/sci/ - Science & Math

>>11059158
Not adopting LEM does not mean
>for all statements A, it's undecidable whether "A or not A" holds
A lot of statements are decidable (You are able to prove that A is true, or that not A is true) and for a lot of statements you can find out whether or not "A or not A" holds.
The modus operandi is collecting statements that are provable. If you want to use that word, you do logic/math and generate statements that are "true". The fact that you can judge "oh, that's a statement I've prove" doesn't contradict that there are statements A out there for which you can't find a proof of "A or not A". I.e. (unless you adopt LEM for all A), there are statements for which "A or not A" isn't provable.

Also, at least in constructive logic, it's noteworthy that for all statements, indeed you can proove that "not(A and not A)" is provable. I.e. the law of non-contradiction holds in even Minimal Logic. Classically, the law of contradiction is equvalent to LEM, but it clearly has different semantics (unless you collapse it, by adopting LEM as an axiom).

It's not like LEM gives you a theory that lets you prove more statements (as seen from the classical perspecitve).

LEM says something about the "truth" of statement (which is a "redundant" notion if you have provability), and that there are only two cases (A is true or not true)
Noncontradition says something about...well contradition: That not two truths apply. Why should those notions be collapsed.
Let excluded middle ("A or not A") be something which has to be established.

>>11057303
not necessary