Degrees of the finite model property: the antidichotomy theorem

TL;DR

This paper introduces the degree of finite model property (fmp) in modal logic, proving an antidichotomy theorem that shows a wide range of degrees of fmp can be realized in extensions of intuitionistic and modal logics.

Contribution

It defines the degree of finite model property and establishes an antidichotomy theorem for extensions of IPC, S4, and K4, contrasting with the classic dichotomy for K.

Findings

Degree of fmp can be any countable or continuum cardinal in certain logic extensions.

Antidichotomy theorem shows the diversity of degrees of fmp in these logics.

Contrast with Blok Dichotomy for degree of incompleteness.

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 $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. 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 $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $\kappa$ such that $\kappa \leq \aleph_0$ or $\kappa = 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$.