# 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 1

By the tableau method of natural deduction:

$p \uparrow p \vdash \neg p$
Line Pool Formula Rule Depends upon Notes
1 1 $p \uparrow p$ Premise (None)
2 1 $\neg \paren {p \land p}$ Sequent Introduction 1 Definition of Logical NAND
3 1 $\neg p$ Sequent Introduction 2 Rule of Idempotence: Conjunction

$\Box$

By the tableau method of natural deduction:

$\neg p \vdash p \uparrow p$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p$ Premise (None)
2 1 $\neg \paren {p \land p}$ Sequent Introduction 1 Rule of Idempotence: Conjunction
3 1 $p \uparrow p$ Sequent Introduction 2 Definition of Logical NAND

$\blacksquare$

## 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 boolean interpretations.

$\blacksquare$