MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

28 Robert GoldblattThis proposal became a major programme for Prior. He used formulas like p ∧¬P p ∧ ¬F p which can be true at only one point of the linear series of moments,or instants. If M(p ∧ ¬P p ∧ ¬F p) is true at some time, the variable p must itselfbe true at exactly one instant and may be identified with that instant. Then theformula L(p → α) expresses that “it is the case at p that α”, and so if p and qare both such instance-variables, L(p → P q) asserts that it is true at p that it hasbeen q, i.e. p is later than q, and q is earlier than p.Systems having variables identified with unique instants or worlds are developedmost fully in the book of [Prior and Fine, 1977, p. 37], where Prior gives anemphatic statement of his metaphysical propensity:. . . I find myself quite unable to take ‘instants’ seriously as individual entities;I cannot understand ‘instants’, and the earlier-than relation that is supposedto hold between them, except as logical constructions out of tensed facts.Tense logic is for me, if I may use the phrase, metaphysically fundamental,and not just an artificially torn-off fragment of the first-order theory of theearlier-than relation.4.5 KangerA semantics is given by Stig Kanger in [1957b] for a version of modal predicatelogic whose atomic formulas are propositional variables and expressions of theform (x 1 , . . . , x n ) ε y, where n ≥ 1 and the x i and y are individual variables orconstants. The language included a list of modal connectives M 1 , M 2 , . . . .A notion of a system is introduced as a pair (r, V ) where r is a frame andV a primary valuation. Here r is a certain kind of sequence of non-empty setswhose elements provide values of individual symbols of various types. V is abinary operation that assigns a truth value V (r, p), belonging to {0, 1}, to eachpropositional variable p and frame r, as well as interpreting individual symbolsand the symbol ε in each frame in a manner that need not concern us. Thena “secondary” truth valuation T (r, V, α) is inductively specified, allowing eachformula α to be defined to be true in system (r, V ) iff T (r, V, α) = 1. For thispurpose each modality M i is assumed to be associated with a class R i of quadruples(r ′ , V ′ , r, V ), and it is declared thatT (r, V, M i α) = 1 iffT (r ′ , V ′ , α) = 1 for each r ′ and V ′ such thatR i (r ′ , V ′ , r, V )(so M i is a “box” type of modality).Kanger states the following soundness results. The theorems of the Feys–vonWright system T are valid (i.e. true in all systems) iff R i (r, V, r, V ) always holds.S4 is validated iff R i (r, V, r, V ) always holds and so does the conditionR i (r, V, r ′ , V ′ ) and R i (r ′′ , V ′′ , r, V ) implies R i (r ′′ , V ′′ , r ′ , V ′ ).S5 is validated iff the S4 conditions hold along with