NOR is not Associative

Theorem
Let $\downarrow$ signify the NOR operation.

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

That is, NOR is not associative.

Also see

 * NAND is not Associative