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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W.V.O. Quine on Consistency - Dictionary of Arguments
IX 209/10 Consistency/Set Theory/Quine: we have been able to prove it twice as we have a simple model in finite sets - which does not apply if we once have added an axiom of infinity. Consistency is questionable and more difficult and more urgent to prove. And the evidence is even less convincing - problem: the question of whether the methods themselves are consistent. >Set Theory/Quine. II 178 The essence of the Corollary of Goedel's incompleteness theorem is that the internal consistency of a mathematical theory can usually only be proved by resorting to another theory based on further premises and it is therefore less reliable than the original one. This has a melancholic connotation. But this helps us to prove that one theory is stronger than another: This is achieved by proving in one theory that the other is consistent. II 180 Goedel's third great discovery: the consistency of the continuum hypothesis and the axiom of choice. II 210 Possible Worlds/QuineVsKripke: possible worlds allow proof of consistency, but no clear interpretation: when are objects equal? For example Bishop Buttler said that any thing is this thing and "no other thing": Problem/QuineVsButler: Identity does not follow necessarily. >Possible Worlds/Quine. IX 192 Set theory/Modern Type Theory/Consistency/Quine: we can prove the freedom of contradiction of this version of set theory with cumulative types: Def Cumulative Types/Set Theory/Quine: Type 0: only L is of type 0. Type 1: L and {L} and nothing else. Type n: should generally include and only include the 2n sets belonging to type n -1. So each quantification interprets only finitely many cases. Every closed statement can be checked mechanically for truth. Such a simple proof will no longer work if the infinity axiom is added. IX 210 Infinite Classes/Consistency: the proof also becomes less convincing if we have to accept infinite amounts. Problem: whether the methods themselves are consistent (only with infinite classes). The highest thing we can often strive for is that we prove that such a system is consistent when another, corresponding system is less distrusted. IX 239 If the consistency of one set theory can be proven in another, the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory._____________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. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, , Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, , Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
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> Counter arguments against Quine
> Counter arguments in relation to Consistency
Ed. Martin Schulz, access date 2024-04-19