# Logical connective

A logical connective (also called logical operator) is a logical constant used to connect logical formulas.

## Natural language

The standard logical connectives of classical logic have rough equivalents in the grammars of natural languages. However, the meanings of natural language connectives are not precisely identical to their nearest equivalents in classical logic.

The following table shows the standard classically definable approximations for the English connectives.

English Connective Symbol Logic gate
not negation $\neg$ NOT
and conjunction $\land$ AND
or disjunction $\lor$ OR
if…then material implication $\rightarrow$ IMPLY
…if converse implication $\leftarrow$
if and only if biconditional $\leftrightarrow$ XNOR
not both alternative denial $\uparrow$ NAND
neither…nor joint denial $\downarrow$ NOR
but not material nonimplication $\nrightarrow$ NIMPLY
converse nonimplication $\nleftarrow$

## Computer science

A truth-functional approach to logical connectives is implemented as logic gates in digital circuits.

## Unary operations

### Truth table for unary logical operators

$p$ $T$ $F$ $p$ $\neg p$
T T F T F
F T F F T

## Binary operations

### Truth table for binary logical operators

$p$ $q$ $T$ $F$ $\neg p$ $\neg q$ $p \land q$ $p \lor q$ $p \rightarrow q$ $p \leftrightarrow q$ $p \leftarrow q$
T T T F F T T T T T T
T F T F F T F T F F T
F T T F T F F T T F F
F F T F T F F F T T T

where:

• T = true
• F = false