logical equivalence
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{{Unreferenced|date=December 2009}}In
logic, statements
p and
q are
logically equivalent if they have the same logical content.
Syntactically,
p and
q are equivalent if each can be
proved from the other.
Semantically,
p and
q are equivalent if they have the same
truth value in every
model.The logical equivalence of
p and
q is sometimes expressed as
( ≡ q
or
( ⇔ q
.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.
Example
The following statements are logically equivalent:
- If Lisa is in France, then she is in Europe. (In symbols,
f → e
.)
- If Lisa is not in Europe, then she is not in France. (In symbols,
≠g e → ≠g f
.)
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. (Note that in this example
classical logic is assumed. Some
non-classical logics do not deem (1) and (2) logically equivalent.)
Relation to material equivalence
Logical equivalence is different from
material equivalence. The material equivalence of
p and
q (often written
p↔
q) is itself another statement in same
object language as
p and
q, which expresses the idea "
p if and only if
q". In particular, the truth value of
p↔
q can change from one model to another. 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. 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
logically valid).
See also
Ekvivalence (logika)Logische ÄquivalenzÉquivalence logiqueЛогичка еквиваленцијаLogische equivalentieEquivalência lógicaEkvivalencia (logika)逻辑等价
- content above as imported from The Pseudopedia
- "logical equivalence" does not exist on GetWiki
- time: 5:02am EDT - Fri, Mar 19 2010
