Pure Modal Logic of Names and Tableau Systems

By a pure modal logic of names (PMLN) we mean a quantifier-free formulation of such a logic which includes not only traditional categorical, but also modal categorical sentences with modalities de re and which is an extension of Propositional Logic. For categorical sentences we use two interpretations: a “natural” one; and Johnson and Thomason’s interpretation, which is suitable for some reconstructions of Aristotelian modal syllogistic (Johnson in Notre Dame J Form Logic 30(2):271–284, 1989; Thomason in J Philos Logic 22(2):111–128, 1993 and J Philos Logic 26:129–141, 1997. In both cases we use Johnson-like models (1989). We also analyze different kinds of versions of PMLN, for both general and singular names. We present complete tableau systems for the different versions of PMLN. These systems enable us to present some decidability methods. It yields “strong decidability” in the following sense: for every inference starting with a finite set of premises (resp. every syllogism, every formula) we can specify a finite number of steps to check whether it is logically valid. This method gives the upper bound of the cardinality of models needed for the examination of the validity of a given inference (resp. syllogism, formula).