Denjoy–Koksma inequality

Exportation 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:

$((P\wedge Q)\to R)\iff (P\to (Q\to R))$

Where "$\iff$" is a metalogical symbol representing "can be replaced in a proof with."

Formal notation

The exportation rule may be written in sequent notation:

$((P\wedge Q)\to R)⊢(P\to (Q\to R))$

where $⊢$ is a metalogical symbol meaning that $(P\to (Q\to R))$ is a syntactic consequence of $((P\wedge Q)\to R)$ in some logical system;

or in rule form:

$\frac{(P\wedge Q)\to R}{P\to (Q\to R)}.$

where the rule is that wherever an instance of "$(P\wedge Q)\to R$" appears on a line of a proof, it can be replaced with "$P\to (Q\to R)$";

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

$((P\wedge Q)\to R))\to (P\to (Q\to R)))$

where $P$, $Q$, and $R$ are propositions expressed in some logical system.

Natural language

Truth values

At any time, if P→Q is true, it can be replaced by P→(P∧Q).
One possible case for P→Q is for P to be true and Q to be true; thus P∧Q is also true, and P→(P∧Q) is true.
Another possible case sets P as false and Q as true. Thus, P∧Q is false and P→(P∧Q) is false; false→false is true.
The last case occurs when both P and Q are false. Thus, P∧Q is false and P→(P∧Q) is true.

Example

It rains and the sun shines implies that there is a rainbow.
Thus, if it rains, then the sun shines implies that there is a rainbow.

Relation to functions

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

References

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