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An antinomy is a real or apparent contradiction between equally well-based assumptions or conclusions. Contradiction is a generic term for both paradox and antinomy, which are roughly synonymous. However, many recent writers employ paradox as an informal catchall for interesting contradictions of any sort, and antinomy as a technical term for contradictions derivable by sound rules of reasoning from accepted axioms within a science. Antinomies may for convenience be gathered under three headings, depending on whether they arise: (1) in ordinary language; (2) in metaphysics or cosmology; or (3) in logic, mathematics, and kindred formal disciplines.

Ordinary Language. Paradoxes arising in ordinary language have been pondered from early Greek times (cf. Plato, Parm. ). The Megaric philosophers about 400 b.c. studied the well-known Liar Paradox, one version of which runs as follows: If a man says "I always lie," how are we to understand his statement? In a puzzling way it seems to be both true and false, since the truth of his confession conflicts with the fact that it is a confession of never telling the truth. Among the main suggestions offered to resolve the Liar Paradox are: (1) that the man's remark can be called both true and false, but in different respects (Aristotle); (2) that it makes no sense at all (Chrysippus); (3) that in a covert manner it embodies two statements having different levels of reference, and when these are kept separate no paradox arises (Bertrand russell); and (4) that the offending sentence would normally be used as an outpouring of remorse or self-disgust, not as an occasion for drawing inferences, and its paradoxical aspect troubles us only when we fail to notice its normal use (Ludwig wittgenstein). The Liar Paradox and numerous other insolubilia, i.e., statements whose assertion or denial leads to the opposite, were discussed in great detail by medieval logicians (see logic, history of).

Kantian Antinomies. In his Critique of Pure Reason Immanuel Kant claimed to expose four serious antinomies in the then prevailing cosmology. Using valid steps of reasoning, he said, a philosopher can demonstrate on the one hand that the universe is infinite in size and duration, and on the other that it is finite. He constructed similar pairs of rival proofs for and against the infinite divisibility of matter, the possibility of freedom, and the world's ultimate dependence upon a necessary being. The scandal of these antinomies, Kant believed, spelled the doom of metaphysics as a science. He regarded them as a species of "transcendental illusion," the result of a natural and unavoidable human urge to apply the categories of the understanding beyond their safe reach, the realm of sensible experience.

Kant's notion that nature should have built into man a disposition toward illusion has tended to strain his readers' comprehension as much as the antinomies themselves. Part, at least, of the blame can be laid to an obscurity in the tradition Kant criticized, dominated by the philosophy of leibniz and C. wolff. That tradition spoke of the physical universe and everything in it "as appearance," i.e., in terms of the perceiver's mental content. This habit of thought, shared by Kant himself, left unsettled the primary issue between idealism and realism: whether what we call the material world is mind-dependent, like the domain of mathematical inventions, or exists independently of minds. Where that issue is unresolved, the inadvertent risk of thinking about the world in both ways at once is also left open, and with it the possibility of antinomous proofs.

Antinomies in Formal Systems. At the turn of the 20th century, antinomies of a different sort forced logicians and mathematicians to reconsider certain accepted fundamentals, chiefly in the foundations of arithmetic. Russell's antinomy, discovered in 1901, was derived from the efforts of Gottlob Frege to develop number theory in terms of the theory of classes. It can be paraphrased in nontechnical terms as follows: The class of all classes is itself a class, but the class of all lions is not a lion. Some classes, then, appear to be members of themselves while others do not. Now consider the whole class of classes which are not members of themselves. Call it C. Is C a member of itself or not? Let us suppose that it is. That would make C a member of a class that is by definition made up of classes that are not members of themselves. Therefore C is not a member of itself, which contradicts the original supposition. On the other hand, if we now suppose it is not, then on second look C must be included in the membership it was defined as having, and so must be counted a member of itself. Whether we call C a member of itself or not, the result leads to the opposite position.

At one time Russell suggested that expressions of the form "C is either a member of itself or not" be struck out of the theory of classes as meaningless. A more popular solution, his "simple theory of types," instead lays down restrictions on the ways in which references to classes of different levels may be combined in a single assertion. Alternative solutions pioneered by Ernst Zermelo in 1904 can be found in certain axiomatizations of set theory.

Semantical Antinomies. It is now common practice to distinguish between antinomies containing reference to natural language, and those involving no metalinguistic expressions. The first kind are called semantical, the second logical. One example of the former, called Grelling's paradox, runs as follows: Work up two lists of adjectives, the first titled self-describing and containing words that apply to themselves, such as "mispelled," "short," "four-syllabled" the second titled non-selfdescribing and containing words that do not, such as "long," "misspelled," "five-syllabled." Into which list shall we put the term "non-self-describing"? If it is a non-self-describing word, as "short" is a short word, then it belongs under self-describing. But in order to go there, "non-self-describing" would have to be non-selfdescribing, as "short" is short. Put into either list, it switches into the other. This antinomy, like that of Russell, can be solved by employing suitable type restrictions.

Logical Antinomies. Among the second group are certain contradictions arising in theories of ordinal and cardinal number series. In the years from 1895 to 1897, Georg Cantor and C. Burali-Forti independently hit upon a contradiction involving ordinals, i.e., numbers defined in terms of their relations to neighbors in a series. If we suppose the series to be well ordered, which roughly means surveyable in the way that segments of it are, then we can speak of the type of the greatest ordinal. At the same time, as we know, any ordinal can be increased by adding one, so there can be no greatest. As a result of holding to both claims, one is forced to allow the possibility of two different ordinals neither of which is greater. This runs counter to the accepted role of "greater than" in numerical relations. The question of removing this contradiction turns upon the permissibility of assuming that the whole series resembles well-ordered classes within itself.

About 1900, Russell pointed out contradictions arising in connection with Cantor's proofs that there is no greatest cardinal number. Cantor argues, for example, that the number of classes in any class is larger than the number of terms in the class. The contradictions appear, according to Russell, when we assume that Cantor's proofs hold good for nonnumerical classes, such as the class of propositions. If we correlate every class of propositions with its own logical product, and then apply Cantor's argument, we notice something wrong with the phrase "every class" as used here. It fails to include the class of propositions that are logical products but are not members of the class they are products of. Where does the product of that excluded class belong? It can be shown, by the kind of reasoning used in Russell's antinomy above, to be both a member and not a member of the excluded class.

See Also: logic, symbolic; mathematics, philosophy of.

Bibliography: i. m. bocheŃski, A History of Formal Logic, tr. i. thomas (Notre Dame, Ind. 1961). b. russell, Introduction to Mathematical Philosophy (London 1919); Principles of Mathematics (2d ed. New York 1938). a. n. whitehead and b. russell, Principia Mathematica, 3 v. (2d ed. Cambridge, Eng. 1925) v. 1.

[h. a. nielsen]

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