MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

46 Robert Goldblatt6.1 IncompletenessA logic Λ is sound with respect to a class C of frames if every member of C is aΛ-frame, i.e. validates all Λ-theorems. By definition Λ is sound with respect to theclass F r(Λ) of all Λ-frames. In the converse direction, Λ is complete with respectto C if any formula that is valid in all members of C is a Λ-theorem. For example,every normal logic is complete with respect to C = {S Λ }, where S Λ = (K Λ , R Λ )is the canonical frame of Λ as defined in section 5.1. For if a formula is valid inS Λ , then it is true in the canonical model M Λ on S Λ , and so is a Λ-theorem.Whether or not Λ is sound with respect to S Λ is an important issue that will bediscussed in section 6.6.A logic Λ is characterised by a class C if it is both sound and complete withrespect to C. Λ is complete per se if it is complete with respect to some class C ofΛ-frames, in which case it is characterised by that C, as well as by the class F r(Λ)of all Λ-frames. It is important to recognise that a given logic may be characterisedby many different classes. For example, S4 is characterised by each of the class ofall quasi-orderings, the class of finite quasi-orderings, and the class of all partialorderings(but not the finite partial-orderings, which characterise S4Grz as we sawin section 5.3).Lemmon was sufficiently taken with the power of Kripke semantics to conjecturethat every normal logic is characterised by some class of relational frames [Lemmon,1977, p. 74]. It turned out that this was as far from the truth as it couldbe. Wim Blok showed that, in a manner which will be explained below, “most”logics Λ are not characterised by any class of frames, and hence are incompletein the sense that there exist formulas that are valid in all Λ-frames but are notΛ-theorems.The first example of an incomplete logic was devised by Steven Thomason[1972b], and is a readily described tense logic in Prior’s P F -language. In additionto a set of postulates for linearly-ordered frames it has the axiomsGp → F pP p → P (p ∧ ¬P p)GF p → F Gp.The first of these is valid in a frame (K, R) only if the “endless time” condition∀x∃y(xRy) is satisfied. The second axiom is equivalent to H(Hp → p) → Hp,which is Segerberg’s axiom W for the past modality H. Its validity entails thatR is irreflexive. Thus if x 0 is a point in any frame validating the first two axioms,{y : x 0 Ry} is an irreflexive linear ordering with no last element. Interpreting pas a set such that both it and its complement are unbounded in {y : x 0 Ry} thengives a model on the frame that falsifies the third axiom at x 0 . In this model thetruth-value of p alternates forever over time.Thus Thomason’s logic is not valid on any frame whatsoever! In other words itis indistinguishable in terms of frame-validity from the inconsistent logic in which