Logical polygon for propositions about relations: rules of constructing and application

Abstract

In this work we formulate rules of constructing and application for geometric figures that graphically express and allow to deduce the logical relations (contrariety, subcontrariety, contradiction, subalternation) among propositions about n-place relations, where n is a natural number greater than 1 (an example of such proposition for n = 2 is “Every lawyer knows some logician”). Such representations are constructed in a way analogous to that of the square of opposition (= logical square), but, unlike the square, for propositions about relations, not properties. These rules and the suggested graphical representation are based on theoretic ideas also formulated in this work.

In order to formulate the rules and construct the figures we determine the kinds of propositions to be considered, find the features of the logical relations of such propositions for cases of n > 1, including the possibility to express the relations of contrariety and subcontrariety by contradiction and subalternation, we show how these features can be graphically represented and how such graphical representation can be used to gain information about the logical relations of a proposition with other propositions.

The suggested rules allow to deduce the relations among propositions. Being algorithms, those rules make the logical polygon more effective in its field of application than predicate calculus (when used in the same field).

In this work the respective figures are constructed for n = 2 and n = 3. It is shown that for other n they can be constructed in the same way and that the square of opposition can also be seen as an instance of such a figure, although for n = 1. This geometric representation of the relations between propositions, together with the rules of its construction and application, can be called the “Logical polygon” for propositions about relations.

The graphical representation proposed by the author of this work is the first and, at the moment, the only successful solution of the problem of constructing figures (analogous to the square of opposition) for expressing the relations among propositions about many-place relations (for n ⩾ 3), and also of generalization of the obtained results in one figure.

This work, together with other papers by the same author, intends to set the ground for a new field of research — an analogue of syllogistic theories, but for propositions about relations.