MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

36 Robert Goldblattbetween worlds as the basis of a model theory.Kripke had been introduced to Beth by Haskell B. Curry, who wrote to Bethon 24 January 1957 thatI have recently been in communication with a young man in Omaha Nebraska,named Saul Kripke. . . . This young man is a mere boy of 16 years;yet he has read and mastered my Notre Dame Lectures and writes me letterswhich would do credit to many a professional logician. I have suggestedto him that he write you for preprints of your papers which I have alreadymentioned. These of course will be very difficult for him, but he appears tobe a person of extraordinary brilliance, and I have no doubt something willcome of it. 34The Notre Dame Lectures of [Curry, 1950] presented a number of deductive systemsof modal logic, including one equivalent to Lewis’s S4 for which a cut eliminationtheorem was demonstrated in [Curry, 1952]. Other such sources that wereinfluential for Kripke included the McKinsey–Tarski papers and the paper of Lemmon[1957] which showed how to axiomatize the Lewis systems in the style ofGödel.In late 1958 Kripke entered Harvard University as an undergraduate, and encountereda philosophical environment that was hostile to modal logic. He wasadvised to abandon the subject and concentrate on majoring in mathematics.This caused the evident delay in publication of his work until the appearance ofthe major articles of 1963 and 1965.Looking back over the intervening decades we see the strong influence of Kripke’sideas on many areas of mathematical logic, ranging across the foundations ofconstructive logic and set theory, substructural logics (including relevance logic,linear logic), provability logic, the Kripke-Joyal semantics in topos theory andnumerous logics of transition systems in theoretical computer science.A proposition is defined in [Kripke, 1963a] to be a function from worlds to truthvalues, while in [1963b] an n-ary predicate letter is modelled as a function fromworlds to n-ary relations. Those definitions formed a cornerstone of Montague’sapproach to intensional logic, 35 and stimulated the substantial development of formalsemantics for natural languages in the theories of Montague [1974], Cresswell[1973], Barwise [1989] and others. Kripke’s models, and his intuitive descriptions ofthem, also stimulated many philosophical and formal investigations of the natureof possible worlds, and the questions of existence and identity that they generate(see [Loux, 1979]).5 THE POST-KRIPKEAN BOOM OF THE SIXTIESThe 1960’s was an extraordinary time for the introduction of new model theories.At the beginning of the decade Abraham Robinson created nonstandard analy-34 Quoted from [de Jongh and van Ulsen, 1998–1999, pp. 290–291].35 As acknowledged in several places, e.g. [Montague, 1970, fn. 5].