An extension of the traditional logic containing the elementary ontology and the algebra of classes

Abstract

In this paper the term "traditional logic" denotes the system of Aristotelian syl? logistic ? in the axiomatic approach presented by J. Lukasiewicz in the p;psr [4] ? enriched by the nominal negation. Besides the laws of the square of opposition, the law of conversion and the categorical syllogisms there are the laws of obversion, con? traposition and inversion of propositions in this system1. The paper deals with some axiomatic extension of traditional logic. Its main aim is arriving at a calculus of names in which all of the known laws of the categorical propositions are preserved and which would admit the introduction of notions cor? responding semantically to the relation z (... is ...) of St. Lesniewski's ontology, empty and universal sets and such operations of the algebra of classes as addition, multiplication and subtraction of sets. The set of notions employed in syllogistic is quite meager and confinement to four categorical propositions limits the possibilities of making use of it, the more so as they evidently are not the most important forms of propositions appearing in science and everyday usage. Attempts of introduction of such notions as the empty set, the product of sets, the sum of sets etc. to the traditional logic meet with essential obstacles which could be characterized as follows. The algebra of classes is the contemporary logical calculus the traditional logic is closest to. The analogy concerns the Aristotelian a and the set-theoretical relation of inclusion of sets ? ? first of all. There are such common properties in them as reflexivity, asymmetry and transitivity. The relations of equality defined by means of a and ? satisfy the conditions of reflexivity, symmetry and transitivity, and ? to some extent ? the law of extensionality. Besides, the notions of the nominal ne? gation and the complement of a set are characterized in the similar way in both cal? culi.2 There are, however, essential differences between these calculi which become evident when the so called "Problem of empty and universal names" (resp. "Problem of the empty set (A) and the universal set (V)") is dealt with. It is known that the no