HUGH MACCOLL, Symbolic Logic and Its Applications. 257 is meant. But in order to express explicitly the whole of what is meant, it is necessary to add the date, and then the statement is no longer " variable," but always true or always false. It results that a variable statement is merely one whose meaning is am- biguous. Now logic ought not to be concerned with forms of words, but with what such forms mean ; hence it is essential that logic should employ only forms of words which are unambiguous, and when this is done "variable " statements disappear. There is, however, a further distinction, namely that between propositions and prepositional functions ; and in regard to the latter, Mr. MacColl's distinction of certain, impossible and variable does seem to be applicable. We may say that "x is a barrister" or " Mrs. Brown is not at home at the time x " is true for some values of x and false for others. Either of these is a preposi- tional function ; but neither is a proposition. Each is merely a general form into which many propositions fit, namely all those resulting from giving values to x. Such a form may be called a certainty when it is " true for all values oi x," i.e., when, whatever value we give to x, the resulting proposition is true ; it is impossible when it is " false for all values of x," and variable when it is neither certain nor impossible. Thus we shall say that true and false are alone applicable to propositions, while certain, variable and impossible are applicable to ambiguous forms of words and to propositional functions. Mr. MacColl's introduction of the " unmeaning " (p. 10), as a separate class of statement, again illustrates the fact that his system is often concerned only with the verbal expression, not with what is expressed. For what is unmeaning is merely a phrase ; it is by no means nothing, on the contrary, it is a definite form of words. In logic, we ought to adopt such a language, and such rules for its employment, that unmeaning phrases shall not occur. This point is connected with Mr. MacColl's view as to the null-class. He says "The symbol denotes non-existence, so that Op 2 , . . . denote a series of names or symbols which cor- respond to nothing in our universe of admitted realities" (p. 5). Here is supposed to denote non-existence itself, whereas Oj, 2 , . . . denote names. Such a want of homogeneity cannot but breed confusion. Thus he says (p. 42) that is the class of individuals which, in the given circumstances, do not have a real existence. But if 1( 0. 2 . . . are to be members of the class 0, this cannot be the case. For O lt 2 . . . are names, and there- fore do have a real existence ; and since, ex hypothesi, they are not the names of anything, we cannot get a series of non-existent individuals which they name. It is a pity that Mr. MacColl has nowhere in his book discussed the meanings of the word existence. It is also a pity that he has not considered explicitly the relation of a name to the thing named. At present, his seems to be the class of things named by names that do not name anything. This would be an unobjectionable definition of the null-class, if he did
