A logical connective (also called logical operator) is a logical constant used to connect logical formulas.
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 | $\nleftarrow$ |
A truth-functional approach to logical connectives is implemented as logic gates in digital circuits.
$p$ | $T$ | $F$ | $p$ | $\neg p$ | |
---|---|---|---|---|---|
T | T | F | T | F | |
F | T | F | F | T |
$p$ | $q$ | $T$ | $F$ | $\neg p$ | $\neg q$ | $p \land q$ | $p \lor q$ | $p \rightarrow q$ | $p \leftrightarrow q$ | $p \leftarrow q$ | |
---|---|---|---|---|---|---|---|---|---|---|---|
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: