A new logical binary connective

Question

Let $p$, $q$, and $r$ be logical statements. Let us consider the following compound statement: $$s\equiv[p\land\neg(q\lor r)]\lor [q\land\neg(p\lor r)]\lor [r\land\neg(p\lor q)].$$ Is there any logical binary sentential connective such as $\bigstar$ for which the following logical equivalence is valid: $$p\bigstar q \bigstar r\equiv s.$$

Answer 1

Assume left-to-right association for the $\star$. To satisfy $(p\star q)\star r \equiv s$ requires at least $3$ values for $p\star q$:

• Let $A = F\star F$, then for any $r$, $A \star r \equiv r$;
• Let $B = F\star T$, then for any $r$, $B \star r \equiv \neg r$; and
• Let $C = T\star T$, then for any $r$, $C \star r \equiv F$.

Then $p\star q$ can't be just true-or-false.