Disjunction has no Inverse

Theorem

Let $\lor$ denote the disjunction operation of propositional logic.

Then there exists no binary logical connective $\circ$ such that:

$(1): \quad \forall p, q \in \left\{{T, F}\right\}: \left({p \lor q}\right) \circ q = p$

Proof

Let $q$ be true.

Then $p \lor q = T$, whatever truth value $p$ holds.

Either $T \circ T = T$ or $T \circ T = F$, but not both.

So if $q = T$ either:

$\left({p \land q}\right) \circ q = T$

or:

$\left({p \land q}\right) \circ q = F$

If the first, then $(1)$ does not hold when $p = F$.

If the second, then $(1)$ does not hold when $p = T$.

Hence there can be no such $\circ$.

$\blacksquare$