Philosophy Dictionary of ArgumentsHome
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| Axiom: principle or rule for linking elements of a theory that is not proven within the theory. It is assumed that axioms are true and evident. Adding or eliminating axioms turns a system into another system. Accordingly, more or less statements can be constructed or derived in the new system. See also axiom systems, systems, strength of theories, proofs, provability._____________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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S.A. Kripke on Axioms - Dictionary of Arguments
III 389ff Axioms/infinite/Kripke: not all Tarski sentences are derivable anymore. Proof/Kripke: Kripke only has a finite number of steps and cites only a finite number of axioms - otherwise rule (rule of evidence): "implicit definition" (Hilbert: "Which axioms are valid?" >Rule following/Kripke. III 389 Infinitely many axioms/Kripke: one cannot derive Tarski sentences for any kind of f's, from an infinite number of truth sentences T(f) ↔ f, e.g. assuming we add a biconditional to a simple predicate P(x) and take P(0), P(1), P(2)... as number-theoretic axioms. These new axioms have the power that P(x) is valid for every number - does (x)P(x) still follow the normal rules of deduction? No, evidence cites only a finite number of axioms. Reductio ad absurdum: if (x)P(x) was deducible (derivable), it would have to be derived from a finite number of axioms: P(m1)...P(mn). M: m is the number name in the formal language of the biconditional which denotes the number m. It is clear that it cannot be derived from a finite number of axioms. If we define P(x) as true of m1...mn, each of the finite axioms will be true, but (x)P(x) will be false. Every instance is known but not the generalization. This is also applicable to finite systems. III 390 Solution: we must allow an infinity rule (e.g.> omega rule) III 391 KripkeVsWallace: the same problems apply to the >referential quantification._____________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. |
Kripke I S.A. Kripke Naming and Necessity, Dordrecht/Boston 1972 German Edition: Name und Notwendigkeit Frankfurt 1981 Kripke II Saul A. Kripke "Speaker’s Reference and Semantic Reference", in: Midwest Studies in Philosophy 2 (1977) 255-276 In Eigennamen, Ursula Wolf, Frankfurt/M. 1993 Kripke III Saul A. Kripke Is there a problem with substitutional quantification? In Truth and Meaning, G. Evans/J McDowell, Oxford 1976 Kripke IV S. A. Kripke Outline of a Theory of Truth (1975) In Recent Essays on Truth and the Liar Paradox, R. L. Martin (Hg), Oxford/NY 1984 |
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Ed. Martin Schulz, access date 2024-04-19