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In logic, statements and are logically equivalent if they have the same logical content. That is, if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of and is sometimes expressed as , , or . However, these symbols are also used for material equivalence. Proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.

Contents

Logical equivalencesEdit

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 statements:

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

Logical equivalences involving biconditionals:

  1.  
  2.  
  3.  
  4.  

ExampleEdit

The following statements are logically equivalent:

  1. If Lisa is in Denmark, then she is in Europe. (In symbols,  .)
  2. If Lisa is not in Europe, then she is not in Denmark. (In symbols,  .)

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) logically equivalent.)

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 (Copi et at. 2014:348).[full citation needed]

The material equivalence of   and   (often written  ) 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.

The claim that two formulas are logically equivalent is a statement in the metalanguage, expressing a relationship between two statements   and  . The statements are logically equivalent if, in every model, they have the same truth value.

See alsoEdit

ReferencesEdit

  • Irving M. Copi, Carl Cohen, and Kenneth McMahon, Introduction to Logic, 14th edition, Pearson New International Edition, 2014.
  • Elliot Mendelson, Introduction to Mathematical Logic, second edition, 1979.