Fine’s Theorem on First-Order Complete Modal Logics

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 firstorder completeness. The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction. 1 The Canonicity Theorem and Its Impact In his PhD research, completed in 1969, and over the next half-dozen years, Kit Fine made a series of fundamental contributions to the semantic analysis and metatheory of propositional modal logic, proving general theorems about notable classes of logics and providing examples of failure of some significant properties. This work included the following (in order of publication): • A study [6] of logics that have propositional quantifiers and are defined semantically by constraints on the range of interpretation of the quantifiable variables as subsets of a Kripke model. Axiomatisations were given for cases where the range of interpretation is either the definable subsets, or an arbitrary Boolean algebra of subsets. Non-axiomatisability was shown for some cases where the range includes all subsets and the underlying propositional logic is weaker than S5. Decidability and undecidability results were also proved. • A model-theoretic proof [7] of Bull’s theorem (originally proved algebraically) that all normal extensions of S4.3 have the finite model property. It was also shown that these logics are all finitely axiomatisable and decidable, and a combinatorial characterisation was given of the lattice of extensions of S4.3 which showed that it is countably infinite.