Antinomy

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An antinomy ( Greek  ἀντί anti 'against', νόμος nomos 'law'; analogous to "incompatibility of laws") is a special type of logical contradiction in which the contradicting statements are equally well founded or (in the case of formal systems ) proven are.

Disambiguation

Antinomies can be found in the matter, even if not in the literal sense of the word in Plato (cf. Phaedon 102; Rep. 523 ff., Parm. 135 E). The modern usage goes back to a legal term of the 17th century. It receives philosophical significance in Immanuel Kant's Critique of Pure Reason (KrV). In the transcendental dialectic , Kant defines an antinomy as a "conflict of laws" (KrV A407 / B434).

In modern logic , the term is not used in a completely uniform way and is in part not sharply delimited from the term “ paradox ”. In the German-speaking world, however, it is largely customary to reserve the term "antinomy" for those contradictions that can be strictly proven within the framework of a formal system and thus indicate an error in the conception of the inference rules or the axioms of this system (e.g. the antinomies of naive set theory , the best known is Russell's antinomy ). As a paradox or paradox ( Greek  παρά para "against", δόξα doxa "opinion"), in contrast, a well-founded statement is usually referred to, which contradicts popular opinion, but which does not cause any real logical difficulties. Many scientific insights can appear paradoxical in this harmless sense (e.g. the twin paradox in Einstein's theory of relativity or the so-called paradoxes of material implication in formal logic; cf. relevance logic ). Probably under the influence of English where the term antinomy is not particularly widespread and is limited in its application mostly on the Kantian antinomies, the term "paradox" (Engl. Is paradoxical ) but also commonly used in a broad sense, which includes the antinomies includes.

In modern logic, a “ contradiction ” is simply understood to mean the conjunction of a statement and its negation, ie a statement of the form (read: “A and not-A”). This (very broad) term is neutral to the question of provability or justifiability and includes e.g. B. also those contradictions that are derived in the context of an indirect proof specifically for the purpose of negating one of the assumptions involved in the derivation. Not every contradiction is therefore philosophically problematic.

Again independent of this - in the modern sense logical - usage, the ambiguous word “ contradiction ” is also used completely differently in the Hegelian dialectic and there also includes social antagonisms , conflicts and the like.

Antinomies in modern logic, mathematics, and the philosophy of language

Differentiation of semantic and logical antinomies

It is common to distinguish the antinomies into semantic and logical.

Logical antinomies are antinomies that arise from purely formal- logical reasons. (Instead, one also speaks of logical paradoxes or set- theoretical antinomies .)

Semantic antinomies are antinomies that result from the semantics of the expressions used. (Synonymous, linguistic or grammatical antinomies are also used).

Logical antinomies

term

The common characteristic of the logical antinomies is u. a. seen by Alfred Tarski and Bertrand Russell in the "self-relationship" or "back relationship".

Examples

solutions

To overcome logical antinomies, Bertrand Russell introduced the so-called type theory .

It is criticized for the fact that it avoids Russell's antinomy, but does not solve the paradoxes of Epimenides (antinomy of the liar) and Grellings and, moreover, works with an "artificial hierarchy".

Semantic Antinomies

Examples

solutions

One way of solving the semantic antinomies is

According to the modifying view, it is more specifically about a negative self-relationship that is contradicting itself.

The Kantian Antinomies

The four antinomies of pure reason in the Transcendental Dialectic (KrV A 426 / B 454ff.) Are intended to demonstrate the "conflict of transcendental ideas" and thus the antinomic character of pure reason in general, which on the one hand seeks to justify the conditioned by the unconditional, on the other but always wants to find further conditions to infinity. The antinomies consist of "thesis" and "antithesis", for each of which a "proof" is presented:

  1. “The world has a beginning in time, and according to space it is also enclosed in limits.” -
    “The world has no beginning and no limits in space, but is, in terms of both time and space, infinite. "
  2. “Every compound substance in the world consists of simple parts, and nothing exists everywhere but the simple, or that which is composed of it.” -
    “No compound thing in the world consists of simple parts, and nothing exists everywhere Simplicity in the same. "(Infinite divisibility)
  3. “Causality according to the laws of nature is not the only one from which the phenomena of the world as a whole can be derived. It is still necessary to assume a causality through freedom to explain it. ”-
    “ It is not freedom, everything in the world only happens according to the laws of nature. ”
  4. “Something belongs to the world that, either as its part or as its cause, is an absolutely necessary being.” -
    “There is no absolutely necessary being everywhere, neither in the world nor outside the world, as its cause.”

Quote

Tarski : "The appearance of an antinomy is a symptom of illness for me."

See also

Web links

Wiktionary: Antinomy  - explanations of meanings, word origins, synonyms, translations

literature

  • L. Goddard, M. Johnston: The Nature of Reflexive Paradoxes: Part I. In: Notre Dame Journal of Formal Logic. 24, 1983, pp. 491-508.
  • Thomas Kesselring: The productivity of the antinomy. Hegel's dialectic in the light of genetic epistemology and formal logic. Frankfurt am Main 1981.
  • Georg Klaus, Manfred Buhr (Hrsg.): Philosophical dictionary. Volume 1, 7th edition. Leipzig 1970, pp. 91-93.
  • Arend Kulenkampff: Antinomy and Dialectics, On the function of contradiction in philosophy. Stuttgart 1970.
  • Franz von Kutschera , Norbert Hinske: Antinomy. In: Joachim Ritter (Hrsg.): Historical dictionary of philosophy. Volume 1, Darmstadt 1971, Sp. 393-405.
  • Harald Schöndorf : Antinomy. In: Walter Brugger, Harald Schöndorf (Hrsg.): Philosophical dictionary. Alber, Freiburg, Br./ München 2010, ISBN 978-3-495-48213-1 .
  • JF Thomson: On some paradoxes. In: RJ Butler (Ed.): Analytical Philosophy. (First Series). London 1962, pp. 104-119.

Individual evidence

  1. ^ Ludwik Borkowski: Formal logic. Akademie Verlag, Berlin 1976, p. 525.
  2. ↑ a b Harald Schöndorf: Antinomy. In: Walter Brugger, Harald Schöndorf (Hrsg.): Philosophical dictionary. Alber, Freiburg, Br./ München 2010, ISBN 978-3-495-48213-1 .
  3. ^ Bertrand Russell, Alfred North Whitehead: Principia Mathematica. In: Uwe Meixner (Hrsg.): Philosophy of logic. Alber, 2003, ISBN 3-495-48016-1 , p. 117 (122)
  4. ^ A b Douglas R. Hofstadter: Gödel, Escher, Bach. 5th edition. Klett-Cotta, Stuttgart 1985, ISBN 3-608-93037-X , p. 24.
  5. ↑ Alfred Tarski: Truth and Proof. In: Alfred Tarski, Introduction to Mathematical Logic. 5th edition. 1977, ISBN 3-525-40540-5 , p. 244 (256).