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Philosophy Dictionary of Arguments

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Equal sign: a symbol, which states that the terms on the left and right hand side refer to the same object. The equal sign is defined for numbers. It has to be redefined when objects from other domains are to be linked. See also identity, equality.

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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.

 
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W.V.O. Quine on Equal Sign - Dictionary of Arguments

IX 9
Equal Sign/Quine: "=" is a two-digit predicate.
>Predicates/Quine.
IX 10
Sign/Sign Set/Quine: each theory has introduced a basic vocabulary of primitive predicates, perhaps by definition. Mostly there are only finally many, then we do not need to add the equal sign "=". Because we can then define it with the help of the others. ((s) "primitive" does not mean "one-digit").

Equal Sign/Quine: suppose the only basic predicate of a theory is "φ". Then we can define "=" by the following explanation of "x = y":

(1) ∀z[(φxz ‹› φyz) u (φzx ‹› φzy)].

For obviously "x= x" proves to be a simple example of a valid formula schema of quantum logic.
The same applies to all special cases of "(x = y u F) > Fy", as far as they are statements which contain no further predicate except "φ".
This is seen in the following way: first, look at all results in which the statements made by "Fx" and "Fy" differ only in one place.
The immediate context of this single occurrence must then be either "φxv" and "φyv" or "φvx" and "φvy", where "v" denotes any variable, (perhaps either x or y).
IX 23
Individuals/Elemental Relation/Extensionality Axiom/Quine: Suggestion: "x ε y", if x is an individual, be true or false, depending on, b x = y or x unequal to y.
Thus, the problem of applying the extensionality axiom to individuals disappears.
"ε" of individuals has the property of "=". (Elemental relationship of individuals: equality! ("is element of", "is contained": becomes the equal sign before individuals).
IX 26
Until then, the equal sign is only defined between class abstraction terms. Between variables we need further tools ...+....
X 88
Logical Truth/Structure/Definition/Quine: our definition of logical truth inevitably referred to the grammatical structure.
Problem: this view is called into question when we introduce identity (identity predicate "=", equal sign).
Identity/logical truth/Quine: the traceability of logical truth to grammatical structure is questioned when identity is introduced, because e.g. "x = x" or "x = y" may not be a logical truth, because not everything can be used. ((s) >Intension: because of it, not all theorems of identity are logical truths.
Quine: it is about the fact that in one logical truth one predicate must be replaced by another, but the equal sign as a predicate cannot be replaced by another predicate.
Identity/Logic/Quine: Truths of Identity Theory
Example "x = x", "Ey((x = y)" or "~(x = y . ~(y = x))" ((s) symmetry of identity)
are not suitable as logical truths according to our definitions of logical truth.
>Logical Truth/Quine.
Reason: they can be wrong if "=" is replaced by other predicates.
Consequence: So should we not count identity to logic, but to mathematics? Together with ">" and "ε"? >Semantic Ascent.
III 268
Two different names can stand for the same object, if the equal sign is inbetween, the equation is true. It is not claimed that the names are the same!
III 271
Equal Sign/Quine: "=" is a common relative term.
The equal sign is necessary because two variables can refer to the same or to different objects.
From a logical point of view, the use of the equal sign between variables is fundamental, not that between singular terms.
III 293
Equality Sign/expressiveness/stronger/weaker/Quine: we also gain expressiveness by making the equality sign obsolete ((s) when we introduce classes). Instead of "x = y" we say that x and y belong to exactly the same classes. I.e.
(a)(x ε a. bik. y ε a)
Identity/Quantities/Quine: the identity of classes can be explained in a way in reverse: "a = b" means that a and b have exactly the same elements. Then the equal sign is simply a convenient shortcut.
Description/Equal Sign/Quine: if we have the equal sign, we can afford the luxury of introducing descriptions without having to calculate them as primitive basic concepts. Because with the equal sign we can eliminate a description from every sentence.
>Descriptions/Quine.


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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 [Author1]Vs[Author2] or [Author]Vs[term] 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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