Page:Mind (New Series) Volume 12.djvu/377

 SYMBOLIC KEASONING. :'.l'..'i The same diagram will illustrate two other propositions which by most logicians are considered equivalent, but which, ac- cording to my interpretation of the conjunction if, are not equivalent. They are the complex conditional, If A is true, then if B is true x is true, and the simple conditional If A and B are both true x is true. Expressed in my notation, and with my interpretation of the conjunction if, these conditionals are respectively A : (B : x) and AB : x. Giving to the propositions A, B, x, AB the same meanings as before (all having reference to the same subject, the random point P) it is evident that B : x, which asserts that the random point P cannot be in B without being also in x, contradicts our data, and is therefore impossible. The statement A, on the other hand, does not contradict our data, neither does its denial A', for both in the given conditions are possible though uncertain. Hence, A is a variable, and B : x being impossible, the complex conditional A : (B : x) becomes 6 : 77, which is synonymous with 6* and therefore an impossibility. But the simple conditional AB : x, instead of being impossible, is, in the given conditions, a certainty, for it is clear that P cannot be in both A and B without being also in x. Hence, though A : (B : x) always implies AB : x, the latter does not always imply the former, so that the two are not in all cases equiva- lent. In other words, {A : (B : x)} : (AB : x) is a formal certainty ; but its converse (AB : x) : {A : (B : a;)} is not. Whether my interpretation of this troublesome little con- junction if is the most natural and the most in accordance with ordinary usage, I do not undertake to say ; it certainly is the most convenient for the purposes of symbolic logic, and this alone is reason sufficient for its adoption. At the same time I may point out, as I did long ago (see MIND, Jan., 1880), that the usual denial of the conditional If A is true B is true is the categorical proposition A may be true with- out B being true ; that is to say (A : B)' is equivalent to (AB') 111, which asserts that AB' is possible. From this equiva- lence necessarily follows the equivalence A : B = (AB') 17 , which is my definition of the symbol A : B. The implication A : B expresses a general law and asserts that it has no excep- tion. Its denial (A : B)' asserts that the law is not in all "cases valid ; it asserts (AB')^, that an exception AB' is possible. The statement AB' (the denial of A' + B) asserts not merely the possibility of AB', but an instance of its actual occurrence.