Degrees of the finite model property: The antidichotomy theorem
Guram Bezhanishvili, Nick Bezhanishvili, Tommaso Moraschini
Abstract:
A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic K is 1 or the continuum. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as S4 or K4) or for extensions of the intuitionistic propositional calculus IPC (see [11, Prob. 10.5]). In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of K remains 1 or the continuum. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of IPC: each nonzero cardinal κ such that κ is countable or κ is the continuum is realized as the degree of fmp of some extension of IPC. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of S4 and K4. This provides a solution of the reformulation of [11, Prob. 10.5] for the degree of fmp.