1 1 8 Theory of Consistence of Logical Class-frequencies, &e. 



" On the Theory of Consistence of Logical Class- frequencies 

 and its Geometrical Eepresentation." By G. Udny Yule, 

 formerly Assistant Professor of Applied Mathematics in 

 University College, London. Communicated by Professor 

 K. Pearson, F.K.S. Eeceived February 9, — Eead February 

 28, 1901. 



(Abstract.) 



The memoir deals with the theory of the conditions to which a series 

 of logical class-frequencies is subject if the series is to be self-consistent ; 

 i.e., if the class-frequencies are to be such as might be observed within 

 one and the same logical universe. 



The theory has been dealt with to a limited extent by De Morgan, 

 in his ' Formal Logic' (" On the Numerically Definite Syllogism") and 

 by Boole, in the 1 Laws of Thought ' (in the chapter entitled " Of 

 Statistical Conditions "). 



In the present memoir the first section deals with the theory of 

 consistence, by a simple method, up to class-frequencies in five attri- 

 butes, and a general formula is then obtained, giving the conditions 

 for airy case. In the second part of the paper some illustrations are 

 given of the geometrical representations of the conditions obtained in 

 Part I. 



In the case of three second-order frequencies (AB), (AC), and (BC), 

 the complete conditions of consistence may be represented by a tetra- 

 hedron with its edges truncated. The first-order frequencies are treated 

 as constant, (AB), (AC), (BC) as co-ordinates, and the limits to (BC), 

 for example, are given by the points in which the line drawn through 

 the point (AB) (AC) parallel to the (BC)-axis cuts the surface. The 

 general form of the surface depends on the value of the firsr-order 

 frequencies. If 



(A)/(u) = (B)/(u) = (C)/(u) = i 



(u) being the total frequency, the edges are not truncated and the 

 " congruence-surface " becomes a simple equilateral tetrahedron. The 

 limits given to (BC) in terms of (AB) and (AC) in this case are shown 

 to correspond to the limits to the correlation coefficient r- 23 in terms of 

 fi2 and r 13 in the case of normal correlation. The congruence-surface 

 shows very clearly the nature of the approximation towards the 

 syllogism, as conditions of the "universal" type (all A's are B, or 

 no A's are B) are approached. One or two illustrations are also given 

 of congruence-surfaces for third-order frequencies, the first- and second- 

 order frequencies being both treated as constants. 



In the third part of the paper some numerical examples, and sketches 

 of congruence-surfaces for actual cases, are given, in further illustration 

 of the theory. 



