232 
PEOFESSOE BOOLE ON THE THEOEY OE PEOBABILITIES. 
Statement of the Method for the Solution of Questions in the Theory of Prohahilities. 
For the general demonstration of this method the reader is referred to the ‘ Laws of 
Thought,’ chap. xvii. For the purpose of the analytical investigation the statement of 
the method will suffice. 
Let 5, t, V, See. represent the events of which the probabilities are given, p, q, r. Sec. 
those probabilities, and w the event of which the probability is sought ; then, whatever 
the definitions of s,t . . . and w may be, and whatever connecting relations may exist, 
it is always possible by the Calculus of Logic to determine the logical dependence of U' 
upon s, t, Sec. in the following most general form, viz. 
w=A+0B+5c+5D. 
Llere A, B, C, D are logical combinations of the events s, t, Sec., and the connexion in 
which these stand to the event w and to each other is the following : A expresses those 
combinations of s, t, Sec. which are entirely included in w, i. e. which cannot happen 
without our being permitted to say that w happens. B represents those combinations 
wffiich may happen but are not included under w ; so that when they happen we may 
say that w does not happen. C represents those combinations the happening of which 
leaves us in doubt whether lo happens or not. D those combinations the happening of 
which Avould involve logical contradiction. 
It follows from the above that the translated form of the problem is 
Given Prob. s=p, Prob. t—q, Prob. v=r, Sec., s, f, v . . . being regarded as events 
subject to the explicit logical condition 
A+B+C=1. 
Bequired the probability u of the event of which the logical expression is 
w~A-\- yC ; 
and it is shown (Laws of Thought, p. 265), upon grounds essentially the same as those 
expressed in Principles I. and II. of this paper, that the solution of the problem is 
involved in the following algebraic equations, viz. 
y,_v, _ A+cc 
P q ’ ' ' u ’ ^ ' 
in which the functions V, V^, . . . are formed in the following manner, viz., — > 
1st. V is derived from A+B-f-C without change of form by interpreting s, t. Sec. no 
longer as logical symbols, but as symbols of quantity. They represent the probabilities 
of the ideal events of Principle II. 
2ndly. is the sum of those terms in V which contain s as a factor, the sum of 
those which contain ^ as a factor, &c. 
The quantity c is an arbitrary constant, admitting of any value between 0 and 1. 
To effect the solution, the quantities s, t. Sec. are to be eliminated from the system (I.), 
and u then determined as a function of p, q,r . . . and c. The arbitrary constant c may 
