SYMBOLIC REASONING. 79 possible ones) let a collection K be taken at random ; and let K t> K 2 , K 3 respectively assert that K will be K p that K will be K,, that K will be K 3 . We shall then have ^l=J^l +Ji5L=i +1=1- t f 3 3 3 which implies A 11 " 9 . But if, on the other hand, we suppose the collection K 3 (as well as K : and K 2 ) to lead to the con- clusion A" 1 ", and we take one of the three collections at random, we shall have (since the disjunctive statement K! + K 2 + K 3 is a certainty) = A"" (K, + K 2 + K 3 ) = KI A^_ K 2 A^< K 3 AI ' " c ' K 2 e ' ~K^ 1 _1_ 1 - K "^ 3 - a conclusion which is synonymous with A 11 '". Finally, suppose that out of m collections of data, ma collec- tions lead to the conclusion A 71 '", while the remaining m (1 - a) collections lead to the denial of this conclusion ; and that out of the total m collections (all equally probable) we take a collection at random. In this case the chance that A" 1 '" is true will be a, a result which we express symbolically by A 7 '"" 1 ; that is to say, we conclude that A 11 '" belongs to the class of statements whose chance of being true is a. We have thus a concrete illustration of A* 1 '" and of the chance of its being true. The general principle may be stated thus : Let A be a statement of any degree as regards exponents, and of any complexity as regards the number of its consti- tuents and the intricacy of their relations ; and let 1; a.,, a 3 , . . ., a n respectively be the chances of A being true on ~n different hypotheses, all equally probable, and of which one and one only can be true. If we take one of these n hypo- theses at random the chance (i.e., the average chance) of A being true will be (a t + a, 2 + a s + . . . + a n ). If this average chance is unity, then A is a certainty ; if it is zero, A is an impossibility ; if it is a fraction between unity and zero, A is a variable. These three conclusions are respectively asserted by the symbols A', A" 1 , A 9 . Observe that whatever be the degree of A, and however complex the relations of its con- stituents, the conclusion A 9 (that A is a variable) is perfectly consistent with the statement A (that A is true), and also perfectly consistent with the statement A' (that A is false) ; but it is not consistent with A' (that A is certain), nor yet with Ai (that A is impossible).
