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| In [[logic]], statements <var>p</var> and <var>q</var> are '''logically equivalent''' if they have the same logical content. This is a [[semantic]] concept; two statements are equivalent if they have the same [[truth value]] in every [[model (logic)|model]] (Mendelson 1979:56). The logical equivalence of <var>p</var> and <var>q</var> is sometimes expressed as <math>p \equiv q</math>, E''pq'', or <math>p \Leftrightarrow q</math>.
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| However, these symbols are also used for [[material equivalence]]; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.
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| ==Logical equivalences==
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| {| class="wikitable"
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| ! ''Equivalence'' !! ''Name''
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| | p∧'''T'''≡p<br />p∨'''F'''≡p || Identity laws
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| |-
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| | p∨'''T'''≡'''T'''<br />p∧'''F'''≡'''F''' || Domination laws
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| |-
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| | p∨p≡p<br />p∧p≡p || Idempotent laws
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| | ﹁(﹁p)≡p || Double negation laws
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| |-
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| | p∨q≡q∨p<br />p∧q≡q∧p || Commutative laws
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| |-
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| | (p∨q)∨r≡p∨(q∨r)<br />(p∧q)∧r≡p∧(q∧r) || Assocative laws
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| |-
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| | p∨(q∧r)≡(p∨q)∧(p∨r)<br />p∧(q∨r)≡(p∧q)∨(p∧r) || Distributive laws
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| | ﹁(p∧q)≡﹁p∨﹁q<br />﹁(p∨q)≡﹁p∧﹁q || De Morgan's laws
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| |-
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| | p∨(p∧q)≡p<br />p∧(p∨q)≡p || Absorption laws
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| | p∨﹁p≡'''T'''<br />p∧﹁p≡'''F''' || Negation laws
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| |}
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| Logical equivalences involving conditional statements：<br />
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| :#p→q≡﹁p∨q<br />
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| :#p→q≡﹁q→﹁p<br />
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| :#p∨q≡﹁p→q<br />
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| :#p∧q≡﹁(p→﹁q)<br />
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| :#﹁(p→q)≡p∧﹁q<br />
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| :#(p→q)∧(p→r)≡p→(q∧r)<br />
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| :#(p→q)∨(p→r)≡p→(q∨r)<br />
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| :#(p→r)∧(q→r)≡(p∧q)→r<br />
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| :#(p→r)∨(q→r)≡(p∨q)→r<br />
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| <br /><br />
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| Logical equivalences involving biconditionals：<br />
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| :#p↔q≡(p→q)∧(q→p)<br />
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| :#p↔q≡﹁p↔﹁q<br />
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| :#p↔q≡(p∧q)∨(﹁p∧﹁q)<br />
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| :#﹁(p↔q)≡p↔﹁q<br />
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| ==Example==
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| The following statements are logically equivalent:
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| #If Lisa is in [[France]], then she is in [[Europe]]. (In symbols, <math>f \rightarrow e</math>.)
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| #If Lisa is not in Europe, then she is not in France. (In symbols, <math>\neg e \rightarrow \neg f</math>.)
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| Syntactically, (1) and (2) are derivable from each other via the rules of [[contraposition]] and [[double negation]]. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either ''Lisa is in France'' is false or ''Lisa is in Europe'' is true.
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| (Note that in this example [[classical logic]] is assumed. Some [[non-classical logic]]s do not deem (1) and (2) logically equivalent.)
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| ==Relation to material equivalence==
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| Logical equivalence is different from [[material equivalence]]. The material equivalence of ''p'' and ''q'' (often written ''p''↔''q'') is itself another statement, call it ''r'', in the same [[formal system|object language]] as ''p'' and ''q''. ''r'' expresses the idea "''p'' if and only if ''q''". In particular, the truth value of ''p''↔''q'' can change from one model to another.
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| The claim that two formulas are logically equivalent is a statement in the [[metalanguage]], expressing a relationship between two statements ''p'' and ''q''. The claim that ''p'' and ''q'' are semantically equivalent does not depend on any particular model; it says that in every possible model, ''p'' will have the same truth value as ''q''. The claim that ''p'' and ''q'' are syntactically equivalent does not depend on models at all; it states that there is a deduction of ''q'' from ''p'' and a deduction of ''p'' from ''q''.
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| There is a close relationship between material equivalence and logical equivalence. Formulas ''p'' and ''q'' are syntactically equivalent if and only if ''p''↔''q'' is a [[theorem]], while ''p'' and ''q'' are semantically equivalent if and only if ''p''↔''q'' is true in every model (that is, ''p''↔''q'' is [[logical validity|logically valid]]).
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| ==See also==
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| {{Portal|Thinking}}
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| * [[Entailment]]
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| * [[Equisatisfiability]]
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| * [[If and only if]]
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| * [[Logical biconditional]]
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| * [[Logical equality]]
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| == References ==
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| * Elliot Mendelson, ''Introduction to Mathematical Logic'', second edition, 1979.
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| {{DEFAULTSORT:Logical Equivalence}}
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| [[Category:Mathematical logic]]
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| [[Category:Metalogic]]
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| [[Category:Logical consequence]]
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The name of the writer is Figures but it's not the most masucline name out there. One of the extremely best issues in the world for me is to do aerobics and now I'm trying to make cash with it. Bookkeeping is what I do. South Dakota is where me and my husband live.
Feel free to visit my blog; tinyurl.com