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144 NOTES AND CORRESPONDENCE. course are on an equal footing as regards logical reality. The premisses decide what compartments are to be erased, i.e., what members are to be declared non-existent. When the Universe of Discourse is divided into two compartments, the fact that a subdivision of one compartment is destroyed does not place that subdivision in the other compartment. The number of divisions in the Universe of Discourse is limited by the number of terms mentioned in the premisses : it is not allowable, after having arrived at our compartments in this way, to go over them and state that some of them have existence in Mr. Russell's (a) sense and some have not, for to do such a thing would be to assume the presence of a further premiss, giving this additional information. Since, then, in the same argument there cannot be two universes, it is correct, in de- scribing 0, to speak of a class with no members rather than to speak of a universe of unrealities. When Mr. Eussell said that some meanings of existence lie ' wholly outside Symbolic Logic,' he did not mean, we may be quite sure, that the logician cannot manipulate arguments dealing with the various kinds of existence. What was meant was that Symbolic Logic, in occupying itself ' with any question whatever on which it can throw any light ' questions of existence among others does not adopt any special meaning of existence that may be found in Metaphysics. Mr. MacColl believes that his fundamental division into realities and unrealities supplies a method of getting rid of certain paradoxes that ordinary symbolists have to encounter ; but from what I have said above it will be seen that this advantage is realised only by means of a procedure that is based on unjustifiable assumptions. He says that, whereas ordinary symbolists are led to state that ' Every round square (a null-class) is a triangle,' he can say ' No round square is a triangle '. Such a universal negative can be reached only by labelling some of the compartments real, and some unreal, and to do this two premisses are assumed, viz., 4 No round squares are real,' and ' All triangles are real '. My division of the Universe of Discourse, when there are two terms and ' existent,' into four compartments was only intended to show that the second and the third compartment do not constitute all the existing things in the universe. It is quite true, as Mr. MacColl suggests, that, with a premiss as to the meaning of 0, these four compartments might be decreased in number. But, when the reduction commenced, we should not proceed on the same lines, because he would say that the compartment 4 0-existent ' is to find a place in his universe of unrealities, whereas I should say that the compartment in question has no occupants. In the concluding paragraphs of his reply to my note Mr. MacColl, in discussing the implications AB : A and 77 : A, introduces doctrines which cannot be discussed here, but which I have criticised in another place. 1 I can only just state that with Mr. Johnson I think (1) it is not correct to speak of propositions as ' always true,' and (2) the term ' certainly ' refers to a relation in which the thinker stands to an implication, and, where such term occurs, this relation must be definitely stated before the logician has material upon which to work. And, in contradistinction to the way in which Mr. MacColl speaks at the close of his reply to Mr. Russell, I should say that it is unadvisable to think of several symbolic systems, each equal to dealing with a certain class of problems. A generalised logic should embrace all that is correct in all the so-called systems, and should be able to deal with every problem of a logical character. A. T. SHEARMAN. 1 In my paper, " Some Controverted Points in Symbolic Logic," Proc. of the Arist. Soc., N.S., vol. v.).