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
