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 ¢ will be evident, if we give to a and € 
2 2) 1B) 6 iTty 80 
the values 7 and : Then a+e—a e=7+7-g979 ta" That is to say, while 
each of the independent probabilities is less than E and, therefore, in favour of 
2 > 
: 2 : 1 ei 
the negative, their compound force is much above 5; 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 their 
_ speaking truth, is expressed by «and. 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 Ais 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 
ae . > 
———————— Tr mn- 
ae aa erie the proper expression for the co 
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 
_ first disposed to consider 
of the event Eis®. B, under the same conditions, states, that it is =. Supposing 
f qd 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 ? 
(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- 
bility of rain to-morrow is 2 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 ess likely to rain than not, and at the same time to believe on the 
_ authority of B, that it is move likely to rain than not. 
What really takes place is this. The two fractions are received as indications 
