Aspects of the Theory of Probability. 121 



exclusive, referred to data f. Then the following four 

 propositions are mutually exclusive, namely 



p.q; p.^q;^p.g; — />. — q*. 



Then by axiom 3 



P(p:f) = J>(p.q:f) + F(p.~q:f) 

 P(g:f) =P(p ,q:f} + P(~p .q:f) 

 . ' T?(pVq:f)=P(p.q:f) + P(p .-^ q :f) + P(^p .q:f). 

 By addition we find 



P{p:f) + P(q:f) = 7(pVq:f)+P(p.q:f) . . (1) 



which is regarded by Jevons and de Morgan as axiomatic. 



The second axiom yields as an obvious corollary the famous 

 " principle of sufficient reason " ; according to this, equal 

 probabilities are assigned to propositions relative to data when 

 the data give no reason for expecting any one rather than any 

 other. In discussing the problems of probability f Poincare, 

 after disposing of the view of Laplace and Boole, seems in- 

 clined to consider this principle as the only possible basis of 

 the theory. Substantially the same view is held by Jevons. 

 There is, however, an objection to basing the whole theory on 

 the principle of sufficient reason. For the only way of passing 

 from the notion of "more probable'' to the numerical esti- 

 mate of probability in any particular case is to discover some 

 set of mutually exclusive and exhaustive alternatives, from 

 which we can pick out some by our judgment as more 

 probable than others ; the most probable on the data then 

 receives the greatest numerical estimate. But if we restrict 

 ourselves to cases where we can obtain a set of alternatives 

 that shall be all equally probable, we are arbitrarily limiting 

 the field to which the theory can be applied. We could, 

 indeed, only deal with those cases where some proposition 

 that is certain on the data can be expressed as the disjunction 

 of a number of equally probable and mutually exclusive 

 propositions ; the probability of any proposition that can be 

 expressed as, oris implied by, the disjunction of any sub-class 

 of these could then be assessed by means of the principle of 

 sufficient reason and axioms 3 and 4. Now there is no reason 

 to believe that the notion of probability is applicable to no 



* — p denotes the proposition that p is false, and p.q denotes the 

 proposition that p and q are both true. Thus -—p. — q denotes the 

 proposition that p and q are both false. The proposition that at least 

 one ofjs and q is true is denoted \>y p\/ q, or the disjunction oip and q. 



t La Science et fllypothcse, 190i, 213-245. 



