124 



Oll tlic TJieonj of Association 



2. Consistence and Inference. 



Although the positive-class frequencies (including N under that heading) are 

 all indepeiideiit in the sense that no Single one cau be expressed in terms of the 

 others, they are nevertheless subject to certain limiting conditions if they are to 

 be self-consistent, i.e. such as might have been observed in one and the sanie field 

 of Observation or " universe," to use the convenient term of the logicians. Coiisider 

 the case of three attributes, for example. It is evident that wo iiuist liave 



{AB) <t: 



as {AB) must not be negative\ 



<{A) + {B)-N -^s {aß) 

 >(^) as {Aß) 



> {B) a.s {aB) 



(4); 



and similar conditions inu.st hold for (AC) and {BC). But these are not the ouly 

 conditions that nuist hold. The second-ordor frequencies inust not only be such 

 as not to imply negative values for tlic frequencies of other classes of their own 

 aggregates, but also must not iinpli/ negative values for any of the third-vrder 

 frequencies. Expanding all the third-order frequencies in terms of the frequencies 

 of positive classes, and puttiug the resulting expansion <t 0, \ve have 



{ABC) < 



<\;{AB) + {AC)-{A) 



<i;{AB) + {BC)-{B) 



<t;(^C)+(ß(7)-((7) 



>{AB) 



>(AC) 



>{BC) 



> {AB) + {AC) + {BG) - {A) - {B) - (0) + X 



or the freqnency given 

 below will be negative 



{ABC) [l]] 

 {Aßy) [2] 

 {aBy) [3] 

 {aßC) [4] 

 {ABy) [ö] 

 {AßC) [ü] 

 iaBC) [7] 

 {<^ßy) [«] 



(5). 



But if any one of the minor limit.s [1] — [4] be greater than any one of the 

 major limits [.5] — [8] these conditions are impossible of fulfilment. There are 

 four minor limits to be compared with four major limits or sixteen comparisons in 

 all to be made ; but the majority of these, twelve in all, ouly lead back to 

 conditions of the form (4). The four comparisons of expansions due to contrary 

 frequencies alono lead to new conditions — viz. 



{AB) + {AC) + {BC) ^{A) + {B) + (C) - N 

 {AB) + {AC)-{BC)1f>{A) 

 {AB)-{AC) + {BG)1^{B) 

 -{AB) + {AC) + {BC)::^{0 



(6). 



