374 BISHOP TERROT ON PROBABILITIES. 
Now nae when rg > sp, OY rq—rp > sp—rps 
or r(g—p) >p-(s—7)s or > a : 
When the ratios only are given, any conceivable case of the grounds upon 
which the probabilities are given may be represented by mp, mg, nr, and ms. 
mp 
Hence the original probability is et the composite is 
mp+nr 
. and this is greater 
mg+tns 
mp 
than ””. when 
ne q 
m= pq+tm ngr > mM pgqrmnp s 
or rg > ps, as before. 
(10.) Valid objections may, I think, be made to the last paragraph‘of the sec- 
tion in the Encyclopzedia already referred to. As this is not long, I quote it en- 
tire. “The following theorem will be readily admitted on its own evidence. Jf 
any assertion appear neither likely nor unlikely in itself, then any logical argument 
in its favour, however weak the premises, makes it in some degree more likely than 
not. In the manner in which writers on Logic apply the calculus of probabilities, 
this is never the consequence of their suppositions. For what we have called a 
is their resulting probability of the argument. Suppose, for instance, a writer 
on logic presumed that the argument from analogy gave = to the probability 
that there is vegetation in the planets, which must be regarded as a thing neither 
likely nor unlikely in regard of evidence from any other source, he would take 
_ to be the probability of this result, that is, less after an argument in its favour 
: S Lgl ose Lor 
than it was before. We substitute 5+5 . Tp>=a9" 
This numerical equation is the value of the expression (a+¢—ae), when 
1 : : 
a=5€= = I have already shown that this expression does not truly represent 
the composite force of the two probabilities @ and e. But farther than this, the 
Beas 3 oe 
argument from analogy, giving ;5 as the probability of the affirmative, is an 
argument, not in favour of, but against the proposition that there is vegetation 
in the planets. It implies that for every three reasons for believing that there is, 
there are seven for believing that there 7s not; and, consequently, the effect of 
the argument ought to be to diminish our disposition to believe the proposition, 
or, in other words, to diminish its probability. 
(11.) But it may be worth while to examine whether the fraction ; be, after 
all, a true available expression for the probability of an event, which is neither 
likely nor unlikely to happen, or to have happened, there being no evidence, no 
reasons for belief, either for or against it. 
