MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 61of CmS induced by the surjections CmS ↠ CmS j [Goldblatt, 2000, §4.5].The first example of non-canonicity in the modal context occurs in [Kripke,1967], where it is stated that Dummett’s Diodorean axiom✷(✷(p → ✷p) → ✷p) → (✸✷p → ✷p)is not preserved by the Jónsson–Tarski representation of modal algebras. TheMcKinsey axiom ✷✸p → ✸✷p was shown not to be canonical in [Goldblatt, 1991a].The formulas of Sahlqvist(see 6.3) define logics Λ for which the class F r(Λ) iselementary and includes all the canonical frames S Λ η . These formulas have beengeneralized by Maarten de Rijke and Yde Venema [1995], who defined Sahlqvistequations for any type of BAO and showed that the structures S whose complexalgebras CmS satisfy such an equation form a basic elementary class. Jónsson[1994] has refined the techniques of [Jónsson and Tarski, 1951] to develop an elegantalgebraic proof that varieties of BAO’s defined by Sahlqvist equations are closedunder canonical embedding algebras.Fine’s theorem (ii) was strengthened by the present author to show that if Λis characterised by some elementary class then it is valid, not just in any canonicalframe S Λ η , but also in any frame that is elementarily equivalent to a canonicalframe. In fact an even stronger generalization of (ii) can be obtained by restrictingattention to quasi-modal sentences. These are first-order sentences of the syntacticform ∀vϕ, with ϕ being constructed from amongst atomic formulas and theconstants ⊥ (False) and ⊤ (True) using at most ∧ (conjunction), ∨ (disjunction),and the bounded universal and existential quantifiers forms ∀z(yRz → ψ) and∃z(yRz ∧ ψ) with y ≠ z. The relevance of quasi-modal sentences, and the reasonfor the name, is that they are precisely those first-order sentences whose satisfactionis preserved by the basic modal-validity preserving operations of S, H, and Ud[van Benthem, 1983; Goldblatt, 1989]. By the quasi-modal theory of a structureS we mean the set of all quasi-modal first-order sentences that are true in S.It transpires that there is no quasi-modally-expressible property that can differentiatethe canonical frames of a logic Λ: the structures S Λ η have exactly thesame quasi-modal first-order theory for all η. We will denote this unique quasimodaltheory of the canonical Λ-frames by Ψ Λ . Moreover, if Λ is not canonical,then it always has a largest canonical proper sublogic Λ c and a largest elementarysublogic Λ e (with Λ e ⊆ Λ c ), and the quasi-modal theories Ψ Λe and Ψ Λc of theseother logics are identical to Ψ Λ . These results are all proven in [Goldblatt, 2001a].The strengthening of Fine’s result is as follows [Goldblatt, 1993, 11.4.2]:If a modal logic Λ is characterized by some elementary class of frames,then it is characterized by the elementary class of all models of thequasi-modal first-order theory Ψ Λ (which includes all the canonicalframes of Λ).Fine asked if the converse of his (ii) was true: if a logic is canonical, must it becharacterised by an elementary class? The algebraic version of this question asks