166 Prof. Boole on certain Propositions in Algebra 



2nd. That they should lead to results capable of verification, 

 wherever verification is possible. 



3rd. That they should be capable of a systematic development 

 consistent in all its parts and processes, and neither acknow- 

 ledging nor imposing any limitations but those which exist in 

 the nature of things. 



Now the principles in which, and in the mode of the application 

 of which, that doctrine consists, have been fully stated in this 

 Journal (December 1854). In so far as they may be considered 

 novel, they are certainly in no proper sense mathematical. I do 

 not indeed conceive that they are in any respect novel, except 

 that they are brought into a novel connexion, and are made the 

 basis of a new application. I do not think that any person com- 

 petent to form an opinion upon these subjects would dispute 

 such positions as the following : viz. 1st, that probability is rela- 

 tive — that it depends, not upon the actual connexions of things, 

 but upon those connexions as known to us ; 2nd, that the di- 

 stinction of events into simple and compound is relative, and 

 the selection of simple terms arbitrary and the offspring of lan- 

 guage ; 3rd, that merely logical connexions which are founded 

 upon definition possess only a derived necessity, and that from 

 the contemplation of things or events defined, and subject as such 

 to relations founded upon definition, we may ascend to the con- 

 templation of things free, and only to be expressed by signs of 

 larger comprehension, but from which, by introducing the limi- 

 tation again, the conception of the more restricted class may be 

 derived. I say, for instance, that between the term " king " 

 and some other term which we will suppose to designate a mem- 

 ber of a particular legislative body, relations founded upon defi- 

 nition of the respective terms might exist, which relations would 

 cease to exist if we ascended to the larger terms of " ruler " and 

 " legislator," and would be restored when we imposed upon those 

 terms the former limitations. Now these are principles which 

 are not in any proper sense mathematical. They would seem to 

 belong far more to the psychologist and to the inquirer into the 

 nature of language than to the mathematician. I have, however, 

 shown that in connexion with received principles they form a 

 valid basis for the theory of probabilities ; and in my treatise 

 on the Laws of Thought numerous verifications of the results 

 of that theory will be found. I proceed now to that which I 

 consider the most important of all, the proof that this theory 

 satisfies the third and last of the conditions above adverted to. 



The argument which I purpose to develope is the following. 

 In the demonstration of the general method in probabilities, 

 published in this Journal (December 1854) and in the Laws of 

 Thought, we ascend, in accordance with the third of the princi- 



