NOR is not Associative/Proof by Truth Table

Theorem
Let $\downarrow$ signify the NOR operation.

Then there exist propositions $p, q, r$ such that:


 * $p \downarrow \paren {q \downarrow r} \not \vdash \paren {p \downarrow q} \downarrow r$

That is, NOR is not associative.

Proof
Apply the Method of Truth Tables:


 * $\begin{array}{|ccccc||ccccc|} \hline

p & \downarrow & (q & \downarrow & r) & (p & \downarrow & q) & \downarrow & r \\ \hline F & F & F & T & F & F & T & F & F & F \\ F & T & F & F & T & F & T & F & F & T \\ F & T & T & F & F & F & F & T & T & F \\ F & T & T & F & T & F & F & T & F & T \\ T & F & F & T & F & T & F & F & T & F \\ T & F & F & F & T & T & F & F & F & T \\ T & F & T & F & F & T & F & T & T & F \\ T & F & T & F & T & T & F & T & F & T \\ \hline \end{array}$

As can be seen by inspection, the truth values under the main connectives do not match for all boolean interpretations.