The gergonne relations

Extract

In this paper I am going to set forth a formal system based on five inter-class relations. These relations exist respectively between a class of a's and a class of b's.

(i) if and only if every a is a b and every b is an a,

(ii) if and only if every a is a b and not every b is an a,

(iii) if and only if it is not the case that either every a is a b or every b is an a or no a is a b,

(iv) if and only if every b is an a and not every a is a b,

(v) if and only if no a is a b.

These relations between classes, which correspond, as will be seen, to the five relations between two circles a and b shown in the well-known Eulerian diagrams,

are of course connected in an intimate way with the four forms of proposition, A, E, I, O, of the traditional syllogistic logic. The French mathematician, J. D. Gergonne, seems to have been the first to recognize these relations explicitly and to understand their importance in syllogistic theory. It is therefore appropriate that they should be called by his name.

Gergonne first of all showed with reference to these relations what are the sufficient and necessary conditions of the truth of propositions of each of the four traditional forms: for example, an A proposition, ‘All a is b’ is true if and only if either the first or the second relation exists between the class of a's and the class of b's.