From 11, 12 in the book Logic in Computer Science by M. Ryan and M. Huth:
"What we are saying is: let’s make the assumption of ¬q. To do this, we open a box and put ¬q at the top. Then we continue applying other rules as normal, for example to obtain ¬p. But this still depends on the assumption of ¬q, so it goes inside the box. Finally, we are ready to apply →i. It allows us to conclude ¬q → ¬p, but that conclusion no longer depends on the assumption ¬q. Compare this with saying that ‘If you are French, then you are European.’ The truth of this sentence does not depend on whether anybody is French or not. Therefore, we write the conclusion ¬q → ¬p outside the box."
My question is about the scope of assumptions in propositional logic and proving techniques. I am not sure I fully understand what this text is trying to say.
How can an assumption only have scope inside the box, but once you finish what you want to prove it is no more part of the assumption box and is accessible universally in the proof? WHY is this possible? Why does it not break things in the proof? This looks too convenient and random.
Secondly, I do not understand the French and European example connection to what is written in this text. If somebody could please connect this example to what the author is actually trying to explain through this.