Difference between revisions of "Implication"

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==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:
 
'''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 ===
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== Example ==
 
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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):
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
 
  (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).
  
=== Link ===
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==See also==
 
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*[[Antecedent]]
[http://www2.let.uu.nl/UiL-OTS/Lexicon/zoek.pl?lemma=Implication&lemmacode=654 Utrecht Lexicon of Linguistics]
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*[[Denotation]]
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*[[Exemplification]]
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*[[Reference]]
  
=== References ===
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== Link ==
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*[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.