MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 19He defined a truth as being necessary when its opposite implies a contradiction, andalso said that there are as many worlds as there are things that can be conceivedwithout contradiction (see [Mates, 1986, pp. 72–73, 106–107]).This way of speaking has provided the motivation and intuitive explanationfor a mathematical semantics of modality using relational structures that are nowoften called Kripke models. A formula is assigned a truth-value relative to eachpoint of a model, and these points are thought of as being possible worlds or statesof affairs.An account will now be given of the contribution of Saul Kripke, followed by asurvey of some of its “anticipations”.4.1 Kripke’s Relatively Possible WorldsKripke’s first paper [1959a] on modal logic gave a semantics for a quantificationalversion of S5 that included propositional variables as the case n = 0 of n-arypredicate variables. A complete assignment for a formula α in a non-empty set Dwas defined to be any function that assigns an element of D to each free individualvariable in α, a subset of D n to each n-ary predicate variable occurring in α, and atruth-value (⊤ or ⊥) to each propositional variable of α. A model of α in D is a pair(G, K), where K is a set of complete assignments that all agree on their treatmentof the free individual variables of α, and G is an element of K. Each member Hof K assigns a truth value to each subformula of α, by induction on the rules offormation for formulas. The truth-functional connectives and the quantifiers ∀, ∃behave as in standard predicate logic, and the key clause for modality is thatH assigns ⊤ to ✷β iff every member of K assigns ⊤ to β.A formula α is true 21 in a model (G, K) over D iff it is assigned ⊤ by G; valid overD iff true in all of its models in D; and universally valid iff valid in all non-emptysets D.An axiomatisation of the class of universally valid formulas was given, withthe completeness proof employing the method of semantic tableaux introduced in[Beth, 1955]. It was then observed that for purely propositional logic this couldbe turned into a truth table semantics. A complete assignment becomes just anassignment of truth values to the variables in α, i.e. a row of a truth table, anda model (G, K) is just a classical truth table with some (but not all) of the rowsomitted and G some designated row. Formula ✷β is assigned ⊤ in every rowif β is assigned ⊤ in every row of the table; otherwise it is assigned ⊥ in everyrow. The resulting notion of “S5-tautology” precisely characterises the theoremsof propositional S5, a result that Kripke had in fact obtained first, before, as heexplained in [1959a, fn. 4],aquaintance with Beth’s paper led me to generalize the truth tables to semantictableaux and a completeness theorem.21 Actually “valid in a model” was used here, but changed to “true” in [Kripke, 1963a].