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A formula, whether originally made up of constant symbols or through a quantifier's binding or an individual constant substitution, that has no free variables is called closed formula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).
It would be interesting to know which book said this! It seems very poor. For a start, a predicate is never true or false, properly speaking.
