MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

6 Robert Goldblattthe expositors of the algebra of logic have not always taken pains to indicatethat there is a difference between the algebraic and ordinary meanings ofimplication.He observed that the algebraic meaning, as used in the Principia Mathematica ofRussell and Whitehead, leads to the “startling theorems” that a false propositionimplies any proposition, and a true proposition is implied by any proposition.These so-called paradoxes of material implication take the symbolic forms¬α → (α → β)α → (β → α).For Lewis the ordinary meaning of “α implies β” is that β can be validly inferred 6from α, or is deducible 7 from α, an interpretation that he considered was notsubject to these paradoxes. Taking “α implies β” as synonymous with “eithernot-α or β”, he distinguished extensional and intensional meanings of disjunction,providing two meanings for “implies”. Extensional disjunction is the usual truthfunctional“or”, which gives the material (algebraic) implication synonymous with“it is false that α is true and β is false”. Intensional disjunctionis such that at least one of the disjoined propositions is “necessarily”true. 8That reading gives Lewis’ “ordinary” implication, which he also dubbed “strict”,meaning that “it is impossible (or logically inconceivable 9 ) that α is true and β isfalse”.The system of Lewis’s book A Survey of Symbolic Logic [1918] used a primitiveimpossibility operator to define strict implication. This later became the systemS3 of [Lewis and Langford, 1932], which introduced instead the symbol ✸ forpossibility, but Lewis decided that he wished S2 to be regarded as the correctsystem for strict implication. The systems were defined with negation, conjunction,and possibility as their primitive connectives, but he made no use of a symbol forthe dual combination ¬✸¬. 10 For strict implication the symbol 3 was used,with α 3 β being a definitional abbreviation for ¬✸(α ∧ ¬β). Strict equivalence(α = β) was defined as (α 3 β) ∧ (β 3 α).Here now are definitions of S1–S5 in Lewis’s style, presented both to facilitatediscussion of later developments and to convey some of the character of his6 [Lewis, 1912, p. 527]7 [Lewis and Langford, 1932, p. 122]8 [Lewis, 1912, p. 523]9 [Lewis and Langford, 1932, p. 161]10 The dual symbol ✷ was later devised by F. B. Fitch and first appeared in print in 1946 in apaper of R. Barcan. See footnote 425 of [Hughes and Cresswell, 1968, fn. 425].