Some completeness results for intermediate propositional logics.
Craig Graham McKay · Notre Dame Journal of Formal Logic · 1967
In my paper "Implicationless wffs of IC" [4] I made use of one of the results of V. A. Jankov, which were stated by him without proof in [3].R. Harrop in a recent article [2] has commented on their importance, so it is perhaps worthwhile to supply a proof.In addition a new and much simpler proof of an older result of Dummett [1] is presented.2. Intermediate Propositional Logics (IPL's) can be characterized in either of two ways.Firstly syntactically: let S be the set of IPL's, K the set of classically valid wffs and / the set of intuitionistically valid wffs.Then S={L:/CLC^}, If |c is Heyting's axiomatization of / then we can obtain axiomatizations for each LεS by augmenting IC with a set of new axioms A,AczK.We write such axiomatizations as .It is clear that one logic may have a number of different axiomatizations, and for this reason we distinguish between the axiomatizations and the logic.Secondly we can characterize S semantically.By an algegra I mean a pseudo-complemented lattice.Let J be the direct product of all the algebras in the Jaskowski sequence, and B(J) the set of all subalgebras of J.If £εB(J) then JL will be said to admit an interpretation of an IPL,L, if under the normal mapping (for details [5])0:L->.£for each PεLφ{P) vanishes identically in Z.We can say that Z is a model for L. If 0(P) vanishes identically in J_ iff PεL, then we say that / is a characteristic model for L. L will be said to be complete with respect to JL.It is well known that there is an infinite sequence of (Boolean) algebras of 2 k elements, k = i, 2, . . .each of which is characteristic for K. From the viewpoint of logic the difference between these algebras is inessential. 1 To deal with such cases we define an equivalence relation on B(J).We put, for ^i,^2εff(J), -£i =-£.2 if -ίi and JL 2 are characteristic for the same logic L. Let B(J) be the resulting set of equivalence classes.If-£ is a model for a logic L, then we say that Ji is a model-set for L. Similarly if £. is a characteristic model for L, we say that ί is the characteristic model-set for L. S is partially ordered under set inclusion.We can partially order 1. A. S. Troelstra pointed out the need for taking this into account.