Page:Mind (New Series) Volume 12.djvu/375

 SYMBOLIC REASONING. 361 notation; but the expression would be extremely long ;uid intricate. Using A ^ B as the denial of (A = B), as is customary, A* would then be expressed by (A =$=0) (A =f= 1), and A. e0 by { (A * 0) (A * 1) * 0} { (A 4= 0) (A 4- 1) 4= 1}. This example of translation speaks for itself and renders all formal argument superfluous. Let any one try to express in this notation the formal certainty A Me + A w " + A. 90 ". The expression needed would take up several lines, and it would be scarcely possible to extract the intended meaning from the bewildering jungle of symbols in which it would be enveloped. It remains to show that my system also includes all valid formulae of their logic of class inclusion. Their symbol A >- B asserts that every individual of the class A belongs also to the class B. This may be expressed by my symbol A : B on the understanding that the two statements A and B have the same subject P, an individual taken at random out of our universe, P 1? P 2, P 3 , etc. Thus A : B becomes a mere ab- breviation for P A : P B, which asserts that P cannot belong to the class A without also belonging to the class B, an assertion equivalent to the traditional All A. is B and to their statement A -< B. Thus, as I showed in MIND, January, 1880, and in MIND, July, 1902, the syllogism Barbara will become a particular case of my formula (A : B) (B : C) : (A : C) ; in which, let it be observed, the symbol : has the same mean- ing throughout, and A, B, C, as well as (A : B), (B : C), (A : C), are propositions. But as this formula is & formal certainty, it holds good whether the statements A, B, C, have the same subject or not, so that it is more general than the syllogism. Barbara may also be expressed by (A -< B) (B -< C) -c (A -<: C), but only on the condition that the symbol -< (unlike my symbol : ) has not the same meaning throughout. For, though we may say that the class A is contained in the class B, the class B, in the class C, and the class A in the class C, we cannot logically speak of the premisses (A -< B) (B < C) as a class contained in the conclusion A < C. It is just the other way ; if the word contain is to be used at all in this case, it is the conclusion that is contained in the premisses, and not the premisses in the conclusion. If, in the last formula, the letters A, B, C denote proposi- tions instead of classes, and we give A ^c B its second mean- ing A' + B, the symbol -< will then (like my symbol : ) have