Can/Do there exist any quantifiers other than "there exists" and "for all"?

Question

I'm curious about why there are only the two logical quantifiers there exists and for all. Intuition and human language support the idea that these quantifiers make sense, but otherwise it seems arbitrary (at least from a symbolic, formal logic perspective) that there are just these two.

In fact, there is really just one quantifier, because "for all x: P(x)" can be expressed as "there does not exist x: not P(x)." (Or alternatively, we could similarly define there exists in terms of for all.)

What I really want to know is whether there are any other "atomic" logical quantifiers, meaning quantifiers that cannot simply be built out of, say, the existential quantifier and other basic logical symbols like not, and, and or. In particular, there exists a unique doesn't count, since that can be built from more basic symbols.

In short, can there be more than one "atomic" logical quantifier, and if not, why?

Answer 1

Yes - the key term is "generalized quantifiers." They are studied in the contexts of both natural language and in mathematical logic. I'll focus on the logic side, about which I know more.

A name which crops up in both contexts is Jon Barwise, and this article of Vaanaanen describes much of Barwise's work on generalized quantifiers; this paper of Barwise on natural language may be of particular interest. And the SEP article is pretty good too.

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To begin with, it's a standard result that each of the following quantifiers is not definable from the usual ones:

• There exist infinitely many things satisfying p.

• There exist exactly k-many things satisfying p (for some fixed infinite cardinal k).

• There exist at least k-many things satisfying p (for some fixed infinite cardinal k).

• At least as many x satisfy p as satisfy ~p.

• And lots of others.

(Basically, apply compactness and Lowenheim-Skolem as appropriate.)

There are also quantifiers which apply to situations richer than mere first-order structures: e.g. if we're looking at a structure equipped with a topology, we have the quantifier "The set of things satisfying p is dense," and with a measure in hand we have the quantifier "The set of things satisfying p has positive measure.

Additionally, there are generalized quantifiers which are syntactically more complicated than the usual ones - for example, the "same number" (or Hartig) quantifier: for p, q formulas we write Ixy(p(x),q(y)) for "{x: p(x)} has the same cardinality as {y: q(y)}." Even ignoring the semantics of $I$, it just looks different (it binds to two formulas instead of one).

The study of logical systems using generalized quantifiers is part of abstract model theory, which more generally studies logic beyond first-order logic; the standard text on the subject is the collection Model-Theoretic Logics.

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Finally, to whet the appetite for the subject (and provide a kind of negative result), let me mention one theorem - Lindstrom's theorem:

There is no logical system strictly stronger than first-order logic which has both the compactness and downward Lowenheim-Skolem properties.

In particular, the genuinely different generalized quantifiers all have some fundamental "logical wildness:" we can't add any of them to our logic without changing its basic properties.

(I'm being a bit vague here - e.g. what exactly does "logical system" mean here? A precise statement and proof of the theorem can be found in either Chapter 2 of the collection mentioned above or the end of Ebbinghaus-Flum-Thomas.)