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The Info List - 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:

( ( P ∧ Q ) → R ) ⇔ ( P → ( Q → R ) )

displaystyle ((Pland Q)to R)Leftrightarrow (Pto (Qto R))

Where "

displaystyle Leftrightarrow

" is a metalogical symbol representing "can be replaced in a proof with."

Contents

1 Formal notation 2 Natural language

2.1 Truth values 2.2 Example

3 Proof 4 Relation to functions 5 References

Formal notation[edit] The exportation rule may be written in sequent notation:

( ( P ∧ Q ) → R ) ⊣⊢ ( P → ( Q → R ) )

displaystyle ((Pland Q)to R)dashv vdash (Pto (Qto R))

where

⊣⊢

displaystyle dashv vdash

is a metalogical symbol meaning that

( P → ( Q → R ) )

displaystyle (Pto (Qto R))

is a syntactic equivalent of

( ( P ∧ Q ) → R )

displaystyle ((Pland Q)to R)

in some logical system; or in rule form:

( P ∧ Q ) → R

P → ( Q → R )

displaystyle frac (Pland Q)to R Pto (Qto R)

,

P → ( Q → R )

( P ∧ Q ) → R

.

displaystyle frac Pto (Qto R) (Pland Q)to R .

where the rule is that wherever an instance of "

( P ∧ Q ) → R

displaystyle (Pland Q)to R

" appears on a line of a proof, it can be replaced with "

P → ( Q → R )

displaystyle Pto (Qto R)

" and vice versa; or as the statement of a truth-functional tautology or theorem of propositional logic:

( ( P ∧ Q ) → R ) ↔ ( P → ( Q → R ) )

displaystyle ((Pland Q)to R)leftrightarrow (Pto (Qto R))

where

P

displaystyle P

,

Q

displaystyle Q

, and

R

displaystyle R

are propositions expressed in some logical system. Natural language[edit] Truth 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. Proof[edit] 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

P → ( Q → R )

displaystyle Prightarrow (Qrightarrow R)

Given

¬ P ∨ ( Q → R )

displaystyle neg Plor (Qrightarrow R)

Material implication

¬ P ∨ ¬ ¬ ( Q → R )

displaystyle neg Plor neg neg (Qrightarrow R)

double negation

¬ [ P ∧ ¬ ( Q → R ) ]

displaystyle neg [Pland neg (Qrightarrow R)]

De Morgan's law

¬ [ P ∧ ( Q ∧ ¬ R ) ]

displaystyle neg [Pland (Qland neg R)]

Negation of Conditional

¬ [ ( P ∧ Q ) ∧ ¬ R ]

displaystyle neg [(Pland Q)land neg R]

Associativity

¬ [ ¬ ( ( P ∧ Q ) → R ) ]

displaystyle neg [neg ((Pland Q)rightarrow R)]

Negation of Conditional

( P ∧ Q ) → R

displaystyle (Pland Q)rightarrow R

double negation

Relation to functions[edit] Exportation is associated with Currying via the Curry–Howard correspondence. References[edit]

^ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. pp. 364–5.  ^ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. p. 371.  ^ Moore and Parker ^ http://www.philosophypages.com

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