MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

40 Robert Goldblattinterpreting formulas as sets of sequences of truth values. In effect he was dealingwith the complex closure algebra Cm(ω, ≤), where ω = {0, 1, 2, . . .} is the set ofnatural numbers viewed as a sequence of moments of time. The question becameone of identifying the logic that is characterised by this algebra, or equivalentlyby the model structure (ω, ≤). Prior called this logic D. 39In 1957 Lemmon observed that D includes the formula✷(✷p → ✷q) ∨ ✷(✷q → ✷p),which arises from the intuitionistically invalid formula (p → q) ∨ (q → p) byapplying the translation of [McKinsey and Tarski, 1948]. Lemmon’s formula istherefore not an S4-theorem, and when added as an axiom to S4 produces a systemcalled S4.3. In 1958 Michael Dummett showed that the formula✷(✷(p → ✷p) → ✷p) → (✸✷p → ✷p)also belongs to D, and then Prior [1962b] pointed out that this is due to thediscreteness of the ordering ≤ on ω: if time were a continuous ordering thenDummett’s formula would not be valid, but Lemmon’s would. In fact the propertyused by Prior to invalidate Dummett’s formula was density (between any twomoments there is a third) rather than continuity in the sense of Dedekind (no“gaps”).Kripke showed in 1963 that D is exactly the normal logic obtained by addingDummett’s formula as an axiom to S4.3. His proof, using semantic tableaux,is unpublished. Dummett conjectured to Bull that taking time as “continuous”would yield a characterisation of S4.3. 40 Bull proved this in his paper [1965] which,in addition to giving an algebraic proof of Kripke’s completeness theorem for D,showed that S4.3 is characterised by the complex algebra of the ordering (R + , ≤)of the positive real numbers. He noted that R + could be replaced here by thepositive rationals, or any linearly ordered set with a subset of order type ω 2 . Inparticular this shows that propositional modal formulas are incapable of expressingthe distinction between dense and continuous time under the relational semantics.Bull made effective use of Birkhoff’s fundamental decomposition [Birkhoff, 1944]of an abstract algebra into a subdirect product of subdirectly irreducible algebras.Birkhoff had observed that subdirectly irreducible closure algebras are wellconnectedin the sense of [McKinsey and Tarski, 1944] (see section 3.2). Applyingthis to Lindenbaum algebras shows that every normal extension of S4 is characterisedby well-connected closure algebras, and in the case of extensions of S4.3the closed (Cx = x) elements of a well-connected algebra are linearly ordered.Bull used this fact, together with the strategy of McKinsey’s finite algebra construction,to build intricate embeddings of finite S4.3-algebras into Cm(R + , ≤) orCm(ω, ≤). He later refined this technique to establish in [Bull, 1966] one of themore celebrated meta-theorems of modal logic:39 The letter D later became a label for the system K+(✷p → ✸p), or equivalently K+✸⊤,because of its connection with Deontic logic.40 See [Prior, 1967, ch. II] as well as [Bull, 1965] for this historical background.