ANTINOMY (Greek ’αντι- [anti]—opposed; νομος [nomos]—law, principle)—a contradiction in reasoning which occurs when we start from premises the truth of which is not in doubt and using rules of reasoning recognized as correct we arrive at contradictions. In formal logic the definition of antinomy is related to the conception of system; an antinomy is a demonstration of two contradictory sentences in a system. The term “paradox” is often used instead of the term “antinomy”.

The first antinomies appeared in ancient thought in the form of the so-called Megarean paradoxes. The best known is the antinomy of the liar ascribed to Eubulides. From the fact that the liar says that he is lying, it follows both that he is lying and is telling the truth. This antinomy was considered throughout the history of philosophy. There were new discussions upon it in the nineteenth and twentieth century when philosophers analyzed the foundations of logic and mathematics and discovered that it was possible to construct many antinomies. This fact led to a break-through in studies on the foundations of the formal sciences: antinomies undermined confidence in natural language (leading to the precise refinement of natural language) and revealed various presuppositions latent in the foundations of the formal sciences.

We may distinguish two basic types of antinomies: semantic antinomies, namely those associated with semantic concepts such as truth, fulfillment, designation, those concerning the relations between the expressions of language and reality; and logical antinomies which are formulated with terms from the predicate calculus and the set calculus (set theory); the latter are often called set-theory antinomies (in distinction from the aforementioned antinomies, called logical).

Semantic antinomies are based on the use of semantic concepts in contexts associated with self-referential statements. The best known semantic antinomies are: the antinomy of the liar, the antinomy of heterological expression (the concept of designation) called Grelling’s antinomy since 1908 (known in the Middle Ages as the antinomy of expression which does not designate itself—vox non appellans se), Barre’s antinomy (the antinomy of the conception of denotation), an antinomy of self-applicable propositions, and the antinomy of the concept of fulfillment or Richard’s antinomy (concerning the semantic conception of definition, definition of properties or a set by a propositional form). The best known antinomy, the antinomy of the liar, may be presented, following Łukasiewicz, in the following form: In the region I shown below there is the following proposition:
I
The sentence written in the region I is not true

If we accept two premises: (a) The sentence written in region I = “The sentence written in region I is not true” (this is an empirical premise; to recognize its truth we must observe that the sentence written in region I is identical to the sentence whose name we have in quotes in the premise (a) on the right hand of the identity); (b) p is true when and only when p (this is the premise that provides a certain definition of a true sentence). The truth of both premises does not raise doubts. If in (b) we substitute for the propositional variable p the sentence written in region I, and then we use the rule of substitution of members of an identity, we obtain the expression: “The sentence written in region I is true when and only when the sentence written in region I is not true“; and on the basis of this expression we may demonstrate two contradictory expressions.

In formulations of semantic antinomies we are dealing with expressions belonging to a certain language beside which occur semantic terms concerning expressions of the language. Semantic antinomies show that a language (or system) which contains both the expressions of a certain language and semantic terms concerning these expressions is contradictory. Thus one of the ways to avoid these antinomies is to make a precise distinction between language and metalanguage (language of a higher order whose terms concern the expressions of an object language); if a system is non-contradictory, then the metasystem containing semantic terms concerning the expressions of the language cannot be a part of the system.

Logic antinomies are based on the possibility of defining certain properties in the language of the predicate calculus or the set calculus. The best known is the so-called Russell’s antinomy discovered in 1902. Other known antinomies are: the antinomy of the universal set (the antinomy of the largest cardinal number), the antinomy of the set of all sets (Cantor’s antinomy), and the antinomy of the set of all ordinal numbers (the Burali-Fortie antinomy of 1897).

Russell’s antinomy, the antinomy of the class of all classes that are not elements of themselves can be presented as follows:

In set theory we can define a certain class of those and only those sets that are not elements of themselves: ARAA. If in this definition we substitute the constant R for the variable A, we obtain an equivalence: RRRR, on the basis of which we may easily demonstrate two contradictory expressions: RR, RR.

The formulation of this antinomy led to a crisis in the foundations of set theory. This antinomy differed from other logical antinomies because it is not based on any statements of the set calculus apart from the mentioned definition, and thereby it has its source in the very foundations of set theory. It is a so-called definitional axiom (of comprehension) that states that for every meaningful condition there exists a set of objects that fulfil the condition. Therefore if the condition concerns a set not being an element of itself, then such a set should exist, and so Russell’s definition is correct. To avoid the antinomy, we must define the concept of a set to limit the action of the definitional axiom. We may do this in two ways: (1) by limiting the set of meaningful expression (so that expressions such as RR could not be recognized as a meaningful expression; we may mention here different versions of the so-called theory of types); (2) not restricting the concept of a meaningful expression, but limiting the formulation of the definitional axiom, introducing an axiom or several axioms stating that for functions of a certain kind exist sets of objects fulfilling these functions; this leads to different conceptions of the so-called axiomatic set theory.

According to the simple theory of types formulated in 1921 by L. Chwistek (and by F. P. Ramsey, W. Wilkosz, A. Tarski, and R Carnap; earlier, in 1908, Russell built the so-called branching theory of types), we may distinguish several types of logical objects. The lowest type is that of individuals who are not sets; then there are sets of individuals, sets of sets of individuals, etc. A syntactical hierarchy of types of expressions referring to these types of objects corresponds to this ontological hierarchy of logical types of objects. An expression is built in accordance with the principle of the theory of types, where on the left side of the functor there is a constant or variable of type n, and on the right side of the functor is a variable (or constant) of the type n + 1; if it is not so, the expression is not a meaningful expression. Russell’s definition is thus not correct, since both in its definiens and in the definiendum on both sides of the functor there are expressions of the same type. A certain variety of the simple theory of types is S. Leśniewski’s theory of component categories. Set theory based on the theory of types has a certain shortcoming, for it leads to philosophical presuppositions that are difficult to accept and it leads to a so-called typical equivocity of concepts, that is, the same concept (e.g., the concept of the number 2) cannot be applied to different objects, but only to objects of a definite type; instead of one constant (e.g., the number 2) we are dealing therefore with an infinite number of constants (the numbers 2 belonging to different types).

In view of the inconveniences of the theory of types, the second of the ways indicated for attacking Russell’s antinomy has been universally accepted, that is, different versions of the axiomatic set theory. In 1908, E. Zermelo built the first system of axiomatic set theory which was later perfected, and one of its simplest versions is the so-called Zermeli-Fraenkl-Skolem system (or Zermeli-Fraenkl system). In this system instead of an axiomatic definition there are only certain particular cases of the axiom to assure that existence of particular kinds of sets. These cases are sufficient to perform demonstrations of the basic theses of set theory, and they are selected so that they do not affirm the existence of sets which, if accepted, would lead to antinomies. The system constructed by P. Bernays (the so-called von Neuman-Bernays-Gödel system) belongs to another kind of axiomatic system, and in this system a distinction is made between a set and a class.

A. Mostowski, Logika matematyczna [Mathematical logic], Wwa 1948, 207–221, 315–320; A. Fraenkel, Y. Bar-Hillel, Foundation of Set Theory, A 1958, 1973³ L. Borkowski, Logika formalna [Formal logic], Wwa 1970, 1977² 285–314, 357–362; S. Haack, Philosophy of Logics, C 1978, 135–151; L. Borkowski, Wprowadzenie do logiki i teorii mnogoŶci [Introduction to logic and set theory], Lb 1991, 213–223, 353–358; A. Tarski, Prawda [Truth], Wwa 1995, 240–242, 292–317.

Marek Lechniak

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