Page:Mind (New Series) Volume 15.djvu/530

 516 HUGH MACCOLL : by our very definitions of the words certain, impossible, variable, respectively represented by the symbols e, i), 0, is not the first statement A certain, because it follows necessarily from our data, which are here limited to our definitions and linguistic conventions? Is not the second statement B impossible, because it contradicts (or is inconsistent with) our data ? And is not the third statement C a variable, because, though perfectly intelligible, it is neither certain nor impossible ? To say that C is neither true nor false would be incorrect ; for it may be either. It is true when x is greater than 1 ; it is false when x is not greater than 1. 18. To take another case ; suppose two men are playing dice, and that, just before a throw, three spectators make the three following statements, which we will denote by A, B, C : " The number that will turn up is less than 8 " (A), "The number that will turn up is greater than 8" (B), " The number that will turn up is 5 " (C). Since by our data, or tacit conventions, the only numbers possible are 1, 2, 3, 4, 5, 6, is it not clear that we must have A'B^C 9 ? Is not A certain because it follows necessarily from our data ? Is not B impossible because it is inconsistent with our data ? And is not C a variable because it neither follows from nor is inconsistent with our data ? In the language of probability, the chance of A is 1, the chance of B is 0, and the chance of C is neither 1 nor but a proper fraction. What that proper fraction is the statement C* does not say ; but we know it to be. Taking the three denials A', B', C', the chance of A' is 0, the chance of B' is 1, and the chance of C' is | ; so that we have (A')"(B')(C')*. This shows that here, as always, the denial of any certainty A is an im- possibility A', the denial of any impossibility B is a certainty B', and the denial of any variable C is also a variable C'. 19. Other paradoxes arise from the fact that each of the words if and implies is used in different senses. Putting A and B for two propositions, the statements "If A then B" and "A implies B," which, in my symbolic system, I find it convenient to treat as synonymous and as having the mean- ing which I represent symbolically by any of the three synonymous symbols (A:B), (AB')' 1, (A' + B)% are used by some logicians not only in the above sense, but also in the weaker sense which I attach to the mutually synonymous symbols (AB') 1 and (A' + B) r ; because these logicians erroneously consider my e to be equivalent to my T, and my t] to my i. But there is yet another sense in which we all sometimes use the word implies; for when we say "A implies B " we sometimes mean not only (AB')' J, that it is