SYMBOLIC REASONING. 81 that P will be in the circle E, that it will be in the circle A, that it will be in the circle B, that it will be in the circle C. The statement E is a certainty ; the statements A, B, C are all three -variables, whose respective chances of being true are A. i.', V; ar >d * ne statement AB is also a variable whose chance of being true is ^V These four conclusions following necessarily from our data, any statement that contradicts, any one of the four must be an impossibility. Now (AB) 11 , which asserts that the chance of AB being true is zero, con- tradicts the fourth conclusion that the chance of AB being true is ^j. Hence (AB)i is not only a statement which happens to turn out false in a particular case and with re- gard to a particular random point, but it is inconsistent with our data, and therefore (within the limits of our data) an impossibility. Let t) l denote this impossibility (AB)" 1 , and let O l denote the variable statement C. Then we get C : (AB)" = 0, : ,, - % ; for the implication l :i) l , which asserts that an impossibility 17,, is a factor of a variable l} is a second impossibility 7/ 2 . Thus, in the given conditions and with the given data, C : (AB) is an impossibility. Next take C : (AB)'. This, by definition of an implication, is synonymous with (CAB), and only asserts that CAB is an impossibility. Now, in the given conditions of Fig. 4 and within the limits of our data, this assertion, that CAB is an impossibility, is evidently a certainty. Call it e^ We have therefore in this case, within the limits of the same data, C : (AB)" = % C : (AB)' = , ; which shows that the two statements C : (AB)i and C : (AB)' are not synonymous. Cases may be given in which (as here) the first is false and the second true; others in which both are true ; and others in which both are false ; but no case can be adduced in which "the first is true and the second false. Instances may also be given in which both are variables ; but in none of these instances will the chance that the first is true exceed the chance that the second is true ; it will be either less or equal. We will now examine the question of the mutual de- pendence or independence of statements. ^ The symbol 8 ^ expresses the dependence of A upon B. It A A is short for = and denotes the increase or diminution in r> e
