# deduction theorem

## English

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### Noun

deduction theorem (plural deduction theorems)

1. (logic) 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 $P, A \vdash C$ to an inference of the form $P \vdash A \rightarrow C$ or vice versa, where $\vdash$ is the turnstile symbol. The validity of the procedure is a metatheorem of the given logical theory.
The deduction theorem reveals the relationship between logical entailment and material implication: it allows to one to "pack" or "record" an inference into a tautology, and conversely, to "unpack" or "play back" a tautology as an inference process.