Page:Mind (New Series) Volume 12.djvu/371

 SYMBOLIC REASONING. 357 time after I have dropped into my place among the silent people of the past. Let me now descend from generalities into particulars. First with regard to point No. 3. In their logic of class inclusion they use the symbol A -^c B to assert that every in- dividual of the class A belongs also to the class B. In their logic of propositions they abandon this definition and use the same symbol to assert that either A is false or B true. I use the symbol A : B in always one and the same sense, namely, to assert that it is certain that either A is false or B true. Hence, when A and B denote each a proposition, we get the follow- ing comparisons and definitions : A-cB = A' + B= (AB') f A : B = (A' + B) e = (AB') A : B = (A -< B) ; So that my symbol A : B is formally stronger than and im- plies their symbol A -< B, just as A e is formally stronger than and implies A T. Thus, my symbol A : B never coin- cides in meaning with their symbol A -< B, when A and B are propositions. They use the symbols 1 and to denote true and false pro- positions respectively ; so that 1 and denote two mutually exclusive classes of propositions. Hence, consistency of notation requires that the symbol ^c 1 should assert that every false proposition is a true proposition, which is absurd. But, as a matter of fact, the statement -< 1 is supposed in their systems, on the contrary, to be always true ; and if we give its second meaning to the symbol < and suppose and 1 to be single propositions instead of classes, the statement A -< B *s always true, as it then asserts that either is false or 1 true, which is self-evident. My symbol i : r, which is erroneously supposed to be equivalent to their -< 1, does not lead to this inconsistency ; for A ; B, by its very definition, means simply (A T B') r '. Hence i : r = T Similarly, we get 77 : e = (rj T e') 7 ' = (vfif^ = e. Though the symbols T, L, e, ij, B, as exponents (or predicates), denote classes, each denotes a single statement when it is the subject of a proposition. Thus rj T asserts that the impossible proposition rj is true, which is absurd. When it is necessary or convenient to distinguish between different propositions of the same class I use subscripta. Thus, in the propositions A B, A 1 ! 8 , A 2 B . the subject A t or A 2 differs from the subject A pretty much as a proper noun differs from a common noun (see MIND, N.S., No. 43). In one or two places in my Sixth