Natural Deduction and Inconsistency

Question

I am sincerely struggling with where to begin with either of these?

If am am asked to prove that (1) and (2) are inconsistent am I required to prove this formally by contradiction.

Looking at (1) would P & R become my 1st premise?

Perhaps if someone could please just explain what I need to do here or provide an example of how to go about proving this formally by contradiction I will take it from there.

Many thanks for taking time to assist me here!

Answer 1

Your premises are (1) $P\land R$ and (2) $R \to \lnot P$.

Elimination of $\land$ on (1) gives :

• (3) $R$
• (4) $P$

Elimination of $\to$ on (3) and (2) gives :

• (5) $\lnot P$ (ie $P \to \bot$)

Elimination of $\to$ on (4) and (5) gives :

• (6) $\bot$

It should be clearer if you can draw an proof tree.