I just ran across this statement in a logic book
A predicate can’t be true or false until a specific value is substituted for the variables, and the quantifiers ∀ and ∃ “close” over a predicate to give a statement which can be either true or false.
I think I understand the "specific value" part; but can somebody give me the general concept being alluded to here when they say "close?" What does "to close" mean in this context?
A quantifier binds a variable of a predicate to a domain of discourse within its (i.e., quantifier's) scope. The author exploits the image an associated phrasal verb 'close over' evokes in order to convey a better intuition.
Thus, the quantifier forms, so to speak, a closure in the mathematical sense by its scope and its counterpart in the domain. I may illustrate this with an arbitrary formula:
$\forall x(P(x)\rightarrow\exists yQ(y)\wedge R(x))\leftrightarrow\exists x Z(x)$
In this example, the universal quantifier instructs us how to evaluate the open formula
$P(x)\rightarrow \exists y Q(y) \wedge R(x)$,
which does not have force outside its scope.
A formula, whether originally made up of constant symbols or through a quantifier's binding or an individual constant substitution, that has no free variables is called closed formula (alternative term is sentence).