MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 37sis by constructing models of the higher-order theory of the real numbers. ThenPaul Cohen’s invention of forcing revolutionized the study of models of set theory,and freed up the log-jam of questions that had been building since the timeof Cantor. Kripke related forcing to his models of Heyting’s predicate calculus,and Dana Scott and Robert Solovay re-formulated it as the technique of Booleanvaluedmodels. Scott then replaced “Boolean-valued” by “Heyting-valued” andextended the topological interpretation from intuitionistic predicate logic to intuitionisticreal analysis. F. William Lawvere’s search for categorical axioms forset theory and the foundations of mathematics and his collaboration with MilesTierney on axiomatic sheaf theory culminated at the end of the decade in thedevelopment of elementary topos theory. This encompassed, in various ways, bothclassical and intuitionistic higher order logic and set theory, including the modelsof Kripke, Cohen, Scott, and Solovay, as well as incorporating the sheaf theory ofthe Grothendieck school of algebraic geometry. Scott’s construction of models forthe untyped lambda calculus in 1969 was to open up the discipline of denotationalsemantics for programming languages, as well as stimulating new investigations inlattice theory and topology, and further links with categorical and intuitionisticlogic.The introduction of Kripke models had a revolutionary impact on modal logicitself. Binary relations are much easier to visualise, construct, and manipulate thanoperators on Boolean algebras. They fall into many naturally definable classes thatcan be used to define corresponding logics. Here then were the tools that wouldenable an exhaustive investigation of the subject, and some important new ideaswere developed during this period.5.1 The Lemmon and Scott CollaborationPioneers in this investigation were John Lemmon and Dana Scott, who conductedan extensive collaboration. They planned to write a book called Intensional Logic,for which Lemmon had drafted some inital chapters when he died in 1966. Scottthen made this material available in a mimeographed form [Lemmon and Scott,1966] which was circulated informally for a number of years, becoming knownas the “Lemmon Notes”. Eventually it was edited by Scott’s student KristerSegerberg, and published as [Lemmon, 1977]. Scott also investigated broad issuesof intensional logic (individuals and concepts, possible worlds and indices,intensional relations and operators etc.) in discussion with Montague, Kaplan andothers. Some of his ideas were presented in [Scott, 1970]. His considerable influenceon the subject has been disseminated through the publications of Lemmonand Segerberg, and is also reported in [Prior, 1967] in relation to tense logic, andin a number of Montague’s papers.The relationship between modal algebras and model structures was first systematicallyexplored in Lemmon’s two part article [1966a; 1966b]. Here a modelstructure has the form S = (K, R, Q), with Q playing the role of the set of non-