I'm currently enrolled in a introductory course on logic and I'm until now everything has been going great. I'm having some trouble with applying monotonicity and the strengthening/weakening of propositions.

The general of concept of strengthening/weakening seems clear; I understand $\displaystyle P \land Q$ is a stronger proposition than $\displaystyle P \lor Q$, so: $\displaystyle P \land Q \models P \lor Q$, since the former has 'less 1s' in its truth table it's a stronger/more restrictive proposition.

Monotonicity is a bit foggier. How I see it, is that it's basically a form of substitution if we have $\displaystyle P \models Q$, then $\displaystyle P \land R \models Q \land R$. If we substitute the left hand side with the 'same thing' we substitute the right hand side with, the relation with respect to stronger/weaker holds. Any more clarification on this subject is more than welcome.

So much for theory. Now I'm asked to show via a calculation that $\displaystyle (R \land (P \Rightarrow Q)) \Rightarrow ((R \Rightarrow P) \Rightarrow Q)$ is a tautology. To do this I have to show that $\displaystyle (R \land (P \Rightarrow Q)) \equiv((R \Rightarrow P) \Rightarrow Q)$.

Using some basic calculation/rewriting rules for implication, de Morgan and double negation, I ended up here:

$\displaystyle (R \land (\lnot P \lor Q)) \equiv (Q \lor (R \land \lnot P))$

Now it gets messy, but by applying distribution to the left hand side twice I end up with this:

$\displaystyle ((R \land \lnot P) \lor R) \land ((R \land \lnot P) \lor Q) \equiv (Q \lor (R \land \lnot P))$

Since the basic rules of strengthening\weakening tell us that $\displaystyle P \land Q \models P$, we also have that $\displaystyle ((R \land \lnot P) \lor R) \land ((R \land \lnot P) \lor Q) \models (Q \lor (R \land \lnot P))$.

By applying weakening to: $\displaystyle ((R \land \lnot P) \lor R) \land ((R \land \lnot P) \lor Q) \equiv (Q \lor (R \land \lnot P))$, we have that: $\displaystyle ((R \land \lnot P) \lor Q) \equiv (Q \lor (R \land \lnot P))$, which is what we wanted.

Now, for me this makes sense, but is this actually correct?

Edit: I'd like to add another exercise I'm not quite sure of, since it seems to be a bit harder.

Show that: $\displaystyle ((P \Rightarrow (Q \lor R)) \land (Q \Rightarrow R)) \Rightarrow (P \Rightarrow R)$ is a tautology. Again, to show this the left hand side should equal the right hand side.

First step is rewriting the implications:

$\displaystyle ((\lnot P \lor (Q \lor R)) \land (\lnot Q \lor R)) \equiv (\lnot P \lor R) $

By weakening we now have:

$\displaystyle (\lnot P \lor (Q \lor R) \equiv (\lnot P \lor R) $

But I'm kind of stuck here.

I could distribute and apply strengthening, but I'm not sure that would be legal. Any help would be welcome!