# Logical equivalence

(Redirected from Logically equivalent)

In logic, statements ${\displaystyle p}$ and ${\displaystyle q}$ are logically equivalent if they have the same logical content. This is a semantic concept; two statements are equivalent if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of ${\displaystyle p}$ and ${\displaystyle q}$ is sometimes expressed as ${\displaystyle p\equiv q}$, ${\displaystyle {\textsf {E}}pq}$, or ${\displaystyle p\iff q}$. However, these symbols are also used for material equivalence; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.

## Logical equivalences

Equivalence Name
${\displaystyle p\wedge {\textbf {T}}\equiv p}$
${\displaystyle p\vee {\textbf {F}}\equiv p}$
Identity laws
${\displaystyle p\vee {\textbf {T}}\equiv {\textbf {T}}}$
${\displaystyle p\wedge {\textbf {F}}\equiv {\textbf {F}}}$
Domination laws
${\displaystyle p\vee p\equiv p}$
${\displaystyle p\wedge p\equiv p}$
Idempotent 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
${\displaystyle p\vee \neg p\equiv {\textbf {T}}}$
${\displaystyle p\wedge \neg p\equiv {\textbf {F}}}$
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. ${\displaystyle (p\implies r)\wedge (q\implies r)\equiv (p\wedge q)\implies r}$
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. ${\displaystyle \neg (p\iff q)\equiv p\iff \neg q}$

## Example

The following statements are logically equivalent:

1. If Lisa is in France, then she is in Europe. (In symbols, ${\displaystyle f\implies e}$.)
2. If Lisa is not in Europe, then she is not in France. (In symbols, ${\displaystyle \neg e\implies \neg f}$.)

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 France 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. The material equivalence of ${\displaystyle p}$ and ${\displaystyle q}$ (often written ${\displaystyle p\iff q}$) is itself another statement, call it ${\displaystyle r}$, in the same object language as ${\displaystyle p}$ and ${\displaystyle q}$. ${\displaystyle r}$ expresses the idea "'${\displaystyle p}$ if and only if ${\displaystyle q}$'". In particular, the truth value of ${\displaystyle p\iff q}$ 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 ${\displaystyle p}$ and ${\displaystyle q}$. The claim that ${\displaystyle p}$ and ${\displaystyle q}$ are semantically equivalent does not depend on any particular model; it says that in every possible model, ${\displaystyle p}$ will have the same truth value as ${\displaystyle q}$. The claim that ${\displaystyle p}$ and ${\displaystyle q}$ are syntactically equivalent does not depend on models at all; it states that there is a deduction of ${\displaystyle q}$ from ${\displaystyle p}$ and a deduction of ${\displaystyle p}$ from ${\displaystyle q}$.

There is a close relationship between material equivalence and logical equivalence. Formulas ${\displaystyle p}$ and ${\displaystyle q}$ are syntactically equivalent if and only if ${\displaystyle p\iff q}$ is a theorem, while ${\displaystyle p}$ and ${\displaystyle q}$ are semantically equivalent if and only if ${\displaystyle p\iff q}$ is true in every model (that is, ${\displaystyle p\iff q}$ is logically valid).