What is the justification for not believing that from a double negation of A follows A is a valid inference?
I took formal systems in university some years ago and remember something about different axiom systems and how one school of thought rejected one particular rule of inference but was this the one?
I am trying to think of a world where this rule is not valid and not succeeding.
The simplest intuition is to exchange "truth" as the airy concept of "is valid/reflects the world" with "I possess tangible evidence to the case".
So, "2 + 2" is true because I can generate tangible evidence of it, but "the Collatz conjecture" is not true because I cannot. It's also not false. To be "false" I'd have to have tangible evidence of "the Collatz conjecture is false", a counterexample. This "I lack evidence" as neither truth nor falsehood is the characteristic of intuitionistic/constructive logic that makes it so interesting.
To be clear, there is now an element of time involved. Or, as others have stated, constructive logic reflects the "communicative nature" of proof. What I should have said a second ago is "The Collatz conjecture is not true (for me) (right now) (because I personally don't happen to have evidence for its validity yet)". If you had a (valid) proof it'd be true for you and once you communicated that proof to me it'd be true for me.
So in the case of rejecting "not not P => P", it's merely a statement of the fact that "not (not P)" failing to actually generate any evidence for "P", it merely shows that there isn't "not evidence". If you live in a world where things must be true or false (e.g., excluded middle holds) then this is sufficient, but if you demand I show you why "P" holds, then "not (not P)" is unsatisfactory.
I took formal systems in university some years ago and remember something about different axiom systems and how one school of thought rejected one particular rule of inference but was this the one?
I am trying to think of a world where this rule is not valid and not succeeding.