In a chapter regarding strenghtening and weakening of propositions, the following is asked:
Show with a calculation that the following is a tautology:
$$( R \land (P \Rightarrow Q)) \Rightarrow ((R \Rightarrow P) \Rightarrow Q) $$
I'm not sure I'm on the right track with my solution or if I'm missing steps. Here goes:
To show this, it is enough to show that
$$( R \land (P \Rightarrow Q)) \vDash ((R \Rightarrow P) \Rightarrow Q) $$
I begin by rewrting the right hand side.
1) Implication, twice:
$$ \neg (\neg R \lor P) \lor Q $$
2) De morgan
$$ (\neg \neg R \land \neg P) \lor Q$$
3) Double negation
$$ (R \land \neg P) \lor Q$$
4) Distribution and implication
$$ (Q \lor R) \land (P \Rightarrow Q) $$
Now, by the standard weakening rules I know that $ R \vDash Q \lor R$
So this concludes the proof that $$( R \land (P \Rightarrow Q)) \vDash ((R \Rightarrow P) \Rightarrow Q) $$
and thus ive shown $$( R \land (P \Rightarrow Q)) \Rightarrow ((R \Rightarrow P) \Rightarrow Q) $$ is a tautology.
Now, does this make any sense :) Especially the weakening part. Can I use that rule like this?
Text used: http://www.amazon.com/Logical-Reasoning-A-First-Course/dp/095430067X in which $P \vDash Q$ is specified as meaning 'P is a stronger proposition than Q'. False being the strongest, true being the weakest: $$ False \vDash P \land Q \vDash P \vDash P \lor Q \vDash True$$