# Definitions for deduction theoremde·duc·tion the·o·rem

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1. deduction theoremnoun

A procedure for "discharging" assumptions from an inference, causing them to become antecedents of the conclusion; or vice versa. Symbolically, the conversion of an inference of the form to an inference of the form or vice versa, where is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.

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1. Deduction theorem

In mathematical logic, the deduction theorem is a metatheorem of first-order logic. It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results. The deduction theorem explains why proofs of conditional sentences in mathematics are logically correct. Though it has seemed "obvious" to mathematicians literally for centuries that proving B from A conjoined with a set of theorems is sufficient to proving the implication A → B based on those theorems alone, it was left to Herbrand and Tarski to show this was logically correct in the general case—another instance, perhaps, of modern logic "cleaning up" mathematical practice. The deduction theorem states that if a formula B is deducible from a set of assumptions, where A is a closed formula, then the implication A → B is deducible from In symbols, implies . In the special case where is the empty set, the deduction theorem shows that implies The deduction theorem holds for all first-order theories with the usual deductive systems for first-order logic. However, there are first-order systems in which new inference rules are added for which the deduction theorem fails.

### Numerology

1. Chaldean Numerology

The numerical value of deduction theorem in Chaldean Numerology is: 8

2. Pythagorean Numerology

The numerical value of deduction theorem in Pythagorean Numerology is: 8

## Translations for deduction theorem

### From our Multilingual Translation Dictionary

• نظرية الاستنباط

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"deduction theorem." Definitions.net. STANDS4 LLC, 2021. Web. 30 Nov. 2021. <https://www.definitions.net/definition/deduction+theorem>.

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##### a decorative musical accompaniment (often improvised) added above a basic melody
• A. descant
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