Exclusive Or with Itself

Theorem

Exclusive or destroys copies of itself:

$p \oplus p \dashv \vdash \bot$


Proof

We apply the Method of Truth Tables.

As can be seen by inspection, the truth values under the main connective match for each boolean interpretations.

$\begin{array}{|ccc|} \hline p & \oplus & p \\ \hline \F & \F & \F \\ \T & \F & \T \\ \hline \end{array}$

$\blacksquare$


Sources

  • 1993: M. Ben-Ari: Mathematical Logic for Computer Science: $\S 2.4$: Logical Equivalence and Substitution
  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$
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