Wouldn’t it be still possible for a constructivist to embrace classical logic and the theoremhood of TND? The constructivist would just have to admit that (A or B) could be true even if neither A nor B is true. (A or B) would still not be meaningless, its truth would imply that there is proof for neither (not A) nor (not B), so this reinterpretation of “or” doesn’t seem to be a big deal.
Constructively, (not ((not A) and (not B))) is weaker than (A or B). While you could call the former “A or B”, you then have to come up with a new name for the latter.
I haven’t been suggesting using (A or B) as a name for (not ((not A) and (not B))) in constructive logic where they aren’t equivalent. Rather, I have been suggesting using classical logic (where the above sentences are equivalent) with a constructivist interpretation, i.e. not making difference between “true” and “theorem”. But since it is possible for (A or B) to be a theorem and simultaneously for both A and B to be non-theorems, logical “or” would not have the same interpretation, namely it wouldn’t match the common language “or” (for when we say “A or B is true”, we mean that indeed one of them must be true).
Wouldn’t it be still possible for a constructivist to embrace classical logic and the theoremhood of TND? The constructivist would just have to admit that (A or B) could be true even if neither A nor B is true. (A or B) would still not be meaningless, its truth would imply that there is proof for neither (not A) nor (not B), so this reinterpretation of “or” doesn’t seem to be a big deal.
Constructively, (not ((not A) and (not B))) is weaker than (A or B). While you could call the former “A or B”, you then have to come up with a new name for the latter.
I haven’t been suggesting using (A or B) as a name for (not ((not A) and (not B))) in constructive logic where they aren’t equivalent. Rather, I have been suggesting using classical logic (where the above sentences are equivalent) with a constructivist interpretation, i.e. not making difference between “true” and “theorem”. But since it is possible for (A or B) to be a theorem and simultaneously for both A and B to be non-theorems, logical “or” would not have the same interpretation, namely it wouldn’t match the common language “or” (for when we say “A or B is true”, we mean that indeed one of them must be true).