100 Prof. Boole (tn the Theory nf Probabilities. 



lloquk^il the ])rohiil)ility P of the conclusion ' '"^ 



*''«*^^ AllZsarcXs. ^ 



Here, by the probfthility 7? of the premiss all Ys are Xs, is meant; 

 the probability that any individual of the class represented by Y, 

 taken at random, is a member of the class Z^ and so in the other 

 cases. The rcsidting probability of the conclusion afforded bjrjt 

 the general method is then .»;^ 



where c is an arbitrary constant expressing the unknown proba- 

 bility, that if the minor premiss is false the conclusion is true. 

 The limiting probabilities of thcr conclusioii dr6 thiis 



pq and;;(/ + l— ^. 



The only published solution of the above problem with which I ^ 

 am acquainted is V=pq, a result which manifestly involves th^*^ 

 hypothesis that the conclusion cannot be true on any other" 

 grounds than are supplied by the premises. ''' '"' " ' " ' 



There are also, I have reason to think_, other cases than the 

 above in which definite numerical results have been assigned; * 

 either by neglecting the arbitrary constants, or by determining 

 them upon grounds not sufficiently explained. I do not, how- 

 ever, purpose to enter into the further consideration of this sub- 

 ject here, nor do 1 offer the above remarks mth any view to 

 depreciate the eminent labours of those from whose writings my 

 illustrations have been drawn. Indeed the results which I have 

 deduced from the new method might all have been obtained by 

 the principles of the received theory, with this principal differ- 

 ence, that the constants, which with their interpretations are • 

 given by the one method, would require to be assumed in the 

 other. While I think it right to make this acknowledgement, I 

 feel it to be just also to say, that it is only to the simpler kind 

 of problems that the remark appears to me to be applicable. 

 Granting even a proper assumption of the arbitraiy constants, I 

 do not see how a solution is to be obtained by the received me- 

 thods when the data are much involved ; not to mention those 

 cases in which the number of the data exceeds or falls short of 

 the number of simple events combined in them, and in the solu- 

 tion of which cases nevertheless arbitrary constants may not be 

 required. Restricting our attention to the ordinaiy theoiy, it 

 appears to me to be certain that the problems which fall under 

 our notice may be resolved into two great classes ; viz. 1st, those 

 in which definite numerical solution is attainable from the data 

 alone, without any determination of arbitrary constants ; 2nd, 

 those in which the data do not suffice to this end, but in which 

 we must either introduce arbitraiy constants, as has been done 

 in this paper, or implicitly determine them as Mr, Be Morgan 



