Binary Logical Connectives with Inverse

== Theorem ==
Let $\circ$ be a binary logical connective.
Then there exists another binary logical connective $*$ such that:
:$\forall p, q \in \set {\F, \T}: \paren {p \circ q} * q \dashv \vdash p \dashv \vdash q * \paren {p \circ q}$
 $\circ$ is either:
:$(1): \quad$ the exclusive or operator
or:
:$(2): \quad$ the biconditional operator.
That is, the only truth functions that have an inverse operation are the exclusive or and the biconditional.
== Proof ==
=== Necessary Condition ===
Let $\circ$ be a binary logical connective such that there exists $*$ such that:
:$\paren {p \circ q} * q \dashv \vdash p$
That is, by definition (and minor abuse of notation):
:$\forall p, q \in \set {\F, \T}: \paren {p \circ q} * q = p$
For reference purposes, let us list from Binary Truth Functions the complete truth table containing all of the binary logical connectives:
$\begin{array}{|r|cccc|} \hline
p & \T & \T & \F & \F \\
q & \T & \F & \T & \F \\
\hline
\map {f_\T} {p, q} & \T & \T & \T & \T \\
p \lor q & \T & \T & \T & \F \\
p \impliedby q & \T & \T & \F & \T \\
\map {\pr_1} {p, q} & \T & \T & \F & \F \\
p \implies q & \T & \F & \T & \T \\
\map {\pr_2} {p, q} & \T & \F & \T & \F \\
p \iff q & \T & \F & \F & \T \\
p \land q & \T & \F & \F & \F \\
p \uparrow q & \F & \T & \T & \T \\
\map \neg {p \iff q} & \F & \T & \T & \F \\
\map {\overline {\pr_2} } {p, q} & \F & \T & \F & \T \\
\map \neg {p \implies q} & \F & \T & \F & \F \\
\map {\overline {\pr_1} } {p, q} & \F & \F & \T & \T \\
\map \neg {p \impliedby q} & \F & \F & \T & \F \\
p \downarrow q & \F & \F & \F & \T \\
\map {f_\F} {p, q} & \F & \F & \F & \F \\
\hline
\end{array}$
Suppose that for some $q \in \set {\F, \T}$:
:$\paren {p \circ q}_{p = \F} = \paren {p \circ q}_{p = \T}$
Then:
:$\paren {\paren {p \circ q} * q}_{p = \F} = \paren {\paren {p \circ q} * q}_{p = \T}$
and so either:
:$\paren {\paren {p \circ q} * q}_{p = \F} \ne p$
or:
:$\paren {\paren {p \circ q} * q}_{p = \T} \ne p$
Thus for $\circ$ to have an inverse operation it is necessary for $\F \circ q \ne \T \circ q$.
This eliminates:
The remaining connectives which may have inverses are:
$\begin{array}{|r|cccc|} \hline
p & \T & \T & \F & \F \\
q & \T & \F & \T & \F \\
\hline
\map {\pr_1} {p, q} & \T & \T & \F & \F \\
p \iff q & \T & \F & \F & \T \\
\map \neg {p \iff q} & \F & \T & \T & \F \\
\map {\overline {\pr_1} } {p, q} & \F & \F & \T & \T \\
\hline
\end{array}$
Suppose that for some $p \in \set {\F, \T}$:
:$\paren {p \circ q}_{q = \F} = \paren {p \circ q}_{q = \T}$
Then:
:$\paren {q * \paren {p \circ q} }_{q = \F} = \paren {q * \paren {p \circ q} }_{q = \T}$
and so either:
:$\paren {q * \paren {p \circ q} }_{q = \F} \ne p$
or:
:$\paren {q * \paren {p \circ q} }_{q = \T} \ne p$
This eliminates:
We are left with exclusive or and the biconditional.
The result follows from Exclusive Or is Self-Inverse and Biconditional is Self-Inverse.
=== Sufficient Condition ===
Let $\circ$ be the exclusive or operator.
Then by Exclusive Or is Self-Inverse it follows that:
:$\paren {p \circ q} \circ q \dashv \vdash p$
Thus $*$ is the inverse operation of the exclusive or operation.
Similarly, let $\circ$ be the biconditional operator.
Then by Biconditional is Self-Inverse it follows that:
:$\paren {p \circ q} \circ q \dashv \vdash p$
Category:Exclusive Or
Category:Biconditional
Category:Propositional Logic