Concept
Logical equivalence
Overview
In logic and mathematics, statements
p
{\displaystyle p}
and
q
{\displaystyle q}
are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of
p
{\displaystyle p}
and
q
{\displaystyle q}
is sometimes expressed as
p
≡
q
{\displaystyle p\equiv q}
,
p
::
q
{\displaystyle p::q}
,
E
p
q
{\displaystyle {\textsf {E}}pq}
, or
p
⟺
q
{\displaystyle p\iff q}
, depending on the notation being used.
However, these symbols are also used for material equivalence, so proper interpretation would depend on the context. Logical equivalence is different from material equivalence, although the two concepts are intrinsically related.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Logical equivalence is.
From the proposition, Some A is not B, we cannot even infer that some B is not A; though some men are not Englishmen, it does not follow that some Englishmen are not men. The only mode usually recognised of converting a particular negative proposition, is in the form, Some A is not B, therefore, something which is not B is A; and this is termed conversion by contraposition. In this case, however, the predicate and…
John Stuart Mill, A System of Logic
States what Logical equivalence is.
Even speech rebels against this idea. A mathematical figure of three dimensions is called a solid--corpus solidum--and hence, according to the Latin, a body capable of being handled. It has a name derived, therefore, by no means from the independent play of imagination but from solid reality.
Friedrich Engels, Landmarks of Scientific Socialism
States what Logical equivalence is.
Philosophers who discussed Logical equivalence
Related ideas
- Truth value
- Mathematics
- Double negation
- Formal system
- Metalanguage
- Tautology (logic)
- Logic
- Classical logic
- De Morgan's laws
- Denmark
- Contraposition
- Europe
- Absorption law
Place in the tradition
Part of Doxastic logic.