What is the formal and rigorous definition of a quantifier?

Question

In mathematical logic, there is the notion of quantifiers. The most commonly used ones are $\forall$ and $\exists$, but there are others, like $\exists!$, the unique existential quantifier, and the quantifier "There exists infinitely many". But that raises the question, what is the formal and rigorous definition of a quantifier? Has someone in some book or paper defined rigorously what a quantifier is?

Answer 1

Incidentally, the relevant term is generalized quantifier.

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Ignoring set/class issues, a unary quantifier is just a map $Q$ assigning to each set $X$ a family of subsets of the powerset of $X$, $Q(X)\subseteq\mathcal{P}(X)$, which is respected by bijections (i.e. if $f:X\rightarrow Y$ is a bijection then $Q(Y)=\{f[A]: A\in Q(X)\}$), so in particular fixed by permutations. If $\mathcal{M}$ is a structure with underlying set $M$, we interpret

$Qx.\varphi(x)$ is true in $\mathcal{M}$

as

the set of $m\in\mathcal{M}$ such that $\varphi(m)$ is true in $\mathcal{M}$ is in $Q(M)$.

This lets us rigorously talk about adding a generalized quantifier to an arbitrary logic.

For example, $\forall$ corresponds to $X\mapsto\{X\}$ and $\exists$ corresponds to $X\mapsto \mathcal{P}(X)\setminus\{\emptyset\}$, while $\exists!$ corresponds to the slightly-more-complicated $X\mapsto\{\{a\}: a\in X\}$. This approach was introduced by Mostowski, On a generalization of quantifiers, although the notation in that paper is a bit archaic; later authors such as Barwise developed it much further.

One key development was the realization that Mostowski's notion of generalized quantifier is usually too broad to study, so instead we typically focus on monotone quantifiers; these are the quantifiers such that $Q(X)$ is an up-set, i.e. $A\in Q(X)$ and $A\subseteq B\subseteq X$ implies $B\in Q(X)$. Both $\forall$ and $\exists$ are monotone in this sense, while $\exists!$ isn't. Barwise's Monotone quantifiers and admissible sets gives a great introduction to this, although it is hard to get a copy of.

Another common modification of the original approach is to have $Q$ also take into account the signature (or similarity type) of the structures under consideration (in which case the "bijection-invariance" above is weakened to "isomorphism-invariance" as one would expect).

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Incidentally, it's also possible to talk about $n$-ary quantifiers for fixed (or even varying) $n$, by appropriately tweaking the codomain of $Q$. E.g. we can think of "$\exists \forall$" as a combination of unary quantifiers or as a single binary quantifier. However, not all higher-arity quantifiers are so decomposable: for instance, consider the $k$-ary Ramsey quantifier $R^k$ whose semantics in natural language is given by

$R^k x_1,...,x_k.\varphi(x_1,...,x_k)$ is true iff there is an infinite subset $H$ of the domain such that $\varphi(a_1,...,a_k)$ holds for any $a_1,...,a_k\in H$.

Note that while the relevant notion of largeness is still somehow unary ($A\subseteq X^k$ is large iff there is an infinite $H\subseteq X$ such that $H^k\subseteq A$) the quantifier itself is still $k$-ary. Unsurprisingly, higher-arity quantifiers can express more in general than lower-arity ones; see e.g. Hella/Luosto/Vaanaanen, The hierarchy theorem for generalized quantifiers.

Responding to a comment below, another example of higher-arity quantifiers is given by Henkin's branching quantifiers - e.g. the $Q_H$ of the wikipedia article is (in the formalism of this answer) the map $$Q_H: X\mapsto \{A\subseteq X^4: \exists f,g\forall a, b((a,b,f(a), g(b))\in A)\}.$$ This is, and is supposed to be, unimpressive: the point is that the natural language definition of the branching quantifier really is a description of when a set of tuples is "large," i.e. a definition of a quantifier in the sense of this answer.