A Study of Some Tense Logics by Gentzen's Sequential Method

Gentzen-style formulations of several fundamental modal logics like T9 S4, S5, etc. are well-known now. See, e.g., Ohnishi and Matsumoto [6], Sato [9], Zeman [10], etc. Especially, Sato has established a close relationship between Gentzen-style formulations of modal calculi and Kripke-type semantics in a decisive way. By the way, there is a strong analogy between classical tense logics and modal logics, which is also well-known. Indeed many techniques originally developed in modal calculi have been applied fruitfully to tense logics. For example, Gabbay [1] has used the so-called Lemmon-Scott or Makinson method to establish the completeness of many tense logics. The main objective of the present paper is to present Gentzen-style formulations of some fundamental tense logics, say, Kt and Kt4, and then to prove the completeness of these logics with due regard to Gentzen-style formulations after the manner of Sato. Since the completeness of Kt and Kt4 is well-known, our main concern here rests in the relationship between our Gentzen-style systems and the ordinal semantics of tense logics. Roughly speaking, traditional tense logics may be regarded as modal logics with two necessity-like operators, say, G and H. However, we will see that the relationship of these two operators is much subtler than that of so-called bimodal logics.