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'''Logical equivalence''' is a concept in [[logic]] and [[mathematics]] that describes a relationship between two [[statements]] or [[propositional logic|propositions]] that are true in the same conditions. This means that the statements are interchangeable in any context without changing the truth value of any [[logical expression]] in which they appear.

== Definition ==
Logical equivalence between two propositions, \( P \) and \( Q \), is denoted by \( P \equiv Q \). This relationship holds if both \( P \) and \( Q \) have the same [[truth value]] in every possible scenario. In terms of [[truth table]]s, \( P \) and \( Q \) are logically equivalent if their truth tables match exactly, column for column.

== Formal Expression ==
The logical equivalence of \( P \) and \( Q \) can be expressed using the [[biconditional]] operator, which is often represented as \( \leftrightarrow \). Thus, \( P \equiv Q \) can be written as \( P \leftrightarrow Q \). This can be further expressed in terms of other logical operators:
\[ P \equiv Q \equiv (P \rightarrow Q) \land (Q \rightarrow P) \]
where \( \rightarrow \) represents the [[implication]] operator, and \( \land \) represents the [[logical conjunction]].

== Properties ==
Logical equivalence has several important properties, including:
* '''Reflexivity''': Every statement is equivalent to itself.
* '''Symmetry''': If \( P \) is equivalent to \( Q \), then \( Q \) is equivalent to \( P \).
* '''Transitivity''': If \( P \) is equivalent to \( Q \), and \( Q \) is equivalent to \( R \), then \( P \) is equivalent to \( R \).

These properties make logical equivalence an [[equivalence relation]] on the set of all logical statements.

== Applications ==
Logical equivalence is fundamental in various areas of mathematics and computer science, particularly in the simplification of [[Boolean algebra|Boolean expressions]], [[proof theory]], and the design of [[digital circuits]]. It is also crucial in the fields of [[philosophy]], especially in the analysis and construction of [[philosophical argument]]s.

== Examples ==
1. \( P \land Q \) is logically equivalent to \( Q \land P \) (Commutativity of conjunction).
2. \( P \lor Q \) is logically equivalent to \( Q \lor P \) (Commutativity of disjunction).
3. \( \neg (P \land Q) \) is logically equivalent to \( \neg P \lor \neg Q \) (De Morgan's Laws).

== See Also ==
* [[Tautology (logic)]]
* [[Contradiction]]
* [[Logical implication]]
* [[Logical connective]]

[[Category:Logic]]
[[Category:Mathematical logic]]
[[Category:Philosophy of logic]]

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Logical equivalence - Revision history

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