Logical consequence in modal logic. II. Some semantic systems for ${\rm S}4$.

Abstract

1 This paper is a continuation of the investigations reported in Corcoran and Weaver [1] where two logics j£Π and -CDD, having natural deduction systems based on Lewis's S5, are shown to have the usually desired properties (strong soundness, strong completeness, compactness). As in [1], we desire to treat modal logic as a clean natural deduction system with a conceptually meaningful semantics. Here, our investigations are carried out for several S4 based logics. These logics, when regarded as logistic systems (cf. Corcoran [2], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be linked. Some of the results of [l] are presupposed here and the more obvious definitions will not be repeated in detail. We consider the logics -Cl, -C2, and -£3. These logics share the same language (^DD) and deductive system (Δ'DD) but each has its own semantics system (Σl, Σ2, Σ3). Σl is an extension of the Kripke [3] semantics for S5 as modified in [l], Σ2 is largely due to Makinson [5], and Σ3 is due to Kripke [4]. -C DD is the usual modal sentential language with D, ~ and 3 as logical constants (see 2 below). Δ'DD (see 3 below), a modification of the natural deduction system given in [l], permits proofs from arbitrary sets of premises. For S a set of sentences and A & sentence, Sv-A means that A is provable from S, i.e., there is a proof (in Δ'DD) of A whose premises are among the members of S. ([-A means S\-A where S is empty.) If S\-A, we sometimes say that the argument (S, A) is demonstrable and when, in addition, S is empty we say that A is provable.