Systems of Demonstrative Syllogistic

  • Piotr Kulicki The John Paul II Catholic University of Lublin, Faculty of Philosophy
Keywords: syllogistic; proof; rejected axiomatization


Aristotle in Analytica Posteriora presented a notion of proof as a special case of syllogism. In the present paper the remarks of Aristotle on the subject are used as an inspiration for developing formal systems of demonstrative syllogistic, which are supposed to formalize syllogisms that are proofs. We build our systems in the style of J. Łukasiewicz as theories based on classical propositional logic. The difference between our systems and systems of syllogistic known from the literature lays in the interpretation of general positive sentences in which the same name occurs twice (of the form SaS). As a basic assumption of demonstrative syllogistic we accept a negation of such a sentence. We present three systems which differ in the interpretation of specific positive sentences in which the same name occurs twice (of the form SiS). The theories are defined as axiomatic systems. For all of them rejected axiomatizations are also supplied. For two of them a set theoretical  model is also defined. 


Arystoteles: Dzieła wszystkie, t. 1, Warszawa 1990.
Borkowski L.: Logika Formalna, Warszawa 1970.
Łukasiewicz J.: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, Clarendon Press, Oxford 1952.
McKinsley J.C.C.: The decision problem for some classes of sentences without quantifiers, „Journal of Symbolic Logic” 8 (1943), s. 61-76.
Pietruszczak A.: O logice tradycyjnej i rachunku nazw dopuszczającym podstawienia nazw pustych, „Ruch Filozoficzny” 44 (1987), nr 2, s. 158-166.
Słupecki J.: Uwagi o sylogistyce Arystotelesa, „Annales UMCS” 1 (1946), nr 3, s. 187-191.