MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

26 Robert GoldblattPrior’s article “Possible Worlds” [1962a, p. 37] gives a fuller exposition of thisU-calculus, saying “This whole symbolism I owe to C. A. Meredith”. He appliesan interpretation of the predicate U, suggested to him by P. T. Geach in 1960, 26as a relation of accessibility. Here is Prior’s account of that interpretation.Suppose we define a ‘possible’ state of affairs or world as one which can bereached from the world we are actually in. What is meant by reaching ortravelling to one world from another need not here be amplified; we mightreach one world from another merely in thought, or we might reach it moreconcretely in some dimension-jumping vehicle dreamed up by science-fiction(the case originally put by Geach), or we might reach it simply by the passageof time (one important sense of ‘possible state of affairs’ is ‘possible outcomeof the present state of affairs’). What I want to amplify here is the idea(the core of Geach’s suggestion) that we may obtain different modal systems,different versions of the logic of necessity and possibility, by making differentassumptions about ‘world-jumping’.Prior was the founder of tense logic (also known as temporal logic). He wanted toanalyse the arguments of the Stoic logician Diodorus Chronos, who had defined aproposition to be possible if it either is true or will be true. Prior conceived theidea of using a logical system with temporal operators analogous to those of modallogic, and thus introduced the connectivesFPGHit will be the case thatit has been the case thatit will always be the case thatit has always been the case that.Here F and P are “diamond” type modalities, with duals G and H respectively.In the paper “The Syntax of Time-Distinctions” [Prior, 1958] a propositional logiccalled the P F -calculus is defined. 27 It is a normal logic with respect to G and H,has the axioms Gp → F p, F F p → F p and F p → F F p, as well as an “interaction”axiom p → GP p and a Rule of Analogy allowing that from any theorem anothermay be deduced by replacing F by P and vice versa.This system is then interpreted into what Prior calls the l-calculus, a first-orderlanguage whose variables x, y, z range over dates, and which has a binary symboll taking dates as arguments, with the expression lxy being read “x is later thany”. 28 Variables p, q, r stand for propositions considered as functions of dates, withthe expression px being read “p at x”. The following interpretations are givenof propositional formulas, using an arbitrarily chosen date variable z to represent“the date at which the proposition under consideration is uttered”.26 This date is given in [Prior, 1962b, p. 140], where the acknowledgement of Meredith isrepeated once more.27 The contents of this paper are reviewed on [Prior, 1967, pp. 34–41].28 Prior notes that the structure of the calculus would be unchanged if l were read “is earlierthan”.