Relevant Analytic Tableaux

Tableau formulations are given for the relevance logics E (Entail? ment), R (Eelevant implication) and RM (Mingle). Proofs of equivalence to modus-ponens-based formulations are via" left-handed" Gentzen sequenzen-kalk?le. The tableau formulations depend on a detailed analysis of the structure of tableau rules, leading to certain "global requirements". Eelevanceis caught by the requirement that each node must be "used"; modality is caught by the requirement that only certain rules can "cross a barrier". Open problems are discussed. In this paper1 we present a tableau-style analysis of the implication negation fragments of the principal relevant logics E, R and RM of [1] as well as the classical propositional calculus TV. This analysis, although similar in certain respects to the one given by Smullyan in [32] for TV, differs substantially in that it is purely proof theoretic in character, as opposed to that of Smullyan, which is semantical. This is then, as far as we know, the first time that a non-semantically based tableau-style analysis has been given for a set of logics. Therefore this paper seems like an appropriate place to distinguish between these two different tableau-style analyses. Henceforth the kinds of structures studied in a semantically based tableau-style analysis will be called analytic semantic tableaux, while the kinds of structures studied in a proof theoretically based tableau-style analysis will be called simply analytic tableaux. Further, logical systems in which the theoremhood (validity) of formulas is de? termined by whether or not analytic tableaux (analytic semantic tableaux) constructed in a prescribed way from these formulas meet certain necess? ary and sufficient conditions will be called analytic tableau systems (ana? lytic semantic tableau systems). Hence the analytic tableau formulations (analytic semantic tableau formulations) of some arbitrary ogic S is simply 1 This paper was delivered by McEobbie at the 1976 Vacation School in Logic, held at the Victoria University of Wellington, New Zealand, from August 15-22, and was announced in [24]. Thanks for encouragement and many helpful discussions are due in particular to Dunn, Martin and Meyer and also to A.S. McEobbie, Mor tensen, Eennie, E. Eoutley and Eubenstein. Thanks also to the National Science Foundation for partial support of Belnap through Grant S0C71 03594 A04. 188 Michael A. McEobbie, Fuel D. Belnap, Jr , an analytic tableau system (analytic semantic tableau system) that is provably equivalent to S.2 In what follows, by 'tableau' we shall be referring to the proof theoretic species and not to the semantic one unless otherwise indicated. All conventions are adopted from [1]. The systems E, R, and RM are defined in ?27 of [1], and their implication-negation fragments E~^, R~>, and BM; are defined in ?14 of [1]. The system TV^ is the implication negation fragment of TV.