642 PROFESSOR BOOLE ON THE COMBINATION 



from which it appears, that if r is the a priori expectation of an event z, and if 

 evidences are presented which severally would change r to p and q, and unitedly 

 would change it to R ; then, reciprocally, if R measured the a priori expectation 

 of the event, and evidences were received which would severally change it to p 

 and q, unitedly they would reduce it to r. Now this is evidently what ought to 

 be the case, since testimonies simply countervailing those by which r was 

 changed to R, would simply undo what was done, and again reduce R to r. 



We see that p and g being the same, R is greater when r is less, and less 

 when r is greater ; and this, though it is contrary to what we might at first ex- 

 pect, is agreeable to reason. For the effect of evidence is to be measured, not by 

 the state of expectation which exists after it has been offered, but by the degree 

 in which the previous state of expectation has been changed by it. Suppose p 

 and q much greater than r, which we will conceive to be a small quantity, then 

 the separate evidences greatly increase the probability of an event which was be- 

 fore very improbable ; and unitedly they do this in a much higher degree than if 

 the separate evidences had merely been such as to raise to the measures p and q, 

 an expectation which was before not much below these measures. 



Now, introducing the principle of the mean already explained, Art. 24, let us 

 in (6) make R-r, we have 



pqQ--r) 



= r 



pg(l-r) + (l~p) (l-q)r 

 and solving this equation relatively to r, we find 



Vj9g 



Vp~q + V{\-p) (1-q) ' (8) 



the formula required. 



37. Upon this result, the following observations may be made : — 

 In the first place it may be shown, from the formula itself, that it always ex- 

 presses a value intermediate between the values p and q. Thus we have 



Vpq 



r—p=y= 



Vpq + V(1 -p) (1 -q) 



s/q (1 — »)— a/<7 (1 — q) , ' 



(9) 



on reduction. In like manner we have 



= _ ^^) -^go^ ) x vw=q) _ (10) 



vpq + Vl—p 1 — q 



As p and q are positive fractions, the values of r-p and r-q, given in (9) and 

 (10), are clearly of opposite signs, whence r must lie between p and q. 



