Philosophy Dictionary of ArgumentsHome
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| Consistency, philosophy, logic: The expression of consistency is applied to systems or sets of statements. From a contradictory system any statement can be derived (see ex falso quodlibet). Therefore, contradictory systems are basically useless. It is characteristic of a consistent system that not every statement can be proved within it. See also systems, provability, proofs, calculus, consistency, theories, completeness, validity, expressiveness.
Within a system, consistency may be demonstrated, but not beyond the boundaries of this system, since the use of the symbols and the set of possible objects are only defined for this system.
Within mathematics, and only there applies that the mathematical objects, which are mentioned in consistent formulas, exist (Hilbert, Über das Unendliche, 1926). See also falsification, verification, existence, well-formed._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||
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Alfred Tarski on Consistency - Dictionary of Arguments
Berka I 401 Consistency Proof/Gödel: a proof of consitency cannot be performed if the metalanguage does not contain higher type variables. >Metalanguage, >Levels, >Provability, cf. >Type theory. Undecidability: is eliminated when the examined theory (object language) is enriched with higher type variables. (1) >Object language. - - - Berka I 474f Consistency/Logical Form/Tarski: is present when - for any statement x: either x ~ε FL(X) or ~x ~ε FL(x). ((s) Either x is not an inference from the system or its negation is not an inference.) But: Completeness: accordingly: if - for any statement x either x ε FL(X) or ~x ε FL(X) ((s) if either any statement or its negation is an inference from the system). I 529 f Law of Contradiction/Tarski: "x ~ε contradiction or ~x ~ε contradiction". We cannot make any generalization from the class of these statement functions. The generalization of these statement functions would itself be a (general) statement, namely the of the law of contradiction. Problem: infinite logical product that cannot be derived with normal methods of inference. I 531 Solution: "rule of infinite induction" - (differs from all other rules of inference by infinitist character).(2) 1. A.Tarski, „Grundlegung der wissenschaftlichen Semantik“, in: Actes du Congrès International de Philosophie Scientifique, Paris 1935, VOl. III, ASI 390, Paris 1936, pp. 1-8 2. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935_____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
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> Counter arguments against Tarski
> Counter arguments in relation to Consistency
Ed. Martin Schulz, access date 2024-04-19