##  [Exportation](/exportation-0) 

 Definition

The logical equivalence that transforms a conditional with a conjunction antecedent into a nested conditional: (A ∧ B) → C is equivalent to A → (B → C). In type-theoretic and computational readings this corresponds to currying, the isomorphism between functions of two arguments and curried functions.

 

 

 

 

 

 





## Principle

Principle

Antecedents can be reorganized from a joint requirement into a sequence of conditional assumptions: proving C from both A and B together is equivalent to proving that A suffices to show that B implies C.

 

 

 

 

 





## Demonstration

Demonstration

If one has a proof that (rain ∧ cold) → cancel, exportation yields a proof of rain → (cold → cancel): assume rain, then assume cold, derive cancel using the original rule. Conversely, given A → (B → C) and both A and B, one derives C.

 

 

 

 

## Misapplication

Misapplication

Treating exportation as valid in contexts where implication is not material (for example in some substructural, relevance, or certain conditional logics) or when modalities change the scope of antecedents can lead to incorrect transformations.

 

 

 

 

 





## Consequence

Consequence

Simplifies the structure of antecedents in proofs, aligns logical implication with function types in programming languages, and permits currying transformations that make higher-order abstraction and reuse straightforward.

 

 

 

 

## Reversal

Reversal

The reverse transformation—uncurrying—combines nested implications into a single implication with a conjunctive antecedent: from A → (B → C) infer (A ∧ B) → C, completing the equivalence pair.

 

 

 

 

 





## Boundary

Boundary

Valid in classical and intuitionistic propositional and predicate logics under standard material implication; may fail or require modification in logics with relevance constraints, linear resource sensitivity, or when modal operators intervene between antecedents and consequent.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between the syntactic convenience of exportation and semantic contexts where the meaning of 'if' is non-material or context-dependent: exportation conflates a joint assumption with sequential commitments, which can be distinct in some logics or in natural language conditionals.

 

 

 

 

 





## Synthesis

Synthesis

Exportation is the equivalence that reinterprets a conjunctive antecedent as a nested conditional, mirroring the currying isomorphism between functions of multiple arguments and chains of single-argument functions, valid under standard material implication.