Disjunction has no Inverse
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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 $\circ$.
$\blacksquare$