Binary Logical Connective is Self-Inverse iff Exclusive Or

Source Work

 * Chapter $2$: Propositional Calculus
 * $2.1$. Boolean Operations
 * $2.1$. Boolean Operations

Mistake
Let $\circ$ be a binary logical connective.

Then:
 * $\paren {p \circ q} \circ q \dashv \vdash p$

$\circ$ is the exclusive or operator.


 * ''$\mathsf{XOR}$ is also essential since it is the only operator having an inverse, namely itself
 * $\paren {p \oplus q} \oplus q = p$''

Incorrect, as the biconditional operator has the same properties:
 * $\paren {\paren {p \iff q} \iff q} = p$

Proof
See Binary Logical Connectives with Inverse.