Propositional quantifiers in modal '

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 . The paper is in two parts: in the first part I consider theories whose non-quantificational part is ; 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 V of proposi- tional variables pl, pa, ... , the operators v (or), -(not) and 0 (necessarily),the universal (p), p a propositional , and brackets ( and ). The formulas of L are then defined in the usual way.

1.1. In Kripke's semantics for S5 [6],a 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 of possible worlds. Formally, a structure A is an ordered triple (W, P, u), where W (worlds) is a non-, P (proposi- tions) is a non-empty set of subsets of Wand v () is a from Vinto P. The -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 . 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 in the Summer term of 1969. PROPOSITIONAL QUANTIFIERS IN 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 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 .) (iii) P is the 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 -schemes and rules of : 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 , 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.

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 to AxSSx + (AxS5x), we may carry out the elimination above in the axiomatized theories. This gives a new proof of . 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 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. Given that S5x+ does justice to a Platonist conception of proposition, then the theory provides the common ground between Platonists and non- Platonists.

PROPOSITION 4. This theory is the weakest (normal) theory to have the following property: If A is a formula each of whose variables p is within the scope of 0 or (p), then A e OA is a . (iv) P infinite. The theory for this condition is equivalent to the modal theory with propositional quantifiers proposed by Kripke in [6]. If the condition is added to those for the theories under (i) and (ii), then the resulting theories are complete in the sense that for each closed formula A either A or - A is a theorem. The three theories considered here are not finitely axiomatiz- able, i.e. they cannot be axiomatized by adding a finite number of axioms to AxSSx.

I1 2.1. Semantics A structure is now an ordered quadruple A=(W, R, P, v), where (W, P, v) is a structure in the old sense and R (accessibility) is a on W.The truth-definition now runs: (i), (ii) and (iii), as before;

(iv) EmB iff B for all y such that xRy;

b’) such that v’(pj)=v(pj)for all j #i, i= 1, 2, .. .. is defined as before. We say A = (W,R, P, v) is closed under formulas if for each structure A’ = (W, R, P, v’) and each formula

E P. Given a modal propositional logic S with PROPOSITIONAL QUANTIFIERS IN MODAL LOGIC 343 corresponding condition C on R, let Sx - (Sx; Sx +) be the set of formulas valid in each structure A=(W, R, P, v) such that R satisfies C and P is Boolean (Ais closed under formulas; P is the power set of W).So, e.g., S4x is the set of formulas valid in each structure A = (W, R, P, v) such that R is reflexive and transitive and A is closed under formulas.

2.2. Axiomatizability Suppose AxS is an axiomatization of S above (but in the enlarged language L). Let AxSx - be AxS with the axiom-schemes 5,6 and 7, the axiom 13. (P>OA+O.(P)A, and the rules MP, Nec., and Gen. Let AxSx be AxSx- with axiom-scheme 8 instead of axiom-scheme 5.

PROPOSITION 5.4 AxKx - , AXKX,AxTx - , AxTx, AxS4x - , AxS4x are axiomatizations of Kx - , Kx, Tx - , Tx, S4x - , S4x respectively. These results are again proved by the method of maximally consistent theories. Indeed, this method shows that AxSx - and AxSx are axiomatizations of Sx - and Sx respectively, whenever the canonical model for S is based upon a frame for S.6 In contrast to propositions 1 and 5, we have:

PROPOSITION 6. The theories Kx+, Tx+, K4x+, S4x+, S4.2x+ and Bx + are not axiomatizable. For each of the theories, this result can be obtained by finding a translation T from second-order into the theory. T is constructed in three steps. First, we express within the theory the assumption that there exists a certain one-one correspondence from worlds into triples of worlds. Second, we define quantifica- tion over sets of triples of worlds (relations) by quantification over

R. Bull was the first to establish this result for s4K [2]. D. Gabbay indepen- dently established these results for the x-theories, but used a different seman- tics [4]. ' For an explanation of this terminology, see [7]. 344 KIT PINE the sets of corresponding worlds (propositions). Third, we give the standard Peano definitions for addition and multiplication.

2.3. Decidability Let a theory S be any set of formulas of L which contains Kx - and is closed under MP and Gen. Two theories S1and S, are compatible if {A : for some B, B E S1 and B +A E S2} is consistent. Let 'p be the conjuction of the axioms of Robinson's finitely axiomatized and essentially undecidable arithmetic; and let

KR= {A : T'p -+A E Kx- }. Then we may show that KR is essentially undecidable, so that any theory compatible with KR is undecidable (cf. [S]). In particular, we may show that:

PROPOSITION 7. Kx-, Kx, Tx-, Tx, K4x-, K4x, S4x-, S4x, S4.2~-, S4.2~~Bx - and Bx are undecidable. The above method does not work for S4.3~-and S4.375 and 1 do not know whether these theories are decidable or not. Given these negative results, it is worth mentioning a positive result. A structure A = (W, R, P, u) is Diodorean if W= {0,1,2, . . .}, R is the natural ordering on W, and P is the power set of W. Let

for all Diodorean A

PROPOSITION 8. Dx+ is decidable. This result can be proved by translating Dx+ into the second- order monadic theory of the successor , which is known to be decidable [l]. From a result of Rabin's, it also follows that S4.3~+ is decidable.

2.4. Theories with variable domain So far we have assumed that the domain P of propositions is the same in each . But one may argue that a proposition PROPOSITIONAL QUANTIFIERS IN MODAL LOGIC 345 exists in a given world iff the individuals which the proposition is about exist in that world. Consequently, if the domain of indi- viduals varies from world to world, so does the domain of propo- sitions. As is well-known, there are several semantics for variable do- main in modal logic. Let me, for illustration, give one. A struc- ture is now an ordered quintuple (x, W, R, P, v) where x E W, R is a binary relation on W, P maps each world y E W into P,, a non- empty set of subsets of W, such that if yRz then P,rIT,, and, finally, v is a map from V into P.. The clauses for 0 and (pi) are now:

v) such that xRy; and

(v) (pi)B iffF B for all structures A’ = (x, W,R, P, X X v’) such that v‘(p,) = u(pj) for all j # i, i = 1, 2, .. .. All of the results of the previous two sections can be extended to the present semantics. E.g., suppose K’x - = (A : E-A for all A = (x, W, R, P, v) such that P, is Boolean for each y in W}. Then by previous methods we may show that K’x - is axiomatiz- able (drop axiom 13 from AxKx - ) and that K’x - is undecidable.

References

ill J. R. BUCHI.“On a decision method in restricted second order arithmetic.” In Logic, methodology and of science: Proceedings of the 1960 International Congress, edited by E. Nagel, P. Suppes, and A. Tarski, pp. 1-1 1. Stanford, Calif.: Stanford University Press, 1962. [2] R. A. BULL.“On modal logic with propositional quantifiers.” The journal of symbolic logic, vol. 34 (1969), pp. 257-263. [3] M. J. CRESSWELL.“A Henkin completeness theorem for T.”Notre Dame jour- nal of formal logic, vol. 8 (1967), pp. 186-190. 23-Theoris. 3:1970 346 KIT PINE

[4] D. M. GABBAY."Montague-type semantics for modal with proposi- tional quantifiers."Unpublished. [5] D. KAPLAN."S5 with quantifiable propositional variables." The journal of symbolic logic, vol. 35 (1970), p. 355. [6] S. A. KRIPKE."A completeness theorem in modal logic. Ibid., vol. 24 (1959), pp. 1-14. [7] K. SEGERBERC."Decidability of S4.1." Theoria, vol. 34 (1968), pp. 7-20. [8] A. TARSKI,A. MOSTOWSKI,AND R. M. ROBINSON.Undecidable theories. Amster- dam: North-Holland Publishing Co., 1953.

Received February 22, 1970.