# Philosophy Dictionary of Arguments

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X 79
Validity/Sentence/Quantity/Schema/Quine: if quantities and sentences fall apart in this way, there should be a difference between these two definitions of validity (via schema with sentences) or models (with quantities). But it follows from the Loewenheim theorem that the two definitions of validity (via sentences or quantities) do not fall apart as long as the object language is not too weakly (poorly) expressive.
Condition: the object language must be able to express (include) the elementary number theory.

Object Language: in such a language, a scheme that remains true for all sentence implementations is also fulfilled by all models and vice versa.
The demand of elementary number theory is quite weak.
Def Elementary Number Theory/eZT/Quine: is about positive integers using addition, multiplication, identity, truth functions and quantification.
>Number Theory/Quine.
Standard Grammar/Quine: the standard grammar would express the addition, multiplication and identity functions by appropriate predicates.
That is how we get the two sentences:

(I) If a scheme remains true for all implentations of sentences of the elementary number theory sets, then it is fulfilled by all models.
X 80
(II) If a scheme is fulfilled by each model, then e is true for all settings of sets.

Quine: Sentence (I) goes back to Loewenheim 1915:

Sentence of Loewenheim/Quine: every scheme that is ever fulfilled by a model is fulfilled by a model 'U,‹U,β,α...', where U contains only the positive integers.
Loewenheim/Hilbert/Bernays: intensification: the quantities α, β,γ,...etc. may each be determined by a sentence of the elementary number theory: So:

(A) If a scheme is fulfilled by a model at all, it is true when using sentences of the elementary number theory instead of its simple schemes.

Prerequisite for the implentations: the quantifiable variables must have the positive integers in their value range. However, they may also have other values.

(I) follows from (A) that: (A) is equivalent to its contraposition: if a schema is wrong in all the implementations of s of sentences of the elementary number theory, it is not fulfilled by any model. If we speak here about its negation instead of the schema, then "false2" becomes "true" and "from no model" becomes "from every model". This gives us (I).
The sentence (II) is based on the theorem of the deductive completeness of the quantum logic.
II 29
Classes: one could reinterpret all classes in its complement, "not an element of ..." - you would never notice anything! Bottom layer: each relative clause, each general term determines a class.
>Classes/Quine.
V 160
Loewenheim/Quine: there is no reinterpretation of characters - but rather a change of terms and domains - the meanings of the characters for truth functions and for quantifiers remain constant. The difference is not that big and can only play a role with the help of a new term: "ε" or "countable". For quantifiers and truth functions only the difference finite/infinte plays a role. Uncountable is not a matter of opinion. Solution: it is all about which term is fundamental: countable or uncountable.

_____________
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.
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:

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

> Counter arguments against Quine

Ed. Martin Schulz, access date 2020-06-07