Logical equivalence

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 ${\displaystyle d\to e}$).
2. If Lisa is not in Europe, then she is not in Denmark (a statement of the form ${\displaystyle \mathrm{\neg }e\to \mathrm{\neg }d}$).

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 ${\displaystyle p}$ and ${\displaystyle q}$ are logically equivalent if and only if the statement of their material equivalence (${\displaystyle p\leftrightarrow q}$) is a tautology.

The material equivalence of ${\displaystyle p}$ and ${\displaystyle q}$ (often written as ${\displaystyle p\leftrightarrow q}$) is itself another statement in the same object language as ${\displaystyle p}$ and ${\displaystyle q}$. This statement expresses the idea "'${\displaystyle p}$ if and only if ${\displaystyle q}$'". In particular, the truth value of ${\displaystyle p\leftrightarrow q}$ 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 ${\displaystyle p}$ and ${\displaystyle q}$. The statements are logically equivalent if, in every model, they have the same truth value.