# Application of Modus Tollens rule in nested proof confusion

From a book I'm reading for logic and computer science, the nested proof is as follows:

1. {$$q \rightarrow r$$ assumption

2. [$$\neg q \rightarrow \neg p$$ assumption

3. ( $$p$$ assumption

4. $$\neg \neg p$$ $$\neg \neg i$$ 3

5. $$\neg \neg q$$ $$MT$$ 2,4 (applying Modus Tollens)

6. $$q$$ $$\neg \neg e$$ 5

7. $$r$$ $$\rightarrow e 1,6$$ )

8. ... ]

9. ...

10. final proof}

note: sorry about the different brackets it was my attempt to show a nested proof.

my question is concerning how r in line number seven is derived. So, my first instinct tells me to apply the Modus Tollen rule to lines 2 and 3 and we can obtain $$q$$ that way, but I don't understand why do they apply the double negation rule to $$p$$ and didn't apply MT rule directly.

So, it would be

given the assumption of $$\neg q \rightarrow \neg p$$ given $$p$$

then we could derive $$\vdash q$$

Remember that modus tollens says that from $$A\to B$$ and $$\neg B$$ you can infer $$\neg A$$. So from $$\neg q\to\neg p$$ and $$\neg \neg p$$ you can infer $$\neg\neg q$$. That's why you need to use the rules to add (and then remove) the double negations even though those rules seem unnecessary to us in natural language.
• OK, so the modus tollens doesn't apply to when a given assumption/premise is not a type of negation? hence why we need to convert to $\neg \neg p$ Commented Oct 6, 2019 at 17:10