MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 41every normal extension of S4.3 has the finite model property.Proofs of this result using relational models were subsequently devised by Kit Fine[1971] and Håkan Franzén (see [Segerberg, 1973]). Fine gave a penetrating analysisof finite S4.3 models to establish that there are exactly ℵ 0 normal extension ofS4.3, all of which are finitely axiomatisable and hence decidable. Segerberg [1975]proved that in fact every logic extending S4.3 is normal.The indistinguishability of rational and real time is overcome by passing to themore powerful language of Prior’s P F -calculus for tense logic (section 4.4). Amodel structure for this language would in principle have the form (K, R P , R F ),with R P and R F being binary relations on K interpreting the modalities P andF . But for modelling tense logic, with its interaction principles p → GP p andp → HF p, the relations R P and R F should be mutually inverse. Thus we continueto use structures (K, R) with the understanding that what we really intend is(K, R −1 , R). For linearly ordered structures, the ability of the two modalitiesto capture properties “in each direction” of the ordering produces formulas thatexpress the Dedekind continuity of R, a fact that was first realised by Montagueand his student Nino Cocchiarella. 41Bull applied his algebraic methodology in the [1968] paper to give completeaxiomatisations of the tense logics characterised by each of the strictly linearlyordered structures (Z,