Answer to a Question Suggested by Schumm

Abstract

in West Lafayette. Indiana (U.S.A.)With wffs built in the usual way from letters p , ~ p , , . . . , parentheses and the connectii-es 1, + and u, a modal logic is t'o be any sentential calculus L with rules substitution and detachment whose theorems include the classical tautologies ; the (proper) extensions of L are those modal logics whose theorem sets (properly) include L's. Saying that a Post-incomplete logic L bounds a, property P if and only if (i) L lacks P but (ii) every proper. Post-con tent extension of L has P, SCHUMM [4] shows that the finite model property (of which the st'andard account is given in 121) is bounded by a t lcast one modal logic and asks whether any such logic bounds lack of t h a t property. I )For each positive integer m . let S,, be the set of wffs in which the only letters occurring are among p , , . . . , p,, and, following MCKINSEY and TARSKI [3], let us say that a logic L is m-reducible just in case each wff all of whose substitution instances in S , are provable in L is also provahlc in L. Two a-ffs LX and j3 are interchangeable in L (for short. LY % L 8) if and only if replacement' of one or more occurrences of either with the other in any theorem (respectively. nontheorem) of L results in a theorem (respectively. nontheorem) of L. lye shall call L rn-restricted just in case the set &',,JgL of equivalence classes in S,,, is finit>e and extract from [71 the following exists a positive integer ni such that (i) L is m-reducible and (ii) L i s m,-restricted.Proof of the necessity of the conditions is straightforward ; for their sufficiency, see 171 where it is shown that (i) guarant.ees that the LINDENBA.UM matrix whose values are the equivalence classes into which E~ partitions 8, is a characteristic model of L while (ii) assures that it is finite. SCHUXM [4] notes t,hat the property of being Characterized by a finite matrix is well-known from [ 5 ] t o be bounded by S5, but he does not consider whether lack of such a mabrix is bounded by any modal logic. To show it is not, it of course suffices to show that if a Post-incomplete modal logic L has a finite characteristic matrix. then so does a t least one of its proper extensions. I n fact, an even stronger theorem of some independent interest can be obtained. TROELSTRA [6] has shown that' e v e r y extension of any superintuitionistic logic with a, finite characteristic matrix is also characterized by such a matrix. His result does not hold for sentential calculi gcnemlly. [9] providing a subsystem of intuitionism l) SCHUNM [4] describes a modal logic as a "class of formulas containing all classical tautologies and closed under the rules of substitution and material detachment." But it is the finite model property which is here under discussion, and since ANDERSON [l] has shown the latter to be a property of sentential calculi -theorems together with rules -rather than merely of their theorem sets, it has seemed judicious to be explicit on this point. L e m m a . A senten,tial calculus L has a finite characteristic m,atrix if and...