SYMBOLIC REASONING. 359 denotes an intelligible proposition that contradicts our data, but by the symbol 0, which (with me) denotes a meaningless proposition. Thus, consistency of notation requires that the formula (0 = 0x1) should assert that every false proposition is meaningless, an assertion which we know to be untrue. But with their other interpretation of the symbol = , and suppos- ing and 1 to denote each a single proposition instead of a whole class, their formula (0 = x 1) is true; for, on this convention, Oxl will then denote not a class of propositions but a single compound proposition which is necessarily false because it contains a false factor 0. If I say " Henry will go to Paris and Richard will go to Berlin," and it turn out that Henry does not go to Paris, though Richard does go to Berlin, I make a false statement, though it is perfectly clear and unambiguous. We can neither call it both true and false nor meaningless. For, by our linguistic conventions, a compound statement is called false, if it contains a single false factor. No inconsistency of this kind, or of any other, will be found in either of my statements (i = IT) and (77 = ?;e), as I always use the symbol = in one and the same sense. With me both statements are formal certainties, for and (jf = 7/e) = fr = (rjT) T } = (77 = 77) = e ; the exponent or predicate r being always understood when not expressed. In most systems I find the formula (A = 1) + (A = 0) = 1, which, like my formula (A T + A l )% is meant to assert that the proposition A is necessarily either true or false. Con- sidering 1 and as single propositions, and adopting the second of their two interpretations of the symbol = , the formula is valid. But with my interpretation of the symbol = , the formula is not valid, whether the symbols 1 and correspond to r and i or to e and 77. For (putting : : for = , to avoid brackets) (A = r) + (A = i) : : r = (A T = T T ) + (A T = t T ) : : T T -= (A = e) + (A = >;) : : e = (A* + A") e . This asserts that it is certain that the statement A is either certain or impossible. Now, this may be true of some parti- cular statement A ; but it is not true of every statement A, for there are numberless statements (those I call variables) that are neither certain nor impossible. In other words, the "statement (A e + A 17 )' is not a formal certainty ; so that the formula of which it has been shown to be the simplification is not valid, or is only valid conditionally and within very
