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Deduction theorem: The deduction theorem is a metatheorem in logic that states that if a proposition B can be deduced from a proposition A, then the implication A → B is also deducible. In other words, if we can prove that B is true given that A is true, then we can also prove that A implies B. The deduction theorem is a useful tool for proving theorems, as it allows us to break down complex proofs into smaller, more manageable steps.
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 Deduction Theorem - Dictionary of Arguments

Berka I 112
Definition "Deduction Theorem"/Hilbert: if a formula B can be derived from a formula A in such a way that every free variable occurring in A is fixed, i.e. that it is neither used for an insertion, done for it, nor as a designated variable of a shemata (α), (β), then the formula A > B can be derived without using the formula A ((s) elimination of the premise).
, >Premises.
I 116 Note:
Rule of the back generalization/scheme (α)/Hilbert:

A > B(a)
A > (x) B(x)

Rule of the front particularisation/scheme (β)/Hilbert:

B(a) > A
(Ex)B(x) > A

>Particularization, >Existential Generalization, >Universal Generalization.

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

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

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