Page:Mind (New Series) Volume 9.djvu/91

SYMBOLIC REASONING. 77 j^ stood to mean This notation leads necessarily to the formula (we will call it AB B A B A t ' B' which symbolically expresses a well-known and easily proved theorem in probability. This formula will be assumed in what follows. Fie. I. P.O. 3. Let any one of these three figures be assumed, and out of the ten points in the circle E belonging to it let a point P be taken at random. Let A, as a statement, assert that P will turn out to be one of the points in the circle A ; and let E assert that it will be one of the points in the circle E. Now, whatever figure we assume, the point P being, by hypothesis, taken in and restricted throughout to the circle E, the statement E must always be a certainty; whereas the statement A will be a certainty, impossibility, or a variable according to the figure we assume. In Fig. 1 we have A a certainty (A') ; in Fig. 2 we have A an impossibility (A 71 ) ; and in Fig. 3 we have A a variable (A 9 ) ; the exact chance of the truth of A in the last case being T ^. But suppose we neither assume Fig. 1, nor Fig. 2 nor Fig. 3, but take one of the three figures at random ; and then, in whatever figure happens to turn up, take (as before) a point P at random out of the ten points in the circle E. In these circumstances what are the respective chances of A, A% A 71 , A 11 ' being true ? Let F 1( as a symbolic statement, assert that Fig. 1 will turn up ; F., that Fig. 2 will turn up ; and F 3 that Fig. 3 will turn up. There being no other figures in our hypo- thetical universe, the disjunctive statement F x + F 2 + F 3 is a certainty ; so that we have A = A (FJ + F 2 + F 3 ) = AF X + AF 2 + AF 3 .