Logical equivalence

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In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of and is sometimes expressed as , , , 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.

Contents

Logical equivalences

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

EquivalenceName

Identity laws

Domination laws

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 statements

Logical equivalences involving biconditionals

Examples

In logic

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.)

Relation to material equivalence

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. [2]

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 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 also

Related Research Articles

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<span class="mw-page-title-main">Logical connective</span> Symbol connecting sentential formulas in logic

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<span class="mw-page-title-main">De Morgan's laws</span> Pair of logical equivalences

In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation.

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<span class="mw-page-title-main">Exclusive or</span> True when either but not both inputs are true

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<span class="mw-page-title-main">Negation</span> Logical operation

In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", standing for " is not true", written , or . It is interpreted intuitively as being true when is false, and false when is true. Negation is thus a unary logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition is the proposition whose proofs are the refutations of .

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<span class="mw-page-title-main">Logical biconditional</span> Concept in logic and mathematics

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<span class="mw-page-title-main">Material conditional</span> Logical connective

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References

  1. Mendelson, Elliott (1979). Introduction to Mathematical Logic (2 ed.). pp.  56. ISBN   9780442253073.
  2. Copi, Irving; Cohen, Carl; McMahon, Kenneth (2014). Introduction to Logic (New International ed.). Pearson. p. 348.