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.

In this case $(1)$ simplifies to:

$(2): \quad \forall p \in \left\{{T, F}\right\}: T \circ T = p$

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

If $p = F$, then it must be that $T \circ T = F$.

If $p = T$, then it must be that $T \circ T = T$.

Hence there can be no such $\circ$.

$\blacksquare$