deduction theorem

Noun

Meaning

deduction theorem (plural deduction theorems)

(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
⊢
C


{\displaystyle P,A\vdash C}

to an inference of the form



P
⊢
A
→
C


{\displaystyle P\vdash A\rightarrow C}

or vice versa, where



⊢


{\displaystyle \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.

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