Tableau methods of proof for modal logics

1 Introduction: In [1] Fitch proposed a new proof proceedure for several standard modal logics. The chief characteristic of this was the inclusion in the object language, of symbols representing worlds in Kripke models. In this paper we incorporate the device into a tableau proof system and it is seen that the resulting (propositional) proof system is highly analogous to a classical first order tableau system, with the modal operators behaving like quantifiers. Exploiting this similarity, a tableau completeness proof for first order logic directly becomes a Kripke completeness proof for modal logic, and Smullyan's fundamental theorem of quantification theory (a Herbrand-like theorem) [7] has its analog. Indeed, more than analogy is at work here; from an appropriate abstract point of view certain modal logics, first order classical and intuitionistic logic, and various infinitary logics may be treated simultaneously, an approach due to R. Smullyan and developed in a forthcoming monograph (see [8] for a preliminary version). We will treat only tableau proof systems and some familiarity with [7] is presumed. In addition to being metatheoretically interesting, specific tableau systems we give for S5, S4, T, B, DS4, DT, and K are quite simple to use. The extension of these systems to first order systems is straightforward , and is discussed briefly in the last section.