Definition:Truth Table/Characteristic

Definition

Let $\circledcirc$ be a logical connective.

The characteristic truth table for $\circledcirc$ is the truth table describing the truth function of $\circledcirc$:

$\begin{array}{|cc||c|} \hline p & q & p \circledcirc q \\ \hline F & F & x \\ F & T & x \\ T & F & x \\ T & T & x \\ \hline \end{array}$

where $x$ is replaced by either $F$ or $T$ as appropriate on each row.

The characteristic truth tables of the various logical connectives are listed below.

Logical Negation

The characteristic truth table of the negation operator $\neg p$ is as follows:

$\begin{array}{|c||c|} \hline p & \neg p \\ \hline \F & \T \\ \T & \F \\ \hline \end{array}$

Logical Conjunction

The characteristic truth table of the logical conjunction operator $p \land q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \land q \\ \hline \F & \F & \F \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$

Logical Disjunction

The characteristic truth table of the logical disjunction operator $p \lor q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \lor q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \T \\ \hline \end{array}$

Biconditional

The characteristic truth table of the biconditional operator $p \iff q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \iff q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$

Exclusive Disjunction

The characteristic truth table of the exclusive or operator $p \oplus q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \oplus q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \F \\ \hline \end{array}$

Conditional

The characteristic truth table of the conditional (implication) operator $p \implies q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \implies q \\ \hline \F & \F & \T \\ \F & \T & \T \\ \T & \F & \F \\ \T & \T & \T \\ \hline \end{array}$

Logical NAND

The characteristic truth table of the logical NAND operator $p \uparrow q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \uparrow q \\ \hline F & F & T \\ F & T & T \\ T & F & T \\ T & T & F \\ \hline \end{array}$

Logical NOR

The characteristic truth table of the logical NOR operator $p \downarrow q$ is as follows:

$\begin{array}{|cc||c|} \hline p & q & p \downarrow q \\ \hline F & F & T \\ F & T & F \\ T & F & F \\ T & T & F \\ \hline \end{array}$