MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 27F pP pGpHp∃x(lxz ∧ px)∃x(lzx ∧ px)∀x(lxz → px)∀x(lzx → px).Prior observes that the interpretations of some theorems of the P F -calculus areprovable in the l-calculus just from the usual axioms and rules for quantificationallogic. This applies to any P F -theorem derivable from the basis for normal logicstogether with the interaction axiom p → GP p and the rule of Analogy. He thenstates that the interpretation of Gp → F p requires for its proof the axiom ∃x lxz(“infinite extent of the future”), and that F F p → F p depends similarly on transitivity:lxy → (lyz → lxz), while F p → F F p depends on the density conditionlxz → ∃y(lxy ∧ lyz).The modality M of possibility is given a temporal reading by defining Mp to bean abbreviation for p∨F p∨P p, i.e. “p is true at some time, past present or future”.This makes the dual Lp equivalent to p ∧ Gp ∧ F p, “at all times, p”. Prior notesthat to derive the S5-principle M¬Mp → ¬Mp, which is “clearly a law” underthis interpretation of M, requires trichotomy: x = y ∨ lxy ∨ lyx. His explorationshere are quite tentative. For instance he defines asymmetry: lxy → ¬lyx, butmakes no use of it, and he fails to note that the S4-principle MMp → Mp alsodepends on trichotomy and not just transitivity.Why did Prior give such unequivocal credit to Meredith for the 1956 U-calculus?The puzzle about this is that his paper on the l-calculus, although published in1958, was presented much earlier, on 27 August 1954, as his Presidential Addressto the New Zealand Philosophy Congress at the Victoria University of Wellington.Perhaps he was crediting Meredith with the extension of the symbolism to modallogic as he understood it, i.e. the logic of necessity and possibility, as distinct fromtense logic. The l-calculus was intended to describe a very specific situation: anordered system of dates or moments in time that forms an “infinite and continuouslinear series” [1958, p. 115]. In the absence of any corresponding interpretation ofthe U-predicate, the purely formal application of the symbolism by Meredith mayhave been seen by Prior as a significant advance.Prior made much use of l and U calculi in his papers and books on tense logic.He did not however pursue their implicit relational model theory, and would nothave thought it philosophically worthwhile to do so. Although he described thel-calculus as “a device of considerable metalogical utility” [1958, p. 115], he wenton to deny that the interpretation of the P F -calculus within the l-calculus hasany metaphysical significance as anexplanation of what we mean by “is”, “has been” and “will be”.On the contrary he proposed that what was needed was an interpretation in thereverse direction [1958, p. 116]:the l-calculus should be exhibited as a logical construction out of the P F -calculus.