#excluded middle

Hello and welcome to the Logical Fallacy of the False Dilemma! 

Look, friend, I just binged 473,000 words of fic in four days. It’s a series about a character losing his hearing and coping with a worsening and complicated chronic illness--and it’s also a story about a loving, devoted, unshakeable romantic relationship at the center of an ever-growing network of friendships and found family. It made me cry repeatedly; it is a story about real losses and suffering and grief. It is also a story about partners snuggling up together at night and cooking for each other and embarrassing each other with affection and compliments. There are BABIES. 

So do I like angst or do I like fluff?

No. I like both, I like them mixed liberally together. I don’t like angst that ends in tragedy and despair--but, hey, if you do, go find you some fic! It’s out there, and the fact that it exists means you’re not the only person who likes it, by definition. 

If you want to find like-minded people, maybe go talk to the people who write fic you like, or your fellow commenters on that fic. They’re out there! Sometimes you need the darkfic and that’s fine! Sometimes you want to talk to other people who also are in that place, and they exist! I have no earthly idea why you think I am one of them, but they’re out there! Go find your people!

OK, can you explain the difference between what “no proposition is neither true nor false” and “all propositions are either true or false” means for an intuitionist?

I originally wrote a long & rambling post in response to this, which I might rewrite and post later. For now, I’ll try to make this shorter and more to-the-point.

For an intuitionist, a universally quantified statement like ‘∀x, R(x)’ does not mean that 'R(x)' “is the case” for each 'x'; it actually means that there's a (constructible total) function which accepts any 'x' as input and returns a proof of 'R(x)'. From this perspective, to validate “all propositions are either true or false” is to construct a function which decides every proposition. I assume I don’t need to explain why this is not possible.

Intuitionists also don’t mean the same thing as classical mathematicians when they assert the negation of a proposition. Instead of ‘~P’ asserting that ‘P’ “is not the case”, it means that we can derive a contradiction from ‘P’. So double-negation doesn’t cancel out in general, and ‘~~P’ means that ‘P’ is consistent (we can’t get a contradiction by assuming it, assuming our formal system is itself consistent).

Any proposition ‘P’ which is intuitionistically true (i.e. has a proof) is necessarily also consistent, so ‘P -> ~~P’, but some propositions are consistent without being provable, in which case the converse does not hold. LEM is one of those propositions - since classical logic is consistent iff intuitionistic logic is, it must be that ‘~~LEM’, but as per the above we cannot actually validate ‘LEM’ itself.