# Psychology Dictionary of Arguments

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Terminology: This section explains special features of the language used by the individual authors.
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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 Terminology - Dictionary of Arguments

Berka I 58
Normal form/Berka: the normal form is another method to replace truth tables. An excellent (canonical) normal form was introduced by Hilbert/Ackermann (1928).
Berka I 112
Definition convertible/Hilbert/Berka: a formula is convertible into another means when the equivalence of the two is derivable.

Definition pranex/Hilbert: pranex is a formula in which all quantifiers are at the beginning and the ranges extend to the end.

Definiton deduction-equal/Hilbert: two formulas are deduction-equal, if each is derivable from the other.
Each formula is deduction-equal to each such formula, which results from it by replacing any free individual variable (IV) with a bound variable which has not previously occurred, and the universal quantifier belonging to the introduced bound variables (in any order) are placed at the beginning. ("Exchange of free variables against bound ones").
This can also be done in reverse order.

Definition Skolem's normal form/Hilbert: the Skolem's normal form is a prenexic formula (that is, all quantifiers at the beginning, range to the end), where there is nowhere among the previous quantifiers a universal quantifier before an existential quantifier.
Each formula is deduction-equal to a Skolem normal form.
(s) Each formula can be transformed into a Skolem normal form.

Note (I 116)
This Skolem normal form is the "proof-theoretic" one.
Definiton fulfillment theoretic Skolem normal form/Hilbert: the fulfillment of the theoretic Skolem normal form is dual to the proof-theoretic Skolem normal form, i.e. the universal quantifiers and existence quantifiers exchange their roles.
>Duality.

Rename/Hilbert/(s): renaming is used here for bound variables(1).

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 Terminology