Definition:Conditional/Truth Table/Number

Definition

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

Ascending order:

$1101$ or $\T \T \F \T$

Descending order:

$1011$ or $\T \F \T \T$

As $\implies$ is not commutative, it is also instructive to give a truth table number for $p \impliedby q$ (which of course is the same as $q \implies p$).

Hence the truth table numbers of the conditional (implication) operator $p \impliedby q$ and the complements of both $p \implies q$ and $p \impliedby q$ are as follows:

$p \impliedby q$:

Ascending order:

$1011$ or $\T \F \T \T$

Descending order:

$1101$ or $\T \T \F \T$

$\map \neg {p \implies q}$:

Ascending order:

$0010$ or $\F \F \T \F$

Descending order:

$0100$ or $\F \T \F \F$

$\map \neg {q \implies p}$:

Ascending order:

$0100$ or $\F \T \F \F$

Descending order:

$0010$ or $\F \F \T \F$

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants