# Logical equivalence

In logic, statements $p$ and $q$ 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 $p$ and $q$ is sometimes expressed as $p\equiv q$ , ${\textsf {E}}pq$ , or $p\iff q$ . 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.

## Logical equivalences

Equivalence Name
$p\wedge \top \equiv p$ $p\vee \bot \equiv p$ Identity laws
$p\vee \top \equiv \top$ $p\wedge \bot \equiv \bot$ Domination laws
$p\vee p\equiv p$ $p\wedge p\equiv p$ Idempotent laws
$\neg (\neg p)\equiv p$ Double negation law
$p\vee q\equiv q\vee p$ $p\wedge q\equiv q\wedge p$ Commutative laws
$(p\vee q)\vee r\equiv p\vee (q\vee r)$ $(p\wedge q)\wedge r\equiv p\wedge (q\wedge r)$ Associative laws
$p\vee (q\wedge r)\equiv (p\vee q)\wedge (p\vee r)$ $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$ Distributive laws
$\neg (p\wedge q)\equiv \neg p\vee \neg q$ $\neg (p\vee q)\equiv \neg p\wedge \neg q$ De Morgan's laws
$p\vee (p\wedge q)\equiv p$ $p\wedge (p\vee q)\equiv p$ Absorption laws
$p\vee \neg p\equiv \top$ $p\wedge \neg p\equiv \bot$ Negation laws

Logical equivalences involving conditional statements：

1. $p\implies q\equiv \neg p\vee q$ 2. $p\implies q\equiv \neg q\implies \neg p$ 3. $p\vee q\equiv \neg p\implies q$ 4. $p\wedge q\equiv \neg (p\implies \neg q)$ 5. $\neg (p\implies q)\equiv p\wedge \neg q$ 6. $(p\implies q)\wedge (p\implies r)\equiv p\implies (q\wedge r)$ 7. $(p\implies q)\vee (p\implies r)\equiv p\implies (q\vee r)$ 8. $(p\implies r)\wedge (q\implies r)\equiv (p\vee q)\implies r$ 9. $(p\implies r)\vee (q\implies r)\equiv (p\wedge q)\implies r$ Logical equivalences involving biconditionals：

1. $p\iff q\equiv (p\implies q)\wedge (q\implies p)$ 2. $p\iff q\equiv \neg p\iff \neg q$ 3. $p\iff q\equiv (p\wedge q)\vee (\neg p\wedge \neg q)$ 4. $\neg (p\iff q)\equiv p\iff \neg q$ ## Example

The following statements are logically equivalent:

1. If Lisa is in Denmark, then she is in Europe. (In symbols, $d\implies e$ .)
2. If Lisa is not in Europe, then she is not in Denmark. (In symbols, $\neg e\implies \neg d$ .)

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) logically equivalent.)

## Relation to material equivalence

Logical equivalence is different from material equivalence. Formulas $p$ and $q$ are logically equivalent if and only if the statement of their material equivalence ($p\iff q$ ) is a tautology (Copi et at. 2014:348).[full citation needed]

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