Pure Modal Logic of Names and Tableau Systems
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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).
KeywordsPure modal logic of names Semantics Tableaus Decidability Modal syllogistic
- 1.Aristotle, Prior Analytics, Hackett, Indianapolis, 1989. Translated with introduction notes and commentary by R. Smith.Google Scholar
- 2.Jarmużek, T., Tableau system for logic of categorial propositions and decidability, Bulletin of the Section of Logic 37:223–231, 2008.Google Scholar
- 3.Jarmużek, T., and A. Pietruszczak, Decidability methods for modal syllogisms, in A. Indrzejczak, J. Kaczmarek, and M. Zawidzki (eds.), Trends in Logic XIII, Łódź University Press, Łódź, 2014, pp. 95–112.Google Scholar
- 4.Johnson, F., Models for modal syllogisms, Notre Dame Journal of Formal Logic 30(2):271–284, 1989. https://doi.org/10.1305/ndjfl/1093635084
- 7.McCall, S., Aristotle’s Modal Syllogisms, Nort-Holland, Amsterdam, 1993.Google Scholar
- 8.Pietruszczak, A., O logice tradycyjnej i rachunku nazw dopuszczającym podstawienia nazw pustych (On traditional logic and calculi of names allowing substitutions of empty names), Ruch Filozoficzny 44:158-166, 1987.Google Scholar
- 9.Pietruszczak, A., Standardowe rachunki nazw z funktorem Leśniewskiego (Pure calculi of names with Leśniewski’s functor), Acta Universitatis Nicolai Copernici, Logika I:5–29, 1991.Google Scholar
- 10.Pietruszczak, A., Bezkwantyfikatorowy rachunek nazw. Systemy i ich metateoria (Quantifier-free Calculus of Names. Systems and their Metatheory), Wydawnictwo Adam Marszałek, Toruń, 1991.Google Scholar
- 11.Pietruszczak, A., Cardinalities of models for pure calculi of names, Reports on Mathematical Logic 28:87–102, 1994.Google Scholar
- 12.Pietruszczak, A., Cardinalities of models for monadic predicate logic (with equality and individual constants), Reports on Mathematical Logic 30:49–64, 1996.Google Scholar
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