TITLE: What do we call entities (like $\sum$) that bind variables?
QUESTION [11 upvotes]: In logic, we refer to entities like $\forall$ and $\exists$ as quantifiers, because they bind variables. However, variable-binding doesn't just occur at quantifiers. For example, the symbol $i$ denotes a bound variable in the following expression.
$$\sum_{i=0}^n i$$
What do we call entities (like $\sum$) that bind variables? I'd like to know the official name of such entities, so I can read up on how they're formalized in logical systems.

REPLY [6 votes]: I've seen them called 'binders' in the context of automated proof tools.
For the HOL family, there is HOL Light, where section 6.1 Quantifiers in "HOL Light Tutorial (for version 2.20)" says

Constants such as ‘!’ that are parsed and printed in this abbreviated form when
applied to abstractions are called binders, because they appear to bind variables. (But
strictly speaking they don’t: it’s the underlying abstraction that does that.)

Also documents related to HOL4 frequently uses the term 'binder'.
Mizar does not have binders, but there is an article "A proposed syntax for binders in Mizar" by Freek Wiedijk.
In essence, a binder is a higher-order function: it is a function that maps a function from $\;X\;$ to $\;Y\;$ to a value of type $\;Z\;$, where very often $\;Y\;$ and $\;Z\;$ are the same.  So for instance $\;\forall \;:\; (U \to \text{boolean}) \to \text{boolean}\;$ and $\;\sum : (U \to \mathbb C) \to \mathbb C\;$.
(Personally, I don't like to use the term 'quantifier' for operators like $\;\prod\;$ and $\;\int\;$, as Gries-Schneider does.  But I really like Gries-Schneider's general theory of binders/quantifiers.)