Properties of NAND

Theorem

Let $\uparrow$ signify the NAND operation.

The following results hold:

NAND With Equal Arguments

$p \uparrow p \dashv \vdash \neg p$

That is, the NAND of a proposition with itself corresponds to the negation operation.

NAND is Commutative

$p \uparrow q \dashv \vdash q \uparrow p$

NAND is Not Associative

$p \uparrow \left({q \uparrow r}\right) \not \vdash \left({p \uparrow q}\right) \uparrow r$