  If we are to think of an intension as designating different things under different circumstances, we need things. At the propositional level truth values play the role of things, but at the first order level something more is needed. In classical logic each model has a domain, the things of that model, and quantifiers are understood as ranging over the members of that domain. It is, of course, left open what constitutes a thing—any collection of any sort can serve as a domain. That way, if someone has special restrictions in mind because of philosophical or mathematical considerations, they can be accommodated. It follows that the validities of classical logic are, by design, as general as possible—they are true no matter what we might choose as our domain, no matter what our things are.