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:

$p \downarrow p \vdash \neg p$

Line		Pool	Formula	Rule	Depends upon
1		1	$p \downarrow p$	$\mathrm P$	(None)
2		1	$\neg \left({p \lor p}\right)$	By definition	1
3		1	$\neg p$	$\mathrm {Idemp}$	2

$\neg p \vdash p \downarrow p$

Line		Pool	Formula	Rule	Depends upon
1		1	$\neg p$	$\mathrm P$	(None)
2		1	$\neg \left({p \lor p}\right)$	$\mathrm {Idemp}$	1
3		1	$p \downarrow p$	By definition	2

$\blacksquare$

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.

$\blacksquare$