638 PROFESSOR BOOLE ON THE COMBINATION 
the conditions of possible experience being that ¢, c’, p, g, and m, should be posi- 
tive proper fractions, subject to the relation 
e+e <l+m : : } : 5 { . (10) 
that root of (9) being taken, which satisfies the conditions 
4 Se =] 
w <n? me Ws: 
—cl—p —cl-—gq 
LW 5 l-we i: 
Now if in (9) we suppose m to vanish, we find 
ep eg (l—w) =we l—pel—q 
“. pgl—w=w 1—p l=q 
~ pq+(l-p) d-9) 
The condition (10) becomes simply e+¢ 21. The remaining conditions are all 
satisfied by the value of w. 
The formula (11), which in the present investigation appears as a kind of 
limiting value, applicable only to cases in which the presumption for or against 
the event < increases most by the combination of the testimonies given, is usual- 
ly regarded as expressing the general solution. The reasoning by which it is 
supposed to be established is the following. 
Let p be the general probability that A speaks truth, g the general probability 
that B speaks truth; it is required to find the probability, that if they agree in a 
statement they both speak truth. Now, agreement in the same statement im- 
plies that they either both speak truth, the probability of which beforehand is 
pq, or that they both speak falsehood, the probability of which beforehand 
is (I—p)(1—q). Hence the probability beforehand that they will agree is 
pq+(1—p) (1—g), and the probability that if they agree, they will agree in speak- 
ing the truth, is accordingly expressed by the formula (11).* In the case of n, tes- 
timonies whose separate probabilities are p, p, . . .p,, the corresponding formula is 
whence w (11) 
PiP2:- Pn 
PP - - Pat (1—p,) A—p,) «+ (1—Pn) 
In applying which, it is usual to regard one of the testimonies as the initial testi- 
mony of the mind itself} Substantially the same reasoning is applied to deter- 
mine the probability of correctness of a decision pronounced unanimously by a 
jury, the probabilities of a correct decision by each member of the jury being given. 
In this reasoning there is no recognition that it is to the same fact that the 
several testimonies are borne. Take the case of two testimonies, and the problem 
(12) 
* Cournor Exposition de la Theorie des Chances, p. 411. De Morean, Formal Logic, p. 191. 
+ Formal Logic, p. 196. 
