Exportation (logic)

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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:

Exportation
TypeRule of replacement
FieldPropositional calculus
Symbolic statement

Where "" is a metalogical symbol representing "can be replaced in a proof with." In strict terminology, is the law of exportation, for it "exports" a proposition from the antecedent of to its consequent. Its converse, the law of importation, , "imports" a proposition from the consequent of to its antecedent.

Formal notation

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The exportation rule may be written in sequent notation:

 

where   is a metalogical symbol meaning that   is a syntactic equivalent of   in some logical system;

or in rule form:

 ,  

where the rule is that wherever an instance of " " appears on a line of a proof, it can be replaced with " ", and vice versa.

Import-export is a name given to the statement as a theorem or truth-functional tautology of propositional logic:

 

where  ,  , and   are propositions expressed in some logical system.

Natural language

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Truth values

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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

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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.

If my car is on, when I switch the gear to D the car starts going. If my car is on and I have switched the gear to D, then the car must start going.

Proof

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The following proof uses a classically valid chain of equivalences. Rules used are material implication, De Morgan's law, and the associative property of conjunction.

Proposition Derivation
  Given
  material implication
  material implication
  associativity
  De Morgan's law
  material implication

Relation to functions

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Exportation is associated with currying via the Curry–Howard correspondence.[citation needed]

References

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  1. ^ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 364–5. ISBN 9780534145156.
  2. ^ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.
  3. ^ Moore and Parker
  4. ^ "Rules of Replacement".