SYMBOLIC REASONING. 517 impossible for A to be true without B being also true, or the equivalent statement that the affirmation of A coupled with the denial of B is inconsistent with our data, but also that A contains B, that is to say, that B is a particular case of A. In this sense, I think it would be better to use the word contains rather than the word implies. For example, we may say that the formula (x m x n = x m + n ) contains the formula (# 3 ar = a; 5 ) as a particular case, and similarly that (x 3 x* = a; 5 ) contains (6 3 6 2 = 6 5 ). Representing the first and most general of these three statements by the functional symbol <f> (x, ra, ri), it follows from our definition of a logical function that (f> (x, 3, 2) must denote the second, and that <f> (6, 3, 2) must denote the third. It also follows that < (x, ra, n) contains </> (x, 3, 2), and that </> (x, 3, 2) contains <f> (6, 3, 2). Again let <jb (A, B, C), or simply $, denote the Barbara of general logic, namely, (A:B)(B:C):(A:C), in which A, B, C may be any statements whatever state- ments which may or may not have the same subject ; and let 4> s (A, B, C), or simply <f> s , denote the Barbara of the traditional logic, namely, (A:B.)(B.:C.):(A,:G,), in which the statements A, B, C are understood to have the same subject S. We may then say that <f> contains </>., as a particular case. Also, since < and <$> g are both certainties, we can assert not only < : <f> g , that </> implies <f> s , but also <j) s : <f), that fa implies <j), since any certainty e x implies any other certainty e r For, by definition, we have e x : e y = (fx^'y) 11 (^x f n) r> e. We cannot however assert that <f) s contains <j>, for it is <f> s that is a particular case of <f>, and not < that is a particular case of <f> 8 . 20. Misunderstandings and consequent paradoxes also arise from the fact that each of the words because, therefore, prove, and infer has more than one meaning ; but a serious discussion of these would unduly lengthen the present article. Post-scriptum. The preceding was written before I read Mr. Russell's kind and appreciative review of my Symbolic Logic and Its Applications (Longmans) in MIND, No. 58. The points of difference between Mr. Russell's views and mine are, as he says, small in comparison with the points of agree- ment, and the former would I feel sure be smaller still if we could discuss them orally face to face. Mr. Shearman in his recently published book, The Develop- ment of Symbolic Logic, submits my symbolic system to much hostile criticism ; but as he has evidently failed to grasp the
