III. On the theory of consistence of logical class-frequencies, and its geometrical representation

Abstract

1. In the ordinary treatment of logic the field of discussion is strictly limited to premises of a non-numerical character, numerically definite data being rigidly excluded. The statistician obtains no help from ordinary logic towards solving even the simplest problems, e. g ., the deduction of inferences from data of the type “ x percent, of A’s are B, y percent, of A’s are C,” or the inferring of association between B and C from known associations of A with B and with C. It is now more than half a century since De Morgan, in the chapter “On the Numerically Definite Syllogism,” of his ‘Formal Logic’ (1847), laid the foundations of a theory of strictly quantitative logic. Substituting the modified notation of Jevons, employed by me in a recent paper, for De Morgan’s own notation, his Theorem may be expressed in the form “if (AB) + (AC) > (A), (BC) must be at least equal to the difference (AB) + (AC) —(A).” For if, e. g ., we imagine (A) boxes, into which we deal a certain number (viz., (AB)) of cards marked (B), one into each box, leaving only (A) — (AB) unoccupied boxes, and then proceed to deal into the remaining boxes cards marked C, one into each; some of these must fall into boxes already occupied by a B if their number exceed (A) — (AB). From similar simple reasoning De Morgan derived the complete conditions of consistence for (AB), (AC), and (BC). He does not, however, consider the case of more than three attributes, and the whole discussion is rendered very lengthy owing to his standpoint being still that of the older logicians.