Binary Logical Connective is Self-Inverse iff Exclusive Or

Source Work

1993: M. Ben-Ari: Mathematical Logic for Computer Science:

Chapter $2$: Propositional Calculus

$2.1$. Boolean Operations

Mistake

Let $\circ$ be a binary logical connective.

Then:

$\paren {p \circ q} \circ q \dashv \vdash p$

if and only if $\circ$ 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.

$\blacksquare$

Sources

• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.1$: Boolean operators