514 HUGH MACCOLL : A. 9 = (0 T ) = e; for every variable, whether it be a variable of the class T (a statement true now but not always) or a variable of the class 6, (a statement false now but not always) remains still a variable, so that (B T ) e , like (# t ) 9 is a formal certainty. On the other hand, we get (A T / = (0l) e =e' = V ; for 6 means (# T ) T , which is a formal certainty, and a certainty cannot be a variable, since certainties and variables form two mutually exclusive classes by definition. This might also be proved in a slightly different way as follows. Let A = 1 = r r We get A" = 0f = e ; but (A T / = (r^) 6 = e 6 = r,. Similarly we may show that A', though equivalent to A 1 , is not synonymous with A'. For let A = 6 t . We have on the one hand (A')' = A* = 0? = e, and on the other The statements A and A' (like A B and its denial A~ B ) are of the same degree, whereas A* (whether x stands for r or t or e or i) or 0, or any other class of statements) is one degree higher, and may therefore be called the revision of the judgment A. Two contradictory judgments, A and A', are placed before us, and we have to decide which is true. If we decide in favour of the affirmative A, we say that " A is true " and write A T ; if we decide in favour of the negative A', we say that " A is false " and write A 1 . But the question to be decided may be not merely to decide whether A is true or false, but whether A follows necessarily from, or is incon- sistent with, our definitions or admitted and unquestioned data. In that case we write A when we decide that A does follow necessarily from our data ; we write A* 1 when we decide that A is inconsistent with our data ; and we write A 6 when we decide that A neither follows from nor is in- consistent with our data. Similarly, A Z2/ , or its synonym (A. x y, is a revision of the judgment A* ; and so on. It will be noticed that h. xy and A^, which means (A^, are quite different statements, since A* and A* are different. The statement A* asserts that A belongs to the class x; the statement A x takes this for granted. For example, A$, which means (Ag)', asserts that " the variable statement A is false " ; whereas A*', which means (A e ) 1 , asserts that " it is false that A is variable ". The statements A and A* are of the same
