My way of 'getting it' emerged only after few weeks, when we spent quite a lot of time on a proof that not (not A) is equivalent to A in propositional logic. T

My way of 'getting it' emerged only after few weeks, when we spent quite a lot of time on a proof that not (not A) is equivalent to A in propositional logic. Then I realized that I shouldn't think about these in terms of ideas about reasoning I already had in my head, but to internalize that this is a self-contained, rule-based system, where we have some arbitrary statements given as a base and some arbitrary derivation rules, and our only goal is to be able to find what we are able to derive. Does this statement have a proof then means "is there a chain of derivations from base that leads to this statement?" I still got tripped on 'implication' (that in my language means something closer to what equivalence means in logic for some reason) and 'or' (that in my language usually implies either-or), but when I looked these with "Its just a rules system, it doesn't have to mean the same thing I use in general language" I was fine. Then, of course, the usage crept from maths back to the language, and I have had confused many of my less mathematical friends (i.e. when they ask "if you are in the city, would you go to a pub", and to this, my answer always is "yes", but what they really meant is "Are you in the city and will you go to a pub?")