526 Prof. G. Boole on the Theory of Probabilities, 



are subject to most of the laws of the symbols of arithmetical 

 quantity^ satisfying however a peculiar law, to which the symbols 

 of quantity, as such, are not subject. Now I have ascertained 

 that in the formal relations which thus assimilate the laws of 

 thought in logic with the laws of thought in arithmetic, lies the 

 basis of a new and general theory of probabilities. Accordingly, 

 from the purely logical equation to which the present application 

 of our analysis has conducted us, there results a system of alge- 

 braic equations determining the numerical value of the proba- 

 bility sought. 



To give some idea of the generality of this method, I shall 

 add a brief account of some of the results to which I have been 

 led by its application. 



The received theory of probabilities is, so far as it is a general 

 theoiy, essentially based upon the hypothesis that the probabi- 

 lities given are those of independent simple events. To meet a 

 few of the cases in which this hyj)othesis is not realized, Laplace 

 has stated, partly as his own and partly as the result of the in- 

 vestigations of others, certain supplementary principles, of which 

 he also makes frequent use. These relate to such questions as 

 the following : viz. the relative probabilities of causes deduced 

 from the probabilities of an observed event upon the several 

 hypotheses of the different causes operating separately ; the pro- 

 bability of a future event deduced from the probabilities of its 

 separate possible causes, and the probabilities of its following 

 those separate causes, &c. I have vei'ified the whole of Laplace^s 

 general principles of this nature by the application of the method 

 above referred to. 



I have applied it to a considerable number of questions, to 

 which, as it appears to me, the received theoi-y is in its present 

 state inapplicable. The necessity for a more general theory is, 

 I conceive, founded on this circumstance ; that observation, 

 especially of social phsenomena, does not in general present to 

 us the probabilities of simple events, but of events occurring, in 

 particular connexions, whether of causation or of coincidence. 

 To such cases the method I am describing is jieculiarly appli- 

 cable, inasmuch as it imposes no restriction upon either the 

 number or the nature of the data. If the data are insufficient 

 for the definite numerical determination of the probability sought, 

 the solution involves arbitrary constants. These express certain 

 unknown probabilities, which are to be determined from further 

 experience. Their interpretation is given, and hence the nature 

 of that experience determined, by referring to the final step of 

 the logical solution. But it does not hence follow, that, when 

 the experience cannot be obtained, the solution is useless ; for 

 by giving to the constants their extreme values and 1, we 



