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# System prepended metadata

title: Practice Exercise (Rules of Inference)

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# 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)