Jankov's Theorems for Intermediate Logics in the Setting of Universal Models
Dick de Jongh, Fan Yang
Abstract:
In this article we prove the Jankov Theorem for extensions of IPC
([6]) and the Jankov Theorem for KC ([7]) in a uniform frame-theoretic
way in the setting of n-universal models for IPC. In frame-theoretic
terms, the first Jankov Theorem states that for each finite rooted
frame there is a formula \psi with the property that any counter-model
for \psi needs this frame in the sense that each descriptive frame
that falsifies \psi will have this frame as the p-morphic image of a
generated subframe. The second one states that KC is the strongest
logic that proves no negationless formulas beyond IPC. On the way we
give a simple proof of the fact discussed and proved in [1] that the
upper part of the n-Henkin model H(n) is isomorphic to the n-universal
model U(n) of IPC. All these results earlier occurred in a somewhat
different form in [8].