610 PROFESSOR BOOLE ON THE COMBINATION 



that the method was founded. As presented in the Laws of Thought, it rests 

 upon principles which, to my own mind, have something of an axiomatic character, 

 Viewed in this light, its perfect accordance with the requirement above explained 

 may be considered as a verification of it a posteriori. In itself, however, this 

 accordance affords a sufficient ground of confidence in the legitimacy of the hypo- 

 thesis. On the proof of this accordance I shall say something hereafter. At 

 present I will only state the hypothesis, and show in what the accordance 

 consists. 



The hypothesis is the following : — Translating our problem by the aid of the 

 calculus of logic into a language in which the events whose probabilities are given, 

 appear as simple events subject to conditions founded on their definitions, Art. 

 11, we ascend above these simple events to another scheme of simple events, 

 which are free, and which, when actually subjected to the conditions to which 

 the before-mentioned simple events are necessarily subject, shall have the same 

 probabilities, and shall in every respect take their place. The unknown proba- 

 bilities of the free simple events, which form the elements of this hypothesis, must 

 be so determined as to render the substitution possible, and to permit a formal 

 construction of the problem, both in its data and by its qusesitum, out of those 

 new elements. 



The unknown probabilities being thus determined, the problem assumes a 

 form in which its elementary data are the probabilities of simple events unre- 

 stricted by any condition. In this form the solution of the problem is possible by 

 mere consequence of the fundamental definition of probability. The ground 

 upon which this hypothesis was presented in the Laws of Thought was its intrin- 

 sic reasonableness. On this point I will only refer to my observations in the 

 original work. The ground upon which, in the present essay, I wish to rest the 

 hypothesis is, that it is the only one which does not impose upon the data other 

 conditions than those of conformity with a possible experience. The conditions 

 which must be fulfilled in order that p, q, &c, in the substituted and hypothetical 

 data, may be measures of probability at all,— i.e., may be positive proper frac- 

 tions, — are precisely the conditions of possible experience in the original data. 

 (See Appendix.) 



17. The application of this hypothesis is so fully explained in the Laws of 

 Thought, cap. xvii., that I shall here only describe the general method for the so- 

 lution of questions in probabilities to which it leads, and show the connection 

 which exists between the several parts of that method and the foregoing doctrine. 



General Method. 



Representing the problem to be solved under the form — 



Given Prob. (f> l (x,y,z, . . )=p Prob. <p 2 (x,y,z . .) = q,&c. 



Required Pi-ob. ^ (x, y,z . . ) 



