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Algebraic semantics for S20 and necessitated extensions

Algebraic techniques are used to show that Feys' system S2°(c/. [1]) and certain necessitated extensions of S2°, such as Lewis' systems S2 and S3, have the finite model property, and accordingly are decidable. Representation theorems are then used to establish set-theoretical semantics for the modal systems studied. Where the results obtained are not new they improve on earlier results (such as those of Lemmon in [3]) in two respects; first they provide direct algebraic treatments of the systems, and second they furnish better semantical results (see the discussion of theorem J for S2). The techniques used however follow those of McKinsey (in [4]) and Lemmon (in [2] and [3]). Since it is now known that these techniques do not work for all necessitated extensions of S2°, a somewhat piecemeal approach is inevitable. Weak results are also obtained for Feys' system Sl° and Lewis' system SI (for details of these systems see [1]). The sentential systems studied are of interest not so much as systems containing a viable necessity operator *Π', but as intensional logics which axiomatise epistemic or other operators. For instance S2° can be interpreted as an epistemic logic such that 'D' reads 'it is believed reasonably that', and S2 as an epistemic logic where Π' reads 'it is known that'. The set-theoretical semantics established are however independent of these epistemic interpretations.