Philosophy Dictionary of Arguments

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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.

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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.

 
Author Concept Summary/Quotes Sources

Leon Henkin on Consistency - Dictionary of Arguments

Quine IX 224
Henkin: shows the consistency of a ω-contradictory (omega-contradictory) system. (Also Goedel and Tarski).
Just interpret "F" as true for all but those objects x that fulfill (7).
(7) x ε N, x ≠ 0, x ≠ 1, x ≠ 2... ad infinitum
(See Sets/Henkin).
A theory that is ω-contradictory seems unacceptable even if it is consistent. But according to Henkin, it is easy to see that the term and its definition are misleading.
If a system is consistent and yet allows "Ex(x e N u ~Fx)" and "F0", "F1"...all as theorems, and if we guarantee the interpretation of "0" , "1" etc. as names of numbers, then the problem seems to be to interpret "N" as "number" and not more comprehensive.
Henkin: shows that "N" can be interpreted as N containing extras even under the most favorable circumstances. (See Sets/Henkin) If the system is ω-contradictory, N must even be interpreted that way. ((s) "Extras": e.g. "...and their successors").
Sometimes it is then possible to limit "N" so that it avoids the extras, and sometimes this is not possible.
For example, for every formifiable condition that is verifiably met by 0,1,2... ad infinitum, there is another condition that we can prove is also met by 0,1,2... and yet not by all things that meet the first condition. This is the chronic form of ω-contradictoriness that cannot be cured by an improved version of "N". (Quine: "numerically insegregative"). (>Löwenheim).
Def Omega-contradictory/(w)/Goedel: (Goedel 1931) is a system when there is a formula "Fx" such that any one of the statements "F0", "F1", "F2",... can be proved ad infinitum in the system, but also "Ex(x ε N and ~Fx)".


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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.

Henkin I
Leon Henkin
Retracing elementary mathematics New York 1962

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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Ed. Martin Schulz, access date 2022-11-30
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