Theorem

Deduction Theorem

Logic and Foundations

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

Statement

A 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

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Characterization Theorem 1

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Logic and Foundations

Source Deduction Theorem (Wikipedia)

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.
In Branch: Logic and Foundations, Lead sentence
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems t
In Group: Proof Theory and Methods of Proof, Lead section
theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly

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Deduction Theorem

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