Intensional Semantics for Syllogistics: what Leibniz and Vasiliev Have in Common

Abstract

This article deals with an alternative interpretation of syllogistics, different from the classical (extensional) one: an intensional one, in which subject and predicate are not associated with a set of individuals (the extension of the concept) but a set of attributes (the content of the concept). The authors of the paper draw attention to the fact that this approach was first proposed by Leibniz in works on logical calculus, which for a long time remained in the shadow of his other philosophical works. Currently, the intensional approach is gaining more and more popularity due to the development of non-classical logics, and the article will present several existing intensional formal syllogistic semantics. The paper will also consider another historical approach to syllogistics, associated with the name of the Russian logician Nikolai Vasiliev, who is not only one of the founders of non-classical (non-Aristotelian logic) but also of a different intensional interpretation of such logic. The authors, along with the already known formalizations of Vasiliev’s ideas, present two new systems. One of them is a reconstruction of one type of imaginary logic with statements of three qualities: affirmative and two types of negative statements (with absolute and ordinary negation). The second system is the one that is adequate to semantics, in which instead of the four classical ones, only three types of statements are presented (two particular statements are replaced by one - accidental), and their significance is determined through the relation of the classical logical entailment. Both of them are interpreted intensionally. The intensional approach in logic and, in particular, in syllogistics allows us to expand the class of accepted principles (which occurs due to the expansion of the class of correct moods of syllogisms).