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:
- $\begin {array} {|cc||c|} \hline
p & q & p \downarrow q \\ \hline \F & \F & \T \\ \F & \T & \F \\ \T & \F & \F \\ \T & \T & \F \\ \hline \end {array}$
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical ConstantsFunctions