Logical polygon: deducing logical relations among propositions about many-place relations

Abstract

This work develops the earlier research aimed at creating an analogue of syllogistic theories, but for propositions about relations, not properties. The aim of this paper is to justify the rules (which we formulated earlier) that allow to find for a given proposition about an n-place relation with n ⩾ 1, where n is a natural number, the propositions of the same kind which are in the logical relations of contrariety and subcontrariety with the given proposition. Based on these rules (and also on the rules for contradiction and subalternation), it is also possible to immediately tell for two given propositions what the logical relation between them is. These rules are algorithms which allow to deduce the logical relations among propositions from the propositions’ quantity and quality characteristics, without applying predicate logic. Here are proved two theorems, the corollaries of which show the connection of such characteristics to the relations among propositions. Based on these rules, we have constructed the diagram, called “logical polygon”, which is proposed as an analogue of the logical square, but for propositions about n-place relations (and not properties, as the square is). Unlike the square, the polygon not only illustrates, — but also allows to deduce the logical relations among propositions. Logical polygon was constructed by us earlier based on semantic and geometric considerations, in this paper we give the needed proofs based on the forms of the propositions themselves. This work, together with other papers by the same author, intends to be useful in a new field of research — constructing and studying analogues of syllogistic theories, but for propositions about relations.