Exportation (logic)

Exportation

Type	Rule of replacement
Field	Propositional calculus
Symbolic statement	${\displaystyle ((P\land Q)\to R)\Leftrightarrow (P\to (Q\to R))}$

Exportation^[1][2][3][4] is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It is the rule that:

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

Where "${\displaystyle \Leftrightarrow }$" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, ${\displaystyle ((P\land Q)\to R)\Rightarrow (P\to (Q\to R))}$ is the law of exportation, for it "exports" a proposition from the antecedent of ${\displaystyle (P\land Q)\to R}$ to its consequent. Its converse, the law of importation, ${\displaystyle (P\to (Q\to R))\Rightarrow ((P\land Q)\to R)}$, "imports" a proposition from the consequent of ${\displaystyle P\to (Q\to R)}$ to its antecedent.