An embedding of classical logic in S4

Abstract

There are well-known embeddings of intuitionistic logic into S4 and of classical logic into S5. In this paper we give a related embedding of (first order) classical logic directly into (first order) S4, with or without the Barcan formula. If one reads the necessity operator of S4 as ‘provable’, the translation may be roughly stated as: truth may be replaced by provable consistency. A proper statement will be found below. The proof is based ultimately on the notion of complete sequences used in Cohen's technique of forcing [1], and is given in terms of Kripke's model theory [3], [4].