Exportation (logic)

The exportation rule may be written in sequent notation:

${\displaystyle ((P\land Q)\to R)\dashv \vdash (P\to (Q\to R))}$

where ${\displaystyle \dashv \vdash }$ is a metalogical symbol meaning that ${\displaystyle (P\to (Q\to R))}$ is a syntactic equivalent of ${\displaystyle ((P\land Q)\to R)}$ in some logical system;

or in rule form:

${\displaystyle {\frac {(P\land Q)\to R}{P\to (Q\to R)}}}$, ${\displaystyle {\frac {P\to (Q\to R)}{(P\land Q)\to R}}.}$

where the rule is that wherever an instance of "${\displaystyle (P\land Q)\to R}$" appears on a line of a proof, it can be replaced with "${\displaystyle P\to (Q\to R)}$" and vice versa;

or as the statement of a truth-functional tautology or theorem of propositional logic:

${\displaystyle ((P\land Q)\to R)\leftrightarrow (P\to (Q\to R))}$

where ${\displaystyle P}$, ${\displaystyle Q}$, and ${\displaystyle R}$ are propositions expressed in some logical system.

The following proof uses Material Implication, double negation, De Morgan's Laws, the negation of the conditional statement, the Associative Property of conjunction, the negation of another conditional statement, and double negation again, in that order to derive the result.

Proposition	Derivation
${\displaystyle P\rightarrow (Q\rightarrow R)}$	Given
${\displaystyle \neg P\lor (Q\rightarrow R)}$	Material implication
${\displaystyle \neg P\lor (\neg Q\lor R)}$	Material implication
${\displaystyle (\neg P\lor \neg Q)\lor R}$	Associativity
${\displaystyle \neg (P\land Q)\lor R}$	De Morgan's law
${\displaystyle (P\land Q)\rightarrow R}$	Material implication

Exportation is associated with Currying via the Curry–Howard correspondence.