88 Prof. Boole's Reply to some Observations by Mr. Wilbraham 



drawn from them. But though I might, I think, justly complain 

 of the representations which Mr. Wilbraham has made, I very 

 gladly dismiss this part of the question, and desire to consider 

 simply whether Mr. Wilbraham's strictures affect in any way the 

 validity of the method which I have published. Before entering 

 upon this inquiry, I would beg permission to state what is the 

 view which I have been led to take of the theory of probabilities 

 as a science. 



It cannot, I think, be doubtful that the theory of probabilities 

 belongs to that class of sciences which are termed pure sciences. 

 Its fundamental idea or conception is that of probability. From 

 this idea, from the definition of the measure of probability by 

 which it becomes associated with number, and from the laws 

 of thought with which it is connected through its having to do 

 with events capable of logical expression, flow the axioms and 

 first principles of the science. I would refer in partial illustra- 

 tion of this view, to a remarkable paper by Professor Donkin,- 

 published in this Journal (May 1851). He has there announced 

 the following principle, and has shown that it leads at once to 

 (I believe) all the principles before recognized. " If there be any 

 number of mutually exclusive hypotheses, ^i, //g, . . /i„, of which 

 the probabilities relative to a particular state of information are 

 Pv P<i> ' ' Pny and if new information be given which changes the 

 probabilities of some of them, suppose of hm+i and all that follow, 

 without having otherwise any reference to the rest, then the pro- 

 babilities of these latter have the same ratio to one another, after 

 the new information, that they had before ; that is, 



P^l'p'9"'P'm=Pl'P2"'Pfn> 



where the accented letters denote the values after the new infor- 

 mation has been acquired.'^ I am not at present going to dis- 

 cuss this principle, but I adduce it as an instance of the general 

 position maintained, viz. that the ordinary doctrines and prin- 

 ciples of the theory of probabilities do run up into some more 

 general ones, the truth of which, when they are once stated, the 

 mind can hardly refuse to acknowledge, and which seem to be 

 involved in the very nature of expectation and of thought. I go 

 on to observe, that such principles, if truly axiomatic, lead in 

 every pure science, and therefore in the theory of probabilities, to 

 a developed system of truth, or of methods for the attainment of 

 truth, which possess certain invariable characteristics never found 

 unimpaired where error has been permitted to enter. These are, 

 mutual consistcrry, the property of verification wherever verifi- 

 cation is possible, continuity, and perhaps some other qualities 

 to which I cannot refer. Now I propose to show, before I have 

 done, that the theory of probabilities does actually admit of this 



