In mathematical logic, a deduction theorem is a metatheorem that justifies conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, so that to prove an implication it suffices to assume the antecedent and derive the consequent from it. Deduction theorems exist for both propositional logic and first-order logic, and the theorem holds for every first-order theory using the usual deductive systems for first-order logic, though it notably fails in Birkhoff-von Neumann quantum logic because the linear subspaces of a Hilbert space form a lattice that is not distributive.
Facts
StatementA deduction theorem states that to prove an implication A implies B in a formal system where A is not already an explicit axiom, it is enough to add A as a temporary hypothesis, derive B from it, and then discharge the hypothesis, yielding A implies B as a theorem of the original system with no added premise. 1 Sources
1. Deduction Theorem (Wikipedia)
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In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A -> B, it is sufficient to assume A as a hypothesis and then proceed to derive B.
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