BISHOP TERROT ON PROBABILITIES. 371 



are supposed to believe absolutely ; there is no question as to the probability of 

 its truth, or the possibility of its falsehood. The only matter in question is 

 whether A is C, or is not C. 



The falsity of the expression a + e—a e will be evident, if we give to a and e 



23 2 3 6 1 10 



the values = and =•. Then ^ + €-ae= = + ~-j^=^+^Q. That is to say, while 

 each of the independent probabilities is less than -x , and, therefore, in favour of 



the negative, their compound force is much above | ; and, therefore, in favour of 



the affirmative. If then we found from internal evidence and external evidence 

 severally, that the chances were against the truth of the proposition A is C, we 

 ought to conclude from their united force, that the chances are in favour of the 

 proposition. But the human mind is incapable of coming to such a conclusion. 



It may be well to notice in passing, that the problem under consideration is 

 altogether different from that of finding the compound force of two identical as- 

 sertions made by two witnesses, whose veracity, that is, the probability of then- 

 speaking truth, is expressed by a and e. In that problem, we possess among the 

 data the fact, that each witness makes the same assertion. But in the problem 

 we have been considering, there is no such assertion. Neither the argument nor 

 the evidence assert or deny that A is C. What they give as data, is merely that 

 the reasons for believing that A is C, are in a given ratio to those for believing 

 that A is not C. And as the data of the two problems are of totally different 

 character, the methods to be applied must of course be different. I have men- 

 tioned this, because some good mathematicians whom I have consulted, were at 



first disposed to consider == — == as the proper expression for the con- 



joined force of the argument and evidence. 



(4.) Let us now consider the Problem under the following form. A, whose veracity 

 is undoubted, states that, from his knowledge of the facts of the case, the probability 



of the event E is -. B, under the same conditions, states, that it is -. Supposing 



q s 



the facts known by each to be altogether distinct, what is the proper measure of the 

 expectation formed in a third mind by these two statements f 



(5.) Before attempting to show how a solution of the problem ought to be 

 sought, it may be well to observe, that the mind cannot admit two probabilities 

 of the same event as co-existent probabilities. Thus, if A tells me that the proba- 



2 4 



bility of rain to-morrow is ^, and B that it is =■, I cannot admit both of these as 



probabilities; for that would be equivalent to believing on the authority of A, 

 that it is less likely to rain than not, and at the same time to believe on the 

 authority of B, that it is more likely to rain than not. 



What really takes place is this. The two fractions are received as indications 



