I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?
This link did help: Universal and Existential quantifier in Propositional logic
However, the part that confuses here is how are the domains restricted in the answer given in the link above?
Best Answer
Technically, the domain that the quantifiers quantify over is always the same. Or, to be exact, you don't change what the quantifiers quantify over by using conditionals or any other logical operator. Thus, if within some context I say:
$\forall x (P(x) \rightarrow Q(x))$
then the $\forall$ still quantifies over the exact same set of objects as in:
$\forall x Q(x)$
Of course, effectively, the latter claims says that all of the objects have property $Q$, while the former says that all of the $P$'s from that domain have property $Q$, and so it looks like the quantification is happening over a more restricted domain (we go from "all objects" to "all P's").
However, the former really says "For all of the objects from the domain: if you are a P then you are a Q", so the universal still quantifies over the whole domain ... as they always do and always will!