Prove $R$ follows from premises $(\lnot R\rightarrow\lnot Q),\;(P\lor Q,),\; (\lnot(P \lor T))$

Question

I'm preparing for an exam and we weren't given an answer sheet. I'd like to know if my reasoning for the given conclusion is correct?

Premises: $(\lnot R) \rightarrow (\lnot Q),\;\; (P \lor Q),\;\; \lnot(P \lor T)$

Conclusion: R

1. $(\lnot R) \rightarrow (\lnot Q):\;$ Given

2. $P \lor Q:\;$ Given

3. $\lnot(P \lor T):\;$ Given

4. $\lnot P \land \lnot T:\;$ DeMorgans Law #3

5. $\lnot P:\;$ Conjunctive Simplification #4

6. $Q:\;$ Disjunctive Syllogism #2 and #5

7. $R:\;$ Modus Tollens #1 and #6

I'm mostly unsure that #1 and #6 gives the conclusion $R$? I'm kind of guessing that I can put those together and get $R$ instead of $\lnot R,$ although it seems like there should be another step?

Answer 1

Looks good (your direction of reasoning is correct) but your proof may be seen as missing a couple of steps:

I suggest you "play it safe" (and don't skip steps):

Write $\lnot \lnot Q$ (from $Q$ using double negation), to get $\lnot \lnot R$ by Modus Tollens, then use double negation again to get $R$.