Logical equivalence
In logic and mathematics, statements and are said to be logically equivalent if they are provable from each other under a set of axioms,^{[1]} or have the same truth value in every model.^{[2]} The logical equivalence of and is sometimes expressed as , ,^{[3]} , or , 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.
Logical equivalencesEdit
In logic, many common logical equivalences exist and are often listed as laws or properties. The following tables illustrate some of these.
General logical equivalences^{[3]}Edit
Equivalence  Name 


Identity laws 

location 

Idempotent or tautology laws 
Double negation law  

Commutative laws 

Associative laws 

Distributive laws 

De Morgan's laws 

Absorption laws 

Negation laws 
Logical equivalences involving conditional statementsEdit
Logical equivalences involving biconditionalsEdit
ExamplesEdit
In logicEdit
The following statements are logically equivalent:
 If Lisa is in Denmark, then she is in Europe (a statement of the form ).
 If Lisa is not in Europe, then she is not in Denmark (a statement of the form ).
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 Denmark is false or Lisa is in Europe is true.
(Note that in this example, classical logic is assumed. Some nonclassical logics do not deem (1) and (2) to be logically equivalent.)
In mathematicsEdit
In mathematics, two statements and are often said to be logically equivalent, if they are provable from each other given a set of axioms and presuppositions. For example, the statement " is divisible by 6" can be regarded as equivalent to the statement " is divisible by 2 and 3", since one can prove the former from the latter (and vice versa) using some knowledge from basic number theory.^{[1]}
Relation to material equivalenceEdit
Logical equivalence is different from material equivalence. Formulas and are logically equivalent if and only if the statement of their material equivalence ( ) is a tautology.^{[4]}
The material equivalence of and (often written as ) is itself another statement in the same object language as and . This statement expresses the idea "' if and only if '". In particular, the truth value of can change from one model to another.
On the other hand, the claim that two formulas are logically equivalent is a statement in the metalanguage, which expresses a relationship between two statements and . The statements are logically equivalent if, in every model, they have the same truth value.
See alsoEdit
ReferencesEdit
 ^ ^{a} ^{b} "The Definitive Glossary of Higher Mathematical Jargon — Equivalent Claim". Math Vault. 20190801. Retrieved 20191124.
 ^ Mendelson, Elliott (1979). Introduction to Mathematical Logic (2 ed.). pp. 56.
 ^ ^{a} ^{b} "Mathematics  Propositional Equivalences". GeeksforGeeks. 20150622. Retrieved 20191124.
 ^ Copi, Irving; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (New International ed.). Pearson. p. 348.