NAND with Equal Arguments

Theorem
Let $\uparrow$ signify the NAND operation.

Then, for any proposition $p$:
 * $p \uparrow p \dashv \vdash \neg p$

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

Proof by Tableau
Proceed by the Tableau method:

Proof by Truth Table
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 models.