A Mysterious Null Class

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A Mysterious Null Class Max Black Philosophy of Science, Vol. 11, No. 2. (Apr., 1944), p. 122. Stable URL: http://links.jstor.org/sici?sici=0031-8248%28194404%2911%3A2%3C122%3AAMNC%3E2.0.CO%3B2-U Philosophy of Science is currently published by The University of Chicago Press.

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DISCUSSION A MYSTERIOUS NULL CLASS

After reading Professor Peter A. Carmichael's severe '(Animadversion on the Null Class" (April, 1943 issue of this JOURNAL, pp. 90-94) one reader at least has been left wondering whether the object of these strictures is identical with the null class to which contemporary logicians refer. One reason for such doubts is the circumstance that the familiar distinction between membership and inclusion apparently fails to apply to the "null class" censured by Professor Carmichael. It svould be absurd to suggest that he is unacquainted with the distinction and its importance; like all other logicians he would wish, without question, to distinguish between the true statement that Dobbin is a member of the class of horses (i.e. is a horse) and the absurd one that he is included in the class of horses (i.e. is a sub-class of horses). Then what are we to make of such quotations as this? "The null class is said t o be a member of every class". (p. 90, line 6) "When i t is asserted t h a t t h e null class is contained in every class. . . ." (p. 91, lines 11-12) ". . . . What is asserted then as a principle of classes is t h a t the null class is a member of every class." (p. 91, lines 27-28) "The word 'contained' is evidently intended t o mean 'having a place within', 'being of the same kind.' " (p. 91, lines 31-32) . . the relation borne by t h e null class t o any class [as alleged by mathematical logicians] is t h e epsilon or membership relation (p. 91, footnote 2) ". . t h e dictum t h a t t h e null class is contained in, or a member of, every class." (p. 92, lines 6-7)

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It seems that the "null class" to which objection is raised can be said indifferently to be a member of and to be included in every class. Xot only is it a null class, then, but, in virtue of its membership in every class, also a horse, a daisy, a cheese, and so on ad lib. All logicians mould join Professor Carmichael in scorn for such a monstrosity; but they would be bound to add that in their usage the term "null class" stands for no such entity. The difficulty of recognizing Professor Carmichael's null class is increased by his identification of it with the class complementary to a given class (e.g. on page 91: "we should say . . . that the class, man, is coctained in the class, nonman. . . . But it is the very opposite of what is asserted . . . as a principle of classes, vix., that the null class is a member of every clat~s.") And yet he seems, throughout, to wish to refer to the very same entity which is located by the designation "null class" in contemporary logical discussion. Can it be that his null class ("the night in which all co~vsare black," p. 91) is some quite other entity? Perhaps only Professor Carmichael can answer this question. MAX BLACK. University of Illinois.