Where "" is a metalogicalsymbol 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.
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "" and vice versa;
or as the statement of a truth-functional tautology or theorem of propositional logic:
where , , and are propositions expressed in some logical system.
Natural languageedit
Truth valuesedit
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.
Exampleedit
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.
exportation, logic, exportation, valid, rule, replacement, propositional, logic, rule, allows, conditional, statements, having, conjunctive, antecedents, replaced, statements, having, conditional, consequents, vice, versa, logical, proofs, rule, that, exportat. 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 ExportationTypeRule of replacementFieldPropositional calculusSymbolic statement P Q R P Q R displaystyle P land Q to R Leftrightarrow P to Q to R P Q R P Q R 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 P Q R P Q R 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 P Q R displaystyle P land Q to R to its consequent Its converse the law of importation P Q R P Q R displaystyle P to Q to R Rightarrow P land Q to R imports a proposition from the consequent of P Q R displaystyle P to Q to R to its antecedent Contents 1 Formal notation 2 Natural language 2 1 Truth values 2 2 Example 3 Proof 4 Relation to functions 5 ReferencesFormal notation editThe exportation rule may be written in sequent notation P Q R P Q R displaystyle P land Q to R dashv vdash P to Q to R nbsp where displaystyle dashv vdash nbsp is a metalogical symbol meaning that P Q R displaystyle P to Q to R nbsp is a syntactic equivalent of P Q R displaystyle P land Q to R nbsp in some logical system or in rule form P Q R P Q R displaystyle frac P land Q to R P to Q to R nbsp P Q R P Q R displaystyle frac P to Q to R P land Q to R nbsp where the rule is that wherever an instance of P Q R displaystyle P land Q to R nbsp appears on a line of a proof it can be replaced with P Q R displaystyle P to Q to R nbsp and vice versa or as the statement of a truth functional tautology or theorem of propositional logic P Q R P Q R displaystyle P land Q to R leftrightarrow P to Q to R nbsp where P displaystyle P nbsp Q displaystyle Q nbsp and R displaystyle R nbsp are propositions expressed in some logical system Natural language editTruth values edit 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 edit 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 editThe following proof uses the rules of material implication De Morgan s laws and the associative property of conjunction Proposition DerivationP Q R displaystyle P rightarrow Q rightarrow R nbsp Given P Q R displaystyle neg P lor Q rightarrow R nbsp material implication P Q R displaystyle neg P lor neg Q lor R nbsp material implication P Q R displaystyle neg P lor neg Q lor R nbsp associativity P Q R displaystyle neg P land Q lor R nbsp De Morgan s law P Q R displaystyle P land Q rightarrow R nbsp material implicationRelation to functions editExportation is associated with currying via the Curry Howard correspondence citation needed References edit Hurley Patrick 1991 A Concise Introduction to Logic 4th edition Wadsworth Publishing pp 364 5 ISBN 9780534145156 Copi Irving M Cohen Carl 2005 Introduction to Logic Prentice Hall p 371 Moore and Parker Rules of Replacement Retrieved from https en wikipedia org w index php title Exportation logic amp oldid 1186957259, wikipedia, wiki, book, books, library,