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

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].