In logic and mathematics are said to be logically equivalent if they have the same truth value in every model., 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 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

Logical equivalences involving conditional statements

Logical equivalences involving biconditionals

represents XOR.

Examples

In logic

The following statements are logically equivalent:

1. If Lisa is in Denmark, then she is in Europe).

2. ).

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

) is a tautology.

) is itself another statement in the same object language can change from one model to another.

On the other hand, the claim that two formulas are logically equivalent is a statement in metalanguage. The statements are logically equivalent if, in every model, they have the same truth value.

See also

• Entailment

• Equisatisfiability

• If and only if

• Logical biconditional

• Logical equality

• ≡ the iff symbol (U+2261 IDENTICAL TO)

• ∷ the a is to b as c is to d symbol (U+2237 PROPORTION)

• ⇔ the double struck biconditional (U+21D4 LEFT RIGHT DOUBLE ARROW)

• ↔ the bidirectional arrow (U+2194 LEFT RIGHT ARROW)

References