Degrees of the finite model property: the antidichotomy theorem

TL;DR

This work introduces the degree of finite model property (fmp) for superintuitionistic and modal logics as a natural analogue to Blok's degree of incompleteness, and proves an Antidichotomy: for every nonzero κ with either κ ≤ $\aleph_0$ or κ = $2^{\aleph_0}$ there exists a logic whose degree of fmp equals κ. The finite case is realized via the Kuznetsov–Gerću logic $\mathsf{KG}$ and its extensions, while the continuum via width-bounded logics $\mathsf{BW}_n$ and carefully constructed Esakia-space models; CH ensures realization of all such κ. The Blok–Esakia correspondence then transfers these results to normal modal logics, yielding a Modal Antidichotomy for $\mathsf{S4}$, $\mathsf{K4}$, and their Grz-related companions. The paper also provides a detailed structural analysis through Jankov formulas, finite Esakia duality, and the Rieger–Nishimura ladder, establishing a robust framework for understanding when a logic's fmp-degree is 1 or can realize continuum cardinalities. Overall, the work significantly broadens the landscape of possible fmp-degrees and raises further questions about exact degrees for individual logics and other semantic frameworks.

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 $κ$ such that $κ\leq \aleph_0$ or $κ= 2^{\aleph_0}$ 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 $\sf K4$.

Figures (16)

• Figure 1: The Rieger-Nishimura lattice.
• Figure 2: The posets $P_1, P_2$, and $P_3$.
• Figure 3: The poset $F_{n+1}$.
• Figure 4: The eleven posets whose Jankov formulas axiomatize $\mathsf{W}_2$.
• Figure 5: The Rieger-Nishimura ladder $\mathfrak{L}$.

Theorems & Definitions (121)

• Theorem 2.1: Esakia Duality
• Theorem 2.2: Finite Esakia Duality
• Definition 2.3
• Theorem 2.4
• Corollary 2.5
• Definition 2.6
• Corollary 2.7
• Definition 3.1
• Theorem 3.2: Antidichotomy Theorem
• Theorem 3.3: Jankov Theorem