# Logical equivalence

In logic and mathematics, statements ${\displaystyle p}$ and ${\displaystyle q}$ are said to be logically equivalent if they have the same

model.[1]
The logical equivalence of ${\displaystyle p}$ and ${\displaystyle q}$ is sometimes expressed as ${\displaystyle p\equiv q}$, ${\displaystyle p::q}$, ${\displaystyle {\textsf {E}}pq}$, or ${\displaystyle p\iff q}$, 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

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

### Logical equivalences involving conditional statements

1. ${\displaystyle p\implies q\equiv \neg p\vee q}$
2. ${\displaystyle p\implies q\equiv \neg q\implies \neg p}$
3. ${\displaystyle p\vee q\equiv \neg p\implies q}$
4. ${\displaystyle p\wedge q\equiv \neg (p\implies \neg q)}$
5. ${\displaystyle \neg (p\implies q)\equiv p\wedge \neg q}$
6. ${\displaystyle (p\implies q)\wedge (p\implies r)\equiv p\implies (q\wedge r)}$
7. ${\displaystyle (p\implies q)\vee (p\implies r)\equiv p\implies (q\vee r)}$
8. ${\displaystyle (p\implies r)\wedge (q\implies r)\equiv (p\vee q)\implies r}$
9. ${\displaystyle (p\implies r)\vee (q\implies r)\equiv (p\wedge q)\implies r}$

### Logical equivalences involving biconditionals

1. ${\displaystyle p\iff q\equiv (p\implies q)\wedge (q\implies p)}$
2. ${\displaystyle p\iff q\equiv \neg p\iff \neg q}$
3. ${\displaystyle p\iff q\equiv (p\wedge q)\vee (\neg p\wedge \neg q)}$
4. ${\displaystyle \neg (p\iff q)\equiv p\iff \neg q\equiv p\oplus q}$

Where ${\displaystyle \oplus }$ represents

XOR
.

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

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) 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.[2]

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.