Logical consequence in modal logic: Natural deduction in ${\rm S}5$.

Abstract

This paper presents a (modal, sentential) logic -£DD which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator (D) which expresses certain kinds of necessity. The language of ^.DD is LDD, the smallest set of formulas containing a countably infinite set of sentential constants and closed under forming Πφ, ~φ, and (φ z> ψ) from any formulas φ and ψ already in the set. This is essentially the language of S5. The semantics of -£DD is essentially Kripke's semantics for S5. In the semantics for .£DD, however, an interpretation of LDD is defined as an ordered pair (a, P) where a is an ordinary truth-value interpretation of the sentential constants and P is a set of such interpretations with a e P. (α, P) is a model of φ if φ is true under (α, P) and {a, P) is a model of S c LDD if each ψ in S is true under (a,P). This permits logical consequence to be defined: for S c LDD, φe LDD, φ is a logical consequence of S(S\=φ) iff every model of S is a model of φ. The deductive system consists of natural rules permitting proofs from arbitrary sets of premises and we let S\-φ mean that there is a proof of φ from the premises S. Let LD be the sublanguage of LDD containing all formulas of LDD devoid of iterated or nested D. Let -£D be the restriction of ^DD to LD, i.e. £π is the logic with language LD such that, for S <Ξ LD and φe LD, S \=φ relative to -£D iff S 1= φ in «£DD and S hφ relative to -£D iff there is a proof in -£DD of φ from S containing only formulas of LD. Strong completeness (S t= φ implies S \-φ) for -£DD is proved from the following three lemmas: (1) strong soundness (Sϊr-φ implies S t= φ) for -£DD, (2) strong completeness for -£D and (3), a reduction theorem to the effect that every formula in LDD is provably equivalent in -£DD to a formula in LD. From these results, together with Kripke's weak soundness and weak completeness results for S5, it follows that φ is a theorem of S5 iff φ is a theorem Of ZDD.