NOR with Equal Arguments/Proof by Truth Table

Theorem

$p \downarrow p \dashv \vdash \neg p$

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

Proof

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 boolean interpretations.

$\blacksquare$