MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 9Λ are the Λ-theorems, and are also said to be Λ-provable. A logic is called normalif it includes Gödel’s second axiom, which is usually presented (with ✷ in place ofB) as✷(p → q) → (✷p → ✷q),and has the rule of Necessitation: from α infer ✷α. S5 can be defined as thenormal logic obtained by adding the axiom p → ✷✸p to Gödel’s axiomatisationof S4. Following [Becker, 1930], p → ✷✸p is called the Brouwerian axiom. Thesmallest normal logic is commonly called K, in honour of Kripke. The normallogic obtained by adding the first Gödel axiom ✷p → p to K is known as T. Thatsystem was first defined by Feys 16 in 1937 by dropping Gödel’s third axiom fromS4. T is equivalent to the system M of [von Wright, 1951]. TheBrouwerian systemB is the normal logic obtained by adding the Brouwerian axiom to T.The first formulation of the non-normal systems S1–S3 in the Gödel style wasmade in [Lemmon, 1957], which also introduced a series of systems E1–E5 designedto be “epistemic” counterparts to S1–S5. These systems have no theorems of theform ✷α, and in place of Necessitation they have the rule from α → β infer ✷α →✷β. Lemmon suggests that they capture the reading of ✷ as “it is scientificallybut not logically necessary that”.3 MODAL ALGEBRASModern propositional logic began as algebra, in the thought of Boole. We haveseen that the same was true for modern modal logic, in the thought of MacColl.By the time that the Lewis systems appeared, algebra was well-established as apostulational science, and the study of the very notion of an abstract algebra wasbeing pursued [Birkhoff, 1933; Birkhoff, 1935]. Over the next few years, algebraictechniques were applied to the study of modal systems, using modal algebras:Boolean algebras with an additional operation to interpret ✸. During the sameperiod, representation theories for various lattices with operators were developed,beginning with the Stone representation of Boolean algebras [1936], and these wereto have a significant impact on semantical studies of modal logic.3.1 McKinsey and the Finite Model PropertyJ. C. C. McKinsey in [1941] showed that there is an algorithm for deciding whetherany given formula is a theorem of S2, and likewise for S4. His method was toshow that if a formula is not a theorem of the logic, then it is falsified by somefinite model which satisfies the logic. This property was dubbed the finite modelproperty by Ronald Harrop [1958], who proved the general result that any finitelyaxiomatisable propositional logic Λ with the finite model property is decidable.The gist of Harrop’s argument was that finite axiomatisability guarantees that Λ16 Who called it “t”.