Jankov’s Theorems for Intermediate Logics in the Setting of Universal Models

Abstract

In this article we prove two well-known theorems of Jankov in a uniform frame-theoretic manner. In frame-theoretic terms, the first one states that for each finite rooted intuitionistic frame there is a formula ψ with the property that this frame can be found in any counter-model for ψ in the sense that each descriptive frame that falsifies ψ will have this frame as the p-morphic image of a generated subframe ([12]). The second one states that KC, the logic of weak excluded middle, is the strongest logic extending intuitionistic logic IPC that proves no negation-free formulas beyond IPC ([13]). The proofs use a simple frame-theoretic exposition of the fact discussed and proved in [4] that the upper part of the n-Henkin model $\mathcal{H}(n)$ is isomorphic to the n-universal model $\mathcal{U}(n)$ of IPC. Our methods allow us to extend the second theorem to many logics L for which L and L + KC prove the same negation-free formulas. All these results except the last one earlier occurred in a somewhat different form in [16].