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 1

By the tableau method of natural deduction:

$p \downarrow p \vdash \neg p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$p \downarrow p$	Premise	(None)
2		1	$\neg \paren {p \lor p}$	Sequent Introduction	1	Definition of Logical NOR
3		1	$\neg p$	Sequent Introduction	2	Rule of Idempotence: Disjunction

$\Box$

By the tableau method of natural deduction:

$\neg p \vdash p \downarrow p$

Line		Pool	Formula	Rule	Depends upon	Notes
1		1	$\neg p$	Premise	(None)
2		1	$\neg \paren {p \lor p}$	Sequent Introduction	1	Rule of Idempotence: Disjunction
3		1	$p \downarrow p$	Sequent Introduction	2	Definition of Logical NOR

$\blacksquare$

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 boolean interpretations.

$\blacksquare$

Also see

• NAND with Equal Arguments

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.3.3$