Page:Mind (New Series) Volume 9.djvu/98

84 HUGH MACCOLL : SYMBOLIC EEASONING. in which a, a 1 , b, b 1 are the chances of A, A 1 , B, B 1 . The proof by successive and self-evident equivalences is A A _ A a A _ a A, _ B b 1 A) _ _ _ = B B 1 ~ B 6 1 ' A B A a N )~ - a _ _ B V a"B~ 6 1 B Let 2> = probable ; while, as in previous conventions, T = true, e = certain, ei = ttncertain, 77 = impossible, r/i = possible. Let <j> (T, p) denote the implication which may be read : " If it is probable that A and B are both true, then it is true that A and B are both probable ". It is clear that < (p, T) will then denote the converse (or inverse) implication (A?Bp) T : (A T B T )P, in which the words true and probable interchange places. The symbol $' (r, p) $" (p, T) will then assert (what is a fact) that the former implication is necessarily true, but that the latter is not necessarily true. A statement is probable when its chance of being true is greater than one-half. Let ty (T, p) denote the implication (A T + B T )P : (A? + BP) T , we get i/r" (T, p) }r' (p, T), which may be read : " It is not certain that if it is probable that either A or B is true, then it is true that either A or B is probable ; but it is certain that if it is true that either A or B is probable, then it is probable that either A or B is true ".