# Practice Exercise (Rules of Inference)

Prove the validity of the following argument forms:

Some of these are quite challenging so they may take you a long time to solve

- $$
\begin{align*}
&(p \land t) \to (r \lor s) \\
&q \to (u \land t) \\
& u\to p \\
& \neg s\\
\hline
\therefore ~& q \to r
\end{align*}
$$

- $$
\begin{align*}
&p \to q \\
&\neg p \to r\\
& q \to s \\
& r \to s \\
\hline
\therefore ~& s
\end{align*}
$$

- $$
\begin{align*}
&\forall x (p \to Q(x)) \\
&\exists x (\neg p \to R(x)) \\
&\neg Q(a) \\
\hline
\therefore ~& \exists x R(x)
\end{align*}
$$

Translate the following argument forms and prove their validity

- > The law of blood shall prevail if the foundation can protect us, or the law of concrete shall prevail if the foundation can protect us.
>
> The law of blood shall not prevail.
>
> The law of concrete shall not prevail.
>
> The foundation can protect us or the scarlet king awakens.
>
> Therefore, the scarlet king awakens.
>
> [SCP-001 (Tufto)](http://www.scp-wiki.net/tuftos-proposal)

- > Everything the void perceives it consumes.
>
> The void does not consume itself.
>
> If the void cannot perceive itself then it is the imperceptible manifestation of unbeing.
>
> Therefore the void is the manifestation of unbeing.
>
> [SCP-2470](http://www.scp-wiki.net/scp-2470)