Non-finite Axiomatizability of Generalized Medvedev Logics

2026-06-30

AI Summary

This study investigates whether generalized Medvedev logics—intermediate logics generated by products of finite rooted Kripke frames without top elements—admit finite axiomatizations. By integrating Kripke semantics, topological logics (notably the Cheq logic), and algebraic techniques, the authors construct pivotal models and counterexamples that confirm Bezhanishvili’s conjecture: the product logics of any finite chain, and more generally of any finite rooted frame without a top, are not finitely axiomatizable. The main contributions include establishing the non-finite axiomatizability of all such generalized Medvedev logics, showing that any extension containing the Cheq logic likewise lacks a finite axiomatization, and demonstrating that the corresponding lattice of logics has no minimal element and contains at least countably infinitely many distinct logics.

Abstract

We introduce a generalized form of Medvedev logics obtained by removing the greatest element from finite products of rooted Kripke frames with a top. We show that, before removing the top, the intermediate logic characterized by such finite products is exactly KC. Classical Medvedev logic is characterized by topless products of 2-chains, and a theorem of Maksimova, Skvortsov and Shehtman establishes that it is not finitely axiomatizable. Motivated by this result, Nick Bezhanishvili conjectured that non-finite axiomatizability extends to topless products of arbitrary finite chains and, more generally, to topless products of finite rooted frames with a top. We prove that every such generalized Medvedev logic is not finitely axiomatizable, thereby settling both conjectures in the affirmative. In 2003, van Benthem, Guram Bezhanishvili, and Gehrke introduced Cheq, the logic of chequered sets, and we show that whenever Cheq is a sublogic of a generalized Medvedev logic, the latter is not finitely axiomatizable over Cheq. Finally, we investigate the order structure of generalized Medvedev logics. We prove that there are at least countably many distinct generalized Medvedev logics and that no least such logic exists. These results extend the classical theory of Medvedev logic and clarify the behaviour of intermediate logics generated by topless product constructions.

Problem

• Medvedev logic
• finite axiomatizability
• intermediate logic
• Kripke frames
• topless products

Innovation

• generalized Medvedev logics
• non-finite axiomatizability
• topless products
• intermediate logics
• Cheq logic