NAND is not Associative

Theorem
Let $\uparrow$ signify the NAND operation.

Then there exist propositions $p,q,r$ such that:
 * $p \uparrow \left({q \uparrow r}\right) \not \vdash \left({p \uparrow q}\right) \uparrow r$

That is, NAND is not associative.

Also see

 * NOR is not Associative