COCA +p) v C](A --f - p)) where p is the first variable not free in A. Then the following equivalences are valid in all structures: 340 KIT FINE AL(Qqi 8~ O(qi +A))), where qi is the i-th variable not free in A, i = 1, . , 12, k = 1, 2, . .. 11. g q3Pl)(Pl & QPJ By adding 10 (10 and 11) as axioms to AxSSx + (AxS5x), we may carry out the elimination argument above in the axiomatized theories. This gives a new proof of completeness. I presume that the above method could also be used to prove the decidability of S5x- . However, the details would be formi- dable, and, to date, my patience has not been the equal of my presumption. 1.4. Predicate and Boolean analogies S~X+is inter-translatable with the first-order theory of atomic Boolean algebras and also with M, the second-order monadic predicate calculus. For example, the translation T from formulas of S5x + to formulas of M is defined by: Tpj = Pixi; T-B= -TB; T(BvC)=TBvTB; TUB = (xi)TB; T(p i)B = (Pr)TB; where the Pi are quantifiers over predicates. One may easily show that: A. T is an effective map. B. A formula A of L is in S5x + iff TA is valid in M. A and B with the classical result that M is decidable provide a new proof of decidability for S5x + . S5x is inter-translatable with the first-order theory of separable Boolean algebras and, of course, with the second-order calculus defined by {TA : A E S~X}.(A Boolean algebra is separable if the set of all atoms in the algebra has a least upper bound). The former result depends upon: PROPOSITIONAL QUANTIFIERS IN MODAL LOGIC 341 PROPOSITION 3. P is closed under formulas if P is Boolean and separ- able. Syntactically, this proposition says that S5x can be axiomatized by adding to AxS5x - the axiom: 12. (~p1)0(p1og). (g, like Mr, will now be used as an abbreviation). The only proof 1 have of this result is indirect. It consists in showing that the quantifier-elimination argument for AxS5x also goes through for the new axiomatization. The elementary theory of Boolean algebras is translatable into S5x-, and I suspect that there is also a translation in the other direction. 1.5. Some further theories There is the following correspondence between conditions on P and axioms: P atomic g P atomless -g P atomic or atomless g -+Og P infinite MITvO-g, MaTvO-g, ... where T = (pl)(pl -+ pl). This means that the theory for a condition or combination of conditions on the left can be axiomatized by adding to AxS5x- the corresponding axioms on the right. Completeness and decida- bility for all such theories can be proved by quantifier elimination. Some of the theories are of special interest: (i) P atomic. The theory for this condition is the same as S5x + . In other words, the distinction between P being atomic and P being a power set cannot be expressed within L. The stronger condition would most naturally be expressed by using quantifiers over sets of propositions. (ii) P atomless. The theory for this condition does justice to a non-Platonist conception of proposition. (Every proposition is expressed by a sentence.) 342 KIT FINE (iii) P atomic or atomless. The theory for this condition is the intersection of the two theories above.
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