MATHEMATICAL MODAL LOGIC: A VIEW OF ITS EVOLUTION

58 Robert Goldblattfrom a suitably chosen ultrapower of any given frame S onto ExS, yielding theobservation that in generalExC ⊆ HPwC. (2)The proof of this requires the choice of a sufficiently saturated ultrapower of S[Goldblatt, 1989, §3.6] and is motivated by a model construction of [Fine, 1975b]that is discussed further in the next section.Duality can be used to bring methods of universal algebra to bear on relationalsemantics. A notable example is the problem of characterising classes of the formF r(Λ), the class of all frames validating a set Λ of modal formulas. The questionof when F r(Λ) is elementary was discussed in section 6.3. It is natural to ask,conversely, for conditions under which a given elementary class of frames is equalto the class F r(Λ) for some Λ. The following answer was given in [Goldblattand Thomason, 1975], where the Ex construction was first introduced (see also[Goldblatt, 1993, 1.20.6], [Goldblatt, 1989, 3.7.6(2)]).If C is an elementary class of frames, then C is equal to F r(Λ) for someset Λ of modal formulas if, and only if,1. C is closed under disjoint unions, p-morphic images and subframes;and2. the complement of C is closed under canonical extensions, i.e.ExS ∈ C implies S ∈ C.The proof applies the Birkhoff–Tarski analysis of varieties to the variety generatedby CmC, and uses the construction for (2) above to show that if C is elementaryand closed under p-morphic images then it is closed under canonical extensions.Duality theory has been developed for arbitrary relational structures and BAO’sby using suitable generalisations of p-morphisms and subframes, called “bounded”morphisms and “inner” substructures (Goldblatt [1989; 1995]). This provides algebraicand relational semantics for polymodal languages having n-ary connectiveswhich generate formulas ✷(α 1 , . . . , α n ) for n > 1. Most of the ideas and resultswe have discussed about completeness, canonicity, elementarity, class operationsetc. carry over to this broader context and apply to cylindric algebras, relationalgebras and other kinds of BAO’s in addition to modal algebras. This revealsthat, mathematically, much of modal semantics is just the case n = 1 of a broaderstructural theory of finitary operators on lattices. A survey of this general theoryis given in [Goldblatt, 2000].If Λ is a normal logic, then the class V (Λ) of modal algebras that satisfy all Λ-theorems is a variety. Algebraic constructions in V (Λ) provide tools for studyingmetalogical questions about Λ, such as whether it fulfills analogues of the BethDefinability Theorem and the Craig Interpolation Theorem. This is related toamalgamation properties of algebras in V (Λ), as has been shown by Larisa Maksimova,whose article of [1992] gives an account of the subject and further referencesto the literature.