MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

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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
Mathematical Modal Logic: A View of its Evolution 47all formulas are theorems. But it is not itself inconsistent, because it is satisfiedby the algebra which consists of all the finite and cofinite subsets of the structure(ω,
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Page 82 and 83: 82 Robert Goldblatt7.7 Modal Logic
Page 84 and 85: 84 Robert GoldblattThis abstracts t
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