MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 7approach. System S1 has the axioms 11(p ∧ q) 3 (q ∧ p)(p ∧ q) 3 pp 3 (p ∧ p)((p ∧ q) ∧ r) 3 (p ∧ (q ∧ r))((p 3 q) ∧ (q 3 r)) 3 (p 3 r)(p ∧ (p 3 q)) 3 q,where p, q, r are propositional variables, and the following rules of inference.• Uniform substitution of formulas for propositional variables.• Substitution of strict equivalents: from (α = β) and γ infer any formulaobtained from γ by substituting β for some occurrence(s) of α.• Adjunction: from α and β infer α ∧ β.• Strict detachment: from α and α 3 β infer β. 12System S2 is obtained by adding the axiom ✸(p ∧ q) 3 ✸p to the basis for S1.S3 is S1 plus the axiom (p 3 q) 3 (¬✸q 3 ¬✸p). S4 is S1 plus ✸✸p 3 ✸p, orequivalently ✷p 3 ✷✷p. S5 is S1 plus ✸p 3 ✷✸p.The axioms for S4 and S5 were first proposed for consideration as further postulatesin a paper of Oskar Becker [1930]. His motivation was to find axiomsthat reduced the number of logically non-equivalent combinations that could beformed from the connectives “not” and “impossible”. He also considered the formulap 3 ¬✸¬✸p, and called it the “Brouwersche axiom”. The connection withBrouwer is remote: if “not” is translated to “impossible” (¬✸), and “implies” toits strict version, then the intuitionistically acceptable principle p → ¬¬p becomesthe Brouwersche axiom.2.4 Gödel on Provability as a ModalityGödel in [1931] reviewed Becker’s 1930 article. In reference to Becker’s discussionof connections between modal logic and intuitionistic logic he wroteIt seems doubtful, however, that the steps here taken to deal with this problemon a formal plane will lead to success.He subsequently took up this problem himself with great success, and at the sametime simplified the way that modal logics are presented. The Lewis systems containall truth-functional tautologies as theorems, but it requires an extensive analysis11 Originally p 3 ¬¬p was included as an axiom, but this was shown to be redundant byMcKinsey in 1934.12 Lewis used the name “Inference” for the rule of strict detachment. He also used “assert”rather than “infer” in these rules.