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An algebraic approach to subframe logics. Modal case. (English) Zbl 1246.03041

It has been shown by Fine that any subframe logic over K4 enjoys the finite model property (FMP), which then was extended by Zakharyaschev to any cofinal subframe logic over K4. From this follows that any cofinal subframe (and then any subframe) superintuitionistic logic has the FMP, whose algebraic proof was also given by the (first two) authors of this paper. Here the algebraic method is applied for the modal case. As a matter of fact, the result of Fine-Zakharyaschev is generalized so that any cofinal subframe logic over wK4 has the FMP, where wK4 is known according to Esakia as the modal logic of topological derivative and constitutes the subsystem of K4 characterized by weak transitive frames. The main observation is that, if a modal formula is refuted on a wK4 algebra B, then it is refuted on a finite wK4-algebra which is isomorphic to a subalgebra of a relativization of B. The common base to support these algebraic proofs is Diego’s theorem that implicative meet-semilattices are locally finite. For the application, however, some more sophisticated manipulations are required in the case of wK4.

MSC:

03B45 Modal logic (including the logic of norms)
03G25 Other algebras related to logic
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