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