# Psychology Dictionary of Arguments

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Inserting: In a formula, elements can be replaced by others, making the formula simpler or more complex. Thus, variables can in turn be replaced by other formulas, or vice versa. See also Fine-grained/coarse-grained, Logical formulas, Formulas.
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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

David Hilbert on Inserting - Dictionary of Arguments

Berka I 113
Derivation/inserting/"evidence threads"/Hilbert: every derivation can be dissolved into evidence threads, that is, we start with the final formula by applying the schemes (α), (β), (...).
>Derivation
, >Derivability.
I 114
N.B.: then by the dissolution of a derivative into evidence threads, one can put back the insertions into the initial formulas.
Inserting/insertion rules/variables/evidence threads/Hilbert: we can do without rules of insertion by putting back the insertions (by means of evidence threads). From the derivation of formulas which do not contain a formula variable, we can eliminate the formula variables altogether, so that the formally deductive treatment of axiomatic theories can take place without any formula variables.
>Proofs, >Provability.
Hilbert: the rule that identical formulas of the propositional calculus are allowed as initial formulas is modified in such a way that each formula which results from an identical formula of the propositional calculus is permitted as an initial formula.
Evidence threads(s): the rule of insertion also becomes superfluous by the fact that one can study the practical application in the course of time. That is, each case is documented, so you do not need a rule for non-current cases.

Hilbert:
In the place of the basic formula
(x)A(x) > (A(a) is now: (x)A(x) > A(t)
and in the place of
(Ex)A(x) is now: A(t) > (Ex)A(x)
t: term.

Formulas are replaced by formula schemes.
Axioms are replaced by axiom schemata.
In the axiom schemata, the previously free individual variables are given by designations of arbitrary terms, and in the formula schemes, the preceding formula variables are replaced by arbitrary formulas(1).
>Axioms, >Axiom systems.

1. D. Hilbert & P. Bernays: Grundlagen der Mathematik, I, II, Berlin 1934-1939 (2. Aufl. 1968-1970).

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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.

Berka I
Karel Berka
Lothar Kreiser
Logik Texte Berlin 1983

> Counter arguments against Hilbert
> Counter arguments in relation to Inserting