MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

14 Robert Goldblattthat S4 is characterised by the class of (finite) closure algebras, as well as theclosure algebra of any Euclidean space, or of any zero-dimensional dense-in-itselfsubspace of Euclidean space. Hence in view of result (5), the claim of [Gödel,1933] follows: if ✷α ∨ ✷β is an S4-theorem, then so is one of α and β, thereforeso is one of ✷α and ✷β by the rule of Necessitation. Similarly, result (6) gives aproof of the disjunction property for IPC: if α ∨ β is a theorem, then so is one of αand β. The final section of the paper uses the relationships between Brouwerianand closure algebras to verify the correctness of the two translations of IPC intoS4 conjectured in Godël’s paper, and introduced a new one:p ✷p¬α ✷¬αα → β ✷(α → β) (i.e. α 3 β)α ∨ β α ∨ βα ∧ β α ∧ β.It is this translation that inspired Kripke [1965a] to derive his semantics for intuitionisticlogic from his model theory for S4 (see section 7.6).Another significant result of the 1948 paper is that S5 is characterised by theclass of all closure algebras in which each closed element is also open. Structuresof this kind were later dubbed monadic algebras by Halmos in his study of thealgebraic properties of quantifiers [Halmos, 1962]. The connection is natural: themodalities ✷ and ✸ have the same formal properties in S5 as do the quantifiers∀ and ∃ in classical logic. The polyadic algebras of Halmos and the cylindricalgebras of Tarski and his co-researchers [Henkin et al., 1971] have a family ofpairwise commuting closure operators for which each closed element is open.Any Boolean algebra can be made into a monadic algebra by defining C0 = 0and otherwise Cx = 1. These are the simple 19 monadic algebras. Let A n bethe simple monadic algebra defined on the finite Boolean algebra with n atoms,viewed as a matrix with only 1 designated. Then S5 is characterised by the set ofall these A n ’s. This was shown by Schiller Joe Scroggs in his [1951], written as aMasters thesis under McKinsey’s direction, whose analysis established that everyfinite monadic algebra is a direct product of A n ’s. Scroggs used this to prove thateach proper extension of S5 is equal to the logic characterised by some A n , andso has a finite characteristic matrix. By “extension” here is meant any logic thatincludes all S5-theorems and is closed under the rules of uniform substitution forvariables and detachment for material implication. Scroggs was able to show fromthis characterisation that any such extension of S5 is closed under the Necessitationrule as well, and so is a normal logic.Another notable paper on S5 algebras from this era is [Davis, 1954], based on a1950 doctoral thesis supervised by Garrett Birkhoff. This describes the correspondencebetween equivalence relations on a set and S5 operations on its powersetBoolean algebra; a correspondence between algebras with two S5 operations and19 In the technical algebraic sense of having no non-trivial congruences.