MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

Mathematical Modal Logic: A View of its Evolution 55presentations of certain recursive functions that define a complete Π 1 1 set. Theformulas γ m describe structures that encode copies of the iterated powersets ω,P(ω), P(P(ω)),. . . . The proofs of these facts are reminiscent of the arithmetisationprocedures and expressibility results involved in Gödel’s incompleteness theorems,and graphically illustrate the expressive power of T. The facts themselves arequite contrary to the situation in first-order logic, where the logical consequencesof a given sentence are effectively enumerable, and no sentence with an infinitemodel is categorical.A logic L 1 is said to be reducible to a logic L 2 if there exists an L 2 -formulaδ and an effective transformation ψ of L 1 -formulas to L 2 -formulas such that forevery collection Γ ∪ {α} of L 1 -formulas,Γ |= α iff {δ} ∪ {ψ(γ) : γ ∈ Γ } |= ψ(α).This definition captures the idea that L 1 can be regarded as a fragment of the logicL 2 , and is motivated by a notion of interpretation of one first-order theory in anotherthat appears in [Shoenfield, 1967]. Here δ may be thought of as describing acertain structure, with ψ(γ) asserting that γ is valid in that structure. In [Thomason,1974b] it is shown that tense logic T is reducible to modal logic M. Theformula δ used for this has the property that for any T-structure S = (K, R −1 , R)there is an M-structure S ′ that contains within it definable copies of (K, R) and(K, R −1 ) in such a way that “P ” statements about S can be interpreted as “✸”statements about S ′ . Applying this reduction to the results about T from [1975b],Thomason concludes that there is an M-formula whose set of logical consequencesis a complete Π 1 1 set.The full monadic second-order theory S of a binary predicate is shown to bereducible to M in [Thomason, 1975c]. For this purpose the logic T n of n temporalorderings is introduced. It has n pairs of modalities P 1 , F 1 , . . . , P n , F n , and structureshaving n binary relations and their inverses to interpret these connectives. Itis shown that for n > 1, T n is reducible to T n−1 . Since reducibility is a transitiverelation, it follows that each T n is reducible to T (= T 1 ), and hence reducible toM. This is then applied to prove the reducibility of S. The argument involvesdefining a T 15 -formula δ with the property that for each frame S = (K, R) thereis a model of δ with 15 temporal orderings that includes within it definable copiesof S; the powerset P(K); the membership relation from K to P(K); the set of all(codes for) S-formulas, the set of all assignments in K and P(K) to the individualand set variables of S; and the satisfaction relation between S-formulas andassignments in S as a second-order model. This leads to a reduction of S to T 15 ,which can then be combined with the reduction of T 15 to M to give the desiredresult. Thomason concludes thatthe logical consequence relation of propositional modal logic (with the Kripkerelational semantics) is as complex as it could possibly be.