How do I show that these are the same logical statement?

Question

I know that if I wanna show that the following statement are the same, I may use some rules in Logic: $$P\Longrightarrow Q,\quad [P \text{ and } (\sim Q)]\Longrightarrow [R\text{ and }\sim R]$$ Is there another way or do we just use truth tables?

Answer 1

Well the second statement is just

$$[P~ \text{and}~ (\sim Q)]\Longrightarrow \bot$$

which is just

$$\sim[P~ \text{and}~ (\sim Q)]$$

which is just

$$(\sim P)~ \text{or}~ Q$$

which in classical logic can be taken as the very definition of the first statement.

Answer 2

One way to do this is with the method of analytic tableaux. We argue by contradiction by supposing $\neg ((P\to Q)\leftrightarrow ((P\wedge\neg Q)\to (R\wedge\neg R)))$. This branches immediately so we can do it in parts: $$\neg (P\to Q)\text{ with }(P\wedge\neg Q)\to (R\wedge\neg R) $$ and $$P\to Q\text{ with }\neg ((P\wedge\neg Q)\to (R\wedge\neg R)).$$

The second part proceeds like this: $$P\to Q \\ \neg ((P\wedge\neg Q)\to (R\wedge\neg R)) \\ P\wedge\neg Q \\ \neg (R\wedge\neg R) \\ P \\ \neg Q, $$

which, from $P\to Q$, has two branches: $\neg P$ and $Q$, both closed (because they contradict the above).

Try the first part yourself. The tableau is below.

These notes (in pdf form) explain the method in detail.

I hope this helps.