Intuition behind proofs by rules of inference Unfortunately I don't get the intuition behind the proofs by rules of inference.
I do understand example, however I don't see the broad picture, I simply don't understand how to reason about the solution.
Following I will show few example with my notes, which are might be incorrect, please show me what I don't get right.
Exercise 1. Prove ¬p from (p → q), (q → r) and ¬r.
We are given few assumption, (p → q) is true, (q → r) is true and ¬r is true. The question is whether ¬p is true.
If (q → r) is true and ¬r is true, then q should be false, therefore ¬q is true.
If (p → q) is true and ¬q is true that ¬p is true.
Whether the reasoning is true?

By transitivity of implication (p → q) and (q → r) implies (p → r).
By logical rule saying that if ((a → b) and ¬b) then ¬b (modus tollens) and already prooven fact that (p → r) and assumption ¬r you can infer ¬p.
