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].
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de Jongh, D., Yang, F. (2011). Jankov’s Theorems for Intermediate Logics in the Setting of Universal Models. In: Bezhanishvili, N., Löbner, S., Schwabe, K., Spada, L. (eds) Logic, Language, and Computation. TbiLLC 2009. Lecture Notes in Computer Science(), vol 6618. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22303-7_5
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DOI: https://doi.org/10.1007/978-3-642-22303-7_5
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