Degrees of the finite model property: The Antidichotomy Theorem

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 [Formula: see text] is [Formula: see text] or [Formula: see text]. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as [Formula: see text] or [Formula: see text]) or for extensions of the intuitionistic propositional calculus [Formula: see text] (see Problem 10.5 of Chagrov and Zakharyaschev’s “Modal Logic”). 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 the Blok Dichotomy Theorem that the degree of fmp of a normal extension of [Formula: see text] remains [Formula: see text] or [Formula: see text]. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of [Formula: see text]: each nonzero cardinal [Formula: see text] such that [Formula: see text] or [Formula: see text] is realized as the degree of fmp of some extension of [Formula: see text]. We then use the Blok–Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of [Formula: see text] and [Formula: see text]. This provides a solution of the reformulation of Problem. 10.5 of Chagrov and Zakharyaschev’ “Modal Logic” for the degree of fmp.