258 CRITICAL NOTICES : not assume that the various names that do not name anything do nevertheless name a number of different unreal individuals. His view, that there are two classes of individuals, the real and the unreal, seems to assume that every name must name something ; -and this it assumes, I think, because his system oscillates between the name and the thing named. All individuals, I should say, are real ; but some phrases analogous in form to names of individuals do not name any individual. 1 Mr. MacColl appears not to apprehend clearly the usual position about the null-class. He says (p. 77) : "The null-class 0, which they [other symbolists] define as containing no members, and which I, for convenience of symbolic operations, define as consisting of the null or unreal members O x , 2 , 3 , etc., is understood by them to be contained in every class, real or unreal; whereas I consider it to be excluded from every real class ". The "whereas " here is quite misleading, since (as he points out in the next paragraph) the people in question also hold that the null-class is excluded from every real class. They say that a is included in b if a has no member which is not a member of b, and that a is excluded from b if a has no member which is a member of b ; hence if a has no members, a is both included in b and excluded from b. Also the question is hardly to be decided by " convenience of symbolic operations," since one question at issue is whether there are such things at all as "null or unreal members". Again Mr. MacColl says (p. 78) that our view 1 involves the absurdity that every indi- vidual impossibility is a member of the class of certainties. But we hold that there are no impossibilities, i.e., no entities of which it is true that they are impossible. He himself holds that impossible propositions (i.e., those whose truth is impossible) imply certain propositions, and points out (p. 13) that this does not mean that if a statement is impossible it is certain. There is a close connexion between the two cases, and the paradox is in both cases only apparent. The distinction between propositions and prepositional functions solves Mr. MacColl's controversy with Schroder as to the following implication : " If p and q jointly imply r, then either p implies r or q implies r ". Mr. MacColl holds that this is false ; and after giving a proof of its falsehood he sums up as follows : " The Boolian logicians . . . draw no distinction between the true (T) and the certain (e), nor between the false (i) and the impossible (r/). Every proposition is with them either certain or impossible, the propositions which I call variables (6) being treated as non-existent. The preceding illustration makes it clear that this is a serious and fundamental error" (p. 75). The case in which, according to Mr. MacColl, the above implication fails, is when the combination u p and not-r " and the combination "q and not-r " are vari- .able, but the combination "p and q and not-r" is impossible. 1 Cf. MIND, N.S., No. 56, p. 491.
