Use logical deduction to show that the following propositions are unconditionally true

Question

These two questions:

1) $P \to ((Q \lor R) \to P)$

and

2) $(P \to (Q \to R)) \to ( P \land Q \to R)$

It'd be really helpful if you could answer these for. I managed to answer the ones that give me a predetermined value of True or False for P Q R, but I don't know how to answer these using logical deduction.

Thank you if you try!

Answer 1

Here is a proof for the first one:

\begin{align} \cfrac{ \cfrac{ \cfrac{\{ P, Q \lor R \} \cap \{ P \} \not = \emptyset}{P, Q \lor R \vdash P} \text{Assm} } {P \vdash (Q \lor R) \rightarrow P}\rightarrow \text {Right}} {\vdash P \rightarrow ((Q \lor R) \rightarrow P)}\rightarrow \text {Right} \end{align}

The key is to start at the bottom, and work your way back. Keep asking: given that my goal is ...., what rule allows me to get ...?

Answer 2

Hint

1) $P,Q \vdash P$ --- axiom

2) $P,R \vdash P$ --- axiom

3) $P,Q \lor R \vdash P$ --- ($\lor\text{Left }$) from 1) and 2)

4) $P \vdash (Q \lor R) \to P$ --- ($\to \text{Right}$)

5) $\vdash P \to ((Q \lor R) \to P)$ --- ($\to \text{Right}$).