## KISSing semantics: Subregular complexity of quantifiers

🕑 9 min • 👤 Thomas Graf • 📆 July 26, 2019 in Discussions • 🏷 subregular, strictly local, tier-based strictly local, monotonicity, quantifiers, semantics, typology

I promised, and you shall receive: a KISS account of a particular aspect of semantics. Remember, KISS means that the account covers a very narrowly circumscribed phenomenon, makes no attempt to integrate with other theories, and instead aims for being maximal simple and self-contained. And now for the actual problem:

It has been noted before that not every logically conceivable quantifier can be realized by a single “word”. Those are very deliberate scare quotes around word as that isn’t quite the right notion — if it can even be defined. But let’s ignore that for now and focus just on the basic facts. We have every for the universal quantifier $$\forall$$, some for the existential quantifier $$\exists$$, and no, which corresponds to $$\neg \exists$$. English is not an outlier, these three quantifiers are very common across languages. But there seems to be no language with a single word for not all, i.e. $$\neg \forall$$. Now why the heck is that? If language is fine with stuffing $$\neg \exists$$ into a single word, why not $$\neg \forall$$? Would you be shocked if I told you the answer is monotonicity? Actually, the full answer is monotonicity + subregularity, but one thing at a time.