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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[4]Edit

Equivalence Name
 
 
Identity laws
 
 
Domination laws
 
 
Idempotent laws
  Double negation law
 
 
Commutative laws
 
 
Associative laws
 
 
Distributive laws
 
 
De Morgan's laws
 
 
Absorption laws
 
 
Negation laws

Logical equivalences involving conditional statementsEdit

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
  7.  
  8.  
  9.  

Logical equivalences involving biconditionalsEdit

  1.  
  2.  
  3.  
  4.  

ExamplesEdit

In logicEdit

The following statements are logically equivalent:

  1. If Lisa is in Denmark, then she is in Europe (a statement of the form  ).
  2. 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 non-classical 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.[5]

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

  1. ^ a b "The Definitive Glossary of Higher Mathematical Jargon — Equivalent Claim". Math Vault. 2019-08-01. Retrieved 2019-11-24.
  2. ^ Mendelson, Elliott (1979). Introduction to Mathematical Logic (2 ed.). p. 56.
  3. ^ "Mathematics | Propositional Equivalences". GeeksforGeeks. 2015-06-22. Retrieved 2019-11-24.
  4. ^ "Mathematics | Propositional Equivalences". GeeksforGeeks. 2015-06-22. Retrieved 2019-11-24.
  5. ^ Copi, Irving; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (New International Edition ed.). Pearson. p. 348.CS1 maint: extra text (link)