The Logic of Objects

Abstract

Traditional assertoric formal logic — in contrast to Aristotelian syllogistic — usually introduces into its language, apart from so-called positive terms, also negative ones. So it employed the functor of nominal negation, ‘non’, defined with respect to the partial ordering relation described by the functor ‘every... is...’. Inasmuch as the Aristotelian syllogistic (as an inferential system) can be successfully reconstructed as a whole as a theory of partial orderings, as demonstrated by A. Mostowski (1948),2 traditional formal logic with negative terms cannot be, with respect to the same ordering, extended into a theory of Boolean algebras without running counter to colloquial language and the philosophical tradition. If the relationship between objects described by the functor ‘every... is...’ had a concept of ‘object’ as its greatest element — for according to the theory of objects ‘everything is an object’ — then the complement of the last element ‘a non-object’ would be the first element. Such a ‘non-object’, being an object (being ‘everything’ at the same time), would be a contradictory being. With reference to the classical distinction between categorial and transcendental concepts we shall call the consistent beings (as distinguished from the concept of all the objects) — the categorial objects. As a consequence of this we shall be defending two ideas: 1. that treating the word ‘is’ as a synonym of the expression ‘every...is...’ has a long tradition and is solidly grounded in colloquial language; and 2. that traditional logic is a theory of not one but of two orderings: of the partial ordering between objects described by the word ‘is’, and of an other partial ordering, between concepts, defined by the expression ‘the concept... is included in the concept...’.