Is this a good way to (or correct) proof?

Question

Given: P ⇒ Q. To show: ¬Q ⇒ ¬P.

Using modus tollens: Assume that ¬Q. Then ¬P. Thus ¬Q ⇒ ¬P.

Does anybody have a good reference to how I can learn how to proof such formulas (including equivalence, negation etc.)

Thank you!

Answer 1

"Does anybody have a good reference to how I can learn how to prove such formulas?"

There is no single formal proof system for propositional logic. There are different general styles of proof system (axiomatic, natural deduction, tableaux ...), and within each general style there are variations (there are, for example, different natural deduction systems with different basic rules of inference, and different ways of laying out proofs). It is important to realise this or you can get confused!

So: which book is your course using (assuming that this is homework)?

Assuming that your proof system does have modus tollens as a basic rule (not a usual choice, though!), then your proof sketch is fine, though it will need to be properly laid out according to the rules of the proof-system you are officially using. Otherwise you will need something more like this:

$ P \to Q\\ \quad\quad | \quad \neg Q\quad\quad\quad\quad\text{assumption}\\ \quad\quad | \quad \quad | \quad P\quad\quad\ \ \text{assumption}\\ \quad\quad | \quad \quad | \quad Q\\ \quad\quad | \quad \quad | \quad \text{contradiction!}\\ \quad\quad | \quad \neg P\\ \neg Q \to \neg P$

Answer 2

Yes, it works, provided that you have Modus tollens available in your "rule-set".

For a proof of it which does not use modus tollens, according to the rules of Gordon J. Pace, Mathematics of Structures for Computer Science (2012), see page 45.