Lexicon of Arguments

Philosophical and Scientific Issues in Dispute
 
[german]


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Theses I
Theses II

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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 quantifier 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.

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