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HI. On the Theory of Consistence of Logical Class-frequencies, and its 
Geometrical Representation. 
By G. Udny YrrijE, formerly Assista^it Professor of Applied Mathematics in University 
College, London. 
Communicated hy Professor K. Pearson, F.R.S. 
Eeceived February 9,—Read February 28, 1901. 
Contents. 
Introductory—Definitions . . 
Congruence of the Third Degree 
„ „ Fourth „ 
„ „ Fifth 
General Theory. 
Geometrical Representation of the Conditions of Consistence— 
Congruence of Third Degree. 
„ Fourth „ . 
Limits to Associations given by Conditions of Consistence . . 
§ l-§ 3. 
§ 5. 
§ 6 -§ 8 - 
§ 9-^10. 
§ ll-§ 14. 
§ 15-§ 25. 
§ 26-§ 31. 
§ 32. 
§ 1 . In the ordinary treatment of logic the held of discussion is strictly limited to 
premises of a non-numerical character, numerically dehnite 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 per 
cent, of A’s are B, y per cent, 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 Dehnite Syllogism,” of his ‘ Formal Logic ’ (1847), laid the foundations of 
a theory of strictly quantitative logic. Substituting the modihed 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 
* “ On the Association of Attributes in Statistics,” Ac., ‘ Phil. Trans.,’ A, 1900, vol. 194, p. 257. The 
notation is essentially that of Jevons, save that small Greek letters have been substituted for his italics. 
2 3.8.1901 
