A metatheorem relating syntactic provability to implication: if a formula B is provable from assumption A (possibly together with other assumptions), then the implication A → B is provable in the surrounding formal system, subject to system-specific conditions about discharged assumptions.

Principle

Internalization of conditional reasoning: proofs that use a temporary assumption can be converted into proofs of a conditional statement by discharging that assumption, thereby reflecting metatheoretic consequence as an object-level implication.

Demonstration

In propositional logic, if assuming P allows one to derive Q through a finite proof, then one can construct a proof of P → Q without assuming P; for example, a subproof that from 'Suppose P' derives Q yields the theorem 'P implies Q'.

Misapplication

Applying the deduction theorem in systems where it fails or requires restrictions (e.g., some modal logics, systems with non-dischargeable global assumptions, or contexts where the inference rules prevent assumption discharge), leading to invalid object-level implications.

Consequence

Permits modular proof construction, theorem formation from conditional reasoning, and mechanization of hypothesis-based proofs; it enables moving between hypothetical derivations and unconditional theorems.

Reversal

The converse statement—if A → B is provable then B is provable from A—does not follow from the theorem itself; deriving B from A still requires either A to be assumed or A to be provable independently, typically by applying modus ponens within the system.

Boundary

Holds in many standard systems such as classical and intuitionistic propositional and first-order logics with ordinary introduction/elimination rules for implication, but fails or needs modification in certain modal, substructural, or relevance logics and in systems with special inference constraints.

Semantic Tension

Tension arises between syntactic transformability guaranteed by the deduction theorem and semantic entailment: provability-from-assumption is a syntactic notion sensitive to proof rules, whereas semantic implication may hold even when the deduction theorem does not apply in a given calculus.

Synthesis

The Deduction Theorem bridges the meta-level notion of assuming hypotheses to derive conclusions and the object-level representation of that relationship as an implication; it streamlines proof development where assumption discharge is permitted but must be applied with attention to system-specific provisos.