Frege and his little monsters

This is NOT a serious post, in case this is not obvious. 

I don't like quantifiers, I call them Frege's little monsters.

I don't like the universal quantifier  because it brings infinity into models. Don't get me wrong, I love infinity as much as any other mathematician and I still get a twinge of pride when I explain Cantor's basic theorems (enumerability of the rationals and lack of such of the reals)  to anyone who hasn't seen them before, especially young ones. But I once attended a lecture by Phokion Kolaitis, who showed me that finite model theory is really very 'different' from model theory: nothing works the way models are supposed to work. It is not the case (as I thought before) that finite model theory is an easy (because finite and surveyable) instance of usual model theory. And one can do an awful lot without bringing infinite domains into our logic pictures.

Also universal quantification and especially vacuous quantification is horrible and very non-intuitive. The typical example of something that we feel forced to accept simply because, without it, the system is even worse.

But you have a duality, people might object. Maybe simply look at the existential quantifier and let the universal be simply its dual. Well,  this does not work for constructivists, who do not believe that 'for all' and 'there exists' are totally dual. But  also, if you chose the existential that is  not exactly dual to universal quantification, the one that has more content, the one which says which 'x' is that that makes "there exists x such that A(x)" true, well then you have a connective that has complicated binding rules, requires commuting conversions and it's not  proof-theoretically well-behaved.

Semantically it's very simple, right? If you say that there exists something such that the predicate A is true of that something, well, just show me the culprit! Call it 'a', because we have a long tradition of thinking of x's and y's as variables and thinking of a's and b's as constants and  then we know that "A(a)" is true. And that a model of this sentence should have 'a' in its domain. Seems simple and direct, but it is anything but simple.

So they are both little monsters, Frege's monstrinhos, and when we want to model them categorically not only do we need our triple adjunction, we also need the Beck-Chevelley condition (BCC), which everyone does its best not to explain. Of course the whole set-up of free and bound variables, predicates and functions, and how substitution works for them, has to be put in place and made to work, before you can add the little monsters.

Anyways I was about to post here a picture of  Frege by Renee Jorgensen Bolinger, in the style of van Gogh. Her pictures of philosophers are great and I really want to buy a few, you can see them at her site.

But she requires people to ask her for permission and I cannot be bothered, so you just need to go and look up her site!