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 characterization 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.

Localization of tetravalent modal algebras

The main aim of this paper is to define the localization of a tetravalent modal algebra [Formula: see text] with respect to a topology [Formula: see text] on [Formula: see text]. In Sec. 5, we prove that the tetravalent modal algebra of fractions relative to a ∧-closed system (defined in Definition 3.1) is a tetravalent modal algebra of localization.

Categories of frames for modal logic

§1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = {y ∣ (∃x)(y < x Є X)}; and if is a CAMA then is a frame, where is the set of atoms of and b1 < b2 ⇔ b1 ∩ mb2 ≠∅.Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence.The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames.The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].