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Animadversion on the Null Class

Published online by Cambridge University Press:  14 March 2022

Peter A. Carmichael
Peter A. Carmichael*
Affiliation:
Louisiana State University, Baton Rouge
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Extract

A term and its contradictory exhaust the universe of discourse, but when examined for their relation to each other, I believe they are seen to exhaust that universe in a more particular respect than may have come into notice. They seem to exhaust or annul one of the leading notions of mathematical logic. This notion is the one concerning the whereabouts of the null class.

Type
Research Article
Information
Philosophy of Science , Volume 10 , Issue 2 , April 1943 , pp. 90 - 94
Copyright
Copyright © The Philosophy of Science Association 1943

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References

1 C. I. Lewis: A Survey of Symbolic Logic, p. 187. The reasoning here seems fallacious, however. If we cannot accept the premise “All members of any class are members of the class of all things” without accepting the proposition “The null class is contained in every class, in extension,” it must be because of another premise, making a syllogism (since the Second proposition contains a term not in the first, viz., “the null class,” and is therefore not an immediate inference from the first). What will this additional premise be? Evidently it will be a universal affirmative about the null class, inasmuch as the conclusion is a universal affirmative about that class; in other words, this must be a Barbara syllogism. Accordingly, the missing premise is “The null class is a member of the class of all things,” and the syllogism is: “All members of any class are members of the class of all things; the null class is a member of the class of all things; therefore the null class is a member of any (every) class.” But here we have an undistributed middle—and no Barbara. Hence it is not established that, accepting the original premise, we are bound to accept the conclusion.

But perhaps the implication goes farther back than this syllogism; perhaps “the principle that, in extension, the null-class is contained in every class” is a principle such that, if denied, then the proposition, “All members of any class, y, are also members of the class of all things,” must also be denied. In other words, we are to suppose that unless the null class is in every class, it is not true that any class exists; if Socrates is not null, he is not Socrates; if y is y, it is non –y. But this is self-contradictory.

Is this objection beside the point, for being on intensional grounds when the argument was strictly extensional? But that argument, though about something extensional, is in fact intensional. It is an argument saying that a certain proposition implies another proposition, not an argument saying that something existential is the case.

2 It might be said, in criticism, that the relation borne by the null class to any class is the epsilon or membership relation while the relation of any class to the null class is not that; to which, however, it must be replied that this depends on the nature of the null class, which is a question under consideration here. Further, if it is argued that “member of” is not transitive, but is supposed to be transitive here, the answer is that “member of” certainly is transitive in some instances (a member of a company is a member of a regiment, e.g.), and could easily be so in this case. Nor, further, is there an a priori reason for supposing that any class is not a member of the null class—when the null class is satisfactorily defined. Logic cannot be disjoined from metaphysics here, though it sometimes looks as if the mathematical logicians were heroically trying to have it so.

3 Ibid., p. 185.