What is "deduction theorem"

n. (context logic English) 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.

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 (independently) this was logically correct in the general case.

The deduction theorem states that if a formula B is deducible from a set of assumptions Δ ∪ {A}, where A is a closed formula, then the implication A → B is deducible from Δ . In symbols, Δ ∪ {A} ⊢ B implies Δ ⊢ A → B .. In the special case where Δ  is the empty set, the deduction theorem shows that {A} ⊢ B implies  ⊢ A → B.

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. Most notably, the deduction theorem fails to hold in Birkhoff-von Neumann quantum logic, because the linear subspaces of a Hilbert space form a non-distributive lattice.

The deduction rule is an important property of Hilbert-style systems because the use of this metatheorem leads to much shorter proofs than would be possible without it. Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens. In other formal proof systems, the deduction theorem is sometimes taken as a primitive rule of inference. For example, in natural deduction, the deduction theorem is recast as an introduction rule for "→".