NOR with Equal Arguments

Theorem
Let $\downarrow$ signify the NOR operation.

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

That is, the NOR of a proposition with itself corresponds to the negation operator.

Proof by Tableau
Proceed by the tableau method:

Proof by Truth Table
Apply the Method of Truth Tables:


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

p & \downarrow & 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.