Canonical modal logics and ultrafilter extensions

Abstract

In this paper thecanonicalmodal logics, a kind of complete modal logics introduced in K. Fine [4] and R. I. Goldblatt [5], will be characterized semantically using the concept of anultrafilter extension, an operation on frames inspired by the algebraic theory of modal logic. Theorem 8 of R. I. Goldblatt and S. K. Thomason [6] characterizing the modally definable Σ⊿-elementary classes of frames will follow as a corollary. A second corollary is Theorem 2 of [4] which states that any complete modal logic defining a Σ⊿-elementary class of frames is canonical.The main tool in obtaining these results is the duality between modal algebras and general frames developed in R. I. Goldblatt [5]. The relevant notions and results from this theory will be stated in §2. The concept of a canonical modal logic is introduced and motivated in §3, which also contains the above-mentioned theorems. In §4, a kind of appendix to the preceding discussion, preservation of first-order sentences under ultrafilter extensions (and some other relevant operations on frames) is discussed.The modal language to be considered here has an infinite supply of proposition letters (p, q, r, …), a propositional constant ⊥ (the so-calledfalsum, standing for a fixed contradiction), the usual Boolean operators ¬ (not), ∨ (or), ∨ (and), → (if … then …), and ↔ (if and only if)—with ¬ and ∨ regarded as primitives—and the two unary modal operators ◇ (possibly) and □ (necessarily)— ◇ being regarded as primitive. Modal formulas will be denoted by lower case Greek letters, sets of formulas by Greek capitals.