SYMBOLIC SEASONING. 507 false. Hence, in this case a and ft are variables, and so are their denials a and ft. For Fig. 1 therefore we have a 6 ft . Next consider Fig. 2 ; and as before let P be taken at random in the collection E ; while, as before, e, a, ft, respec- tively assert that P will belong to the collection E, that it will belong to the collection A, that it will belong to the collection B. As before, a may turn out true or false, but this time ft cannot be false, since the point P is restricted, by hypothesis, to the collection E, which is wholly included in the collection B. In this case therefore a is a variable, but ft is a constant of the class e or certainties ; so that for this figure we have a? ft*. Next consider Fig. 3. Here neither a nor ft can possibly be true, since the collections A and B are both excluded from our universe of discourse ; so that we have cpfi 1 . Now let us consider the implication a : ft and its implied alternative a + ft. The alternative a + ft is a certainty in Fig. 1, for here its denial aft' is an impossibility, as no point taken at random in the collection E can, at the same time, belong to the class A and not belong to the class B. In Fig. 1 therefore we have (a/3') 1 ', which is equivalent to (a + ft)' and also to a : ft. In Fig. 4 the alternative a + ft, and therefore also its denial aft', are variables ; for in Fig. 4 either statement (i.e. the alternative or its denial) may be true or false ; and if the experiment of the random point P be repeated often enough, each will be sometimes true and sometimes false. The statement a + ft is true, and its denial aft false, every time P happens not to be in A, and also every time it happens to be in B ; and vice versa. In Fig. 1 not only have we a : ft true (with its synonym ap) but also the causal implication -Q, which implies both ap and ft 11 by definition. But this is not the case in Fig. 2 ; for in Fig. 2 we have ft', as already shown, which contra- ct diets ft"- ; so that in Fig. 2 the causal implication ^ is false. In Fig. 4 the general implication a : ft, and therefore also a the causal implication ^, are false. For a : ft means (a/3')" 1 and asserts that aft' is an impossibility ; whereas in Fig. 4 we have the denial of this, namely (aft'} 1 ' 1 , which asserts that aft' is a possibility. a In Fig. 3 the causal implication 3, and therefore also the
