Fine's Theorem on First-Order Complete Modal Logics

TL;DR

This paper reviews Fine's Canonicity Theorem, explores its implications for first-order complete modal logics, and introduces a new characterization of canonical validity highlighting the non-commutativity with ultrapower construction.

Contribution

It provides a new characterization of canonical validity in modal logics, distinguishing it from first-order completeness and analyzing the non-commutativity with ultrapower construction.

Findings

Canonical frame construction does not commute with ultrapower construction.

A new characterization of when a logic is canonically valid.

Clarifies the distinction between first-order completeness and canonicity.

Abstract

Fine's influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames. This article reviews the background and context of this result, and the history of its impact on further research. It then develops a new characterisation of when a logic is canonically valid, providing a precise point of distinction with the property of first-order completeness. The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction.