16 NATURE AND DESIGN OF THIS WORK. [CHAP. I. 



the most general case, of four distinct classes of terms. By the 

 first class are expressed those combinations of the events A, B, C, 

 which both necessarily accompany and necessarily indicate the 

 occurrence of the event X ; by the second class, those combina- 

 tions which necessarily accompany, but do not necessarily imply, 

 the occurrence of the event X ; by the third class, those combi- 

 nations whose occurrence in connexion with the event X is im- 

 possible, but not otherwise impossible ; by the fourth class, 

 those combinations whose occurrence is impossible under any cir- 

 cumstances. I shall not dwell upon this statement of the result 

 of the logical analysis of the problem, further than to remark 

 that the elements which it presents are precisely those by which 

 the expectation of the event X, as dependent upon our know- 

 ledge of the events A, B 9 C, is, or alone can be, affected. General 

 reasoning would verify this conclusion; but general reasoning 

 would not usually avail to disentangle the complicated web of 

 events and circumstances from which the solution above de- 

 scribed must be evolved. The attainment of this object consti- 

 tutes the first step towards the complete solution of the question 

 proposed. It is to be noted that thus far the process of solution 

 is logical, i. e. conducted by symbols of logical significance, and 

 resulting in an equation interpretable into a proposition. Let this 

 result be termed the final logical equation. 



The second step of the process deserves attentive remark. 

 From the final logical equation to which the previous step has 

 conducted us, are deduced, by inspection, a series of algebraic 

 equations implicitly involving the complete solution of the pro- 

 blem proposed. Of the mode in which this transition is effected 

 let it suffice to say, that there exists a definite relation between 

 the laws by which the probabilities of events are expressed as 

 algebraic functions of the probabilities of other events upon which 

 they depend, and the laws by which the logical connexion of 

 the events is itself expressed. This relation, like the other co- 

 incidences of formal law which have been referred to, is not 

 founded upon hypothesis, but is made known to us by observation 

 (1.4), and reflection. If, however, its reality were assumed a priori 

 as the basis of the very definition of Probability, strict deduction 

 would thence lead us to the received numerical definition as a 



