NAND with Equal Arguments/Proof by Truth Table

Theorem

$p \uparrow p \dashv \vdash \neg p$

That is, the NAND of a proposition with itself corresponds to the negation operation.

Proof

We apply the Method of Truth Tables:

$\begin{array}{|ccc||cc|} \hline

p & \uparrow & p & \neg & p \\ \hline \F & \T & \F & \T & \F \\ \T & \F & \T & \F & \T \\ \hline \end{array}$

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

$\blacksquare$

Sources

• 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants