MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 39follows directly from properties of maximally consistent sets that R Λ is reflexive.This gives a technique for proving that various logics are characterised by suitableconditions on models, a technique that is explored extensively in [Lemmon andScott, 1966].If Scott’s representation of modal algebras is applied to the Lindenbaum algebraof Λ, the result is a model structure isomorphic to (K Λ , R Λ ). The constructioncan also be viewed as an adaptation of the method of completeness proof introducedin [Henkin, 1949], and first used for modal logic in [Bayart, 1958] (seesection 4.3). There were others who independently applied this approach to therelational semantics for modal logic, including David Makinson [1966] and MaxCresswell [1967], their work being completed in 1965 in both cases. Makinson dealtwith propositional systems, while Cresswell’s appears to be the first Henkin-styleconstruction of relational models of quantificational modal logic. David Kaplanoutlined a proof of this kind in his review [1966] of [Kripke, 1963a], explaining thatthe idea of adapting Henkin’s technique to modal systems had been suggested tohim by Dana Scott.Another construction of lasting importance from the Lemmon Notes is a techniquefor proving the finite model property by forming quotients of the model M L .To calculate the truth-value of a formula α at points in M Λ we need only know thetruth-values of the finitely many subformulas of α. We can regard two members ofM Λ as equivalent if they assign the same truth-values to all subformulas of α. Ifthere are n such subformulas, then there will be at most 2 n resulting equivalenceclasses of elements of M Λ , even though M Λ itself is uncountably large. Identifyingequivalent elements allows M Λ to be collapsed to a finite quotient model whichwill falsify α if M Λ does. This process, which has become known as filtration, 37was first developed in a more set-theoretic way in [Lemmon, 1966b, p. 209] as analternative to McKinsey’s finite algebra construction. In its model-theoretic formit has proven important for completeness proofs as well as for proofs of the finitemodel property. Some eighteen modal logics were shown to be decidable by thismethod in [Lemmon and Scott, 1966].5.2 Bull’s Tense AlgebraA singular contribution from the 1960’s is the algebraic study by Robert Bull, astudent of Arthur Prior, 38 of logics characterised by linearly ordered structures.Prior had observed that the Diodorean temporal reading of ✷α as “α is and alwayswill be true” leads, on intuitive grounds, to a logic that includes S4 but not S5.In his 1956 John Locke Lectures at Oxford on Time and Modality (published as[Prior, 1957]) he attempted to give a mathematical precision to this reading by37 This term was first used in [Segerberg, 1968a], where “canonical model” was also introduced.38 Initially at Christchurch, New Zealand, and then at Manchester, England. Bull was one oftwo graduate students from New Zealand who studied with Prior at Manchester at the beginningof the 1960’s. The other was Max Cresswell, who later became the supervisor of the presentauthor.