Two theorems of Jankov in the setting of universal models


The talk concerns work with Fan Yang on the intuitionistic propositional calculus IPC. Two well-known theorems of Jankov are proved 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. 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. The proofs use a simple frame-theoretic exposition of the fact discussed and proved in Nick Bezhanishvili's dissertation that the upper part of the n-Henkin model H(n) is isomorphic to the n-universal model 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.