Unary Truth Functions

Theorem
There are $4$ distinct unary truth functions:


 * $(1): \quad$ The constant function $f \left({p}\right) = F$
 * $(2): \quad$ The constant function $f \left({p}\right) = T$
 * $(3): \quad$ The identity function $f \left({p}\right) = p$
 * $(4): \quad$ The logical not function $f \left({p}\right) = \neg p$

Proof
From Count of Truth Functions there are $2^{\left({2^1}\right)} = 4$ distinct truth functions on $1$ variable.

These can be depicted in a truth table as follows:


 * $\begin{array}{|c|cccc|} \hline

p & \circ_1 & \circ_2 & \circ_3 & \circ_4 \\ \hline T & T & T & F & F \\ F & T & F & T & F \\ \hline \end{array}$

$\circ_1$: Whether $p = T$ or $p = F$, $\circ_1 \left({p}\right) = T$.

Thus $\circ_1$ is the constant function $\circ_1 \left({p}\right) = T$.

$\circ_2$: We have:
 * $(1): \quad p = T \implies \circ_2 \left({p}\right) = T$
 * $(2): \quad p = F \implies \circ_2 \left({p}\right) = F$

Thus $\circ_2$ is the identity function $\circ_2 \left({p}\right) = p$.

$\circ_3$: We have:
 * $(1): \quad p = T \implies \circ_3 \left({p}\right) = F$
 * $(2): \quad p = F \implies \circ_3 \left({p}\right) = T$

Thus $\circ_3$ is the logical not function $\circ_3 \left({p}\right) = \neg p$.

$\circ_4$: Whether $p = T$ or $p = F$, $\circ_4 \left({p}\right) = F$.

Thus $\circ_4$ is the constant function $\circ_4 \left({p}\right) = F$.

All four have been examined, and there are no other unary truth functions.