# Difference between revisions of "Implication"

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− | + | ==Definition== | |

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'''Implication''' is a 1. (material implication) the combination in [[propositional logic]] of two formulae with the connective -> (''if ... then ...''), also called conditional. The implication of phi and psi, phi -> psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false: | '''Implication''' is a 1. (material implication) the combination in [[propositional logic]] of two formulae with the connective -> (''if ... then ...''), also called conditional. The implication of phi and psi, phi -> psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false: | ||

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2. (logical implication) the relation that exists between two sentences phi and psi if phi -> psi is a [[tautology]]. In other words, psi is the logical implication or ''logical consequence'' of phi if psi is true in every [[model]] in which phi is true. | 2. (logical implication) the relation that exists between two sentences phi and psi if phi -> psi is a [[tautology]]. In other words, psi is the logical implication or ''logical consequence'' of phi if psi is true in every [[model]] in which phi is true. | ||

− | + | == Example == | |

− | + | That q is a logical implication of (p V q) can be demonstrated by merely setting up the [[truth table]] for the formula in (ii): | |

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(ii) (p V q) -> q | (ii) (p V q) -> q | ||

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This implication is true for every combination of [[truth value]]s for p and q. A logical consequence of [[predicate logic]] is the consequence of ThereIs(x) [ P(x) ] from P(c). | This implication is true for every combination of [[truth value]]s for p and q. A logical consequence of [[predicate logic]] is the consequence of ThereIs(x) [ P(x) ] from P(c). | ||

− | == | + | ==See also== |

− | + | *[[Antecedent]] | |

− | [ | + | *[[Denotation]] |

+ | *[[Exemplification]] | ||

+ | *[[Reference]] | ||

− | === | + | == Link == |

+ | *[http://www2.let.uu.nl/UiL-OTS/Lexicon/zoek.pl?lemma=Implication&lemmacode=654 Utrecht Lexicon of Linguistics] | ||

+ | == References == | ||

* Gamut, L.T.F. 1991. ''Logic, language, and meaning,'' Univ. of Chicago Press, Chicago. | * Gamut, L.T.F. 1991. ''Logic, language, and meaning,'' Univ. of Chicago Press, Chicago. | ||

{{dc}} | {{dc}} | ||

[[Category:Semantics]] | [[Category:Semantics]] |

## Latest revision as of 07:06, 16 August 2014

## Definition

**Implication** is a 1. (material implication) the combination in propositional logic of two formulae with the connective -> (*if ... then ...*), also called conditional. The implication of phi and psi, phi -> psi, is only false if phi (which is called the antecedent) is true while psi (the consequent) is false:

(i) phi psi phi -> psi 1 1 1 1 0 0 0 1 1 0 0 1

2. (logical implication) the relation that exists between two sentences phi and psi if phi -> psi is a tautology. In other words, psi is the logical implication or *logical consequence* of phi if psi is true in every model in which phi is true.

## Example

That q is a logical implication of (p V q) can be demonstrated by merely setting up the truth table for the formula in (ii):

(ii) (p V q) -> q

This implication is true for every combination of truth values for p and q. A logical consequence of predicate logic is the consequence of ThereIs(x) [ P(x) ] from P(c).

## See also

## Link

## References

- Gamut, L.T.F. 1991.
*Logic, language, and meaning,*Univ. of Chicago Press, Chicago.