Our math logic reading group is going through Manzano’s Model Theory as therapy/revision before tackling Hodges’s Shorter Model Theory. I’m not sure that Manzano was, after all, a good choice; though equally it isn’t clear what would have served our purposes better.
Anyway, I was struck again by the still-standard logician’s habit of treating the formal semantics of first-order languages by explaining how to extend an interpretation by assigning values in the domain to each and every variable of the language — and then later proving that e.g. different assignments to variables other than those that appear in the wff being evaluated don’t make any difference. I know this is how Tarski did things, but isn’t there something inelegant about stocking up on assignments of objects to variables only not to use infinitely many of them?
It is reminiscent of the bad old overstocking habits of industry! Toyota-style “just in time” production, where we only stock up with what we actually need next, is better!! Likewise, surely, giving a semantic story where we deal with e.g. “AzFz” by talking about alternative ways of extending an interpretation by assigning an object to “z” (treated as a parameter/temporary name) is more elegant and more intuitive. That way, we just talk about alternative extensions of an interpretation to cover particular variables as and when we need them.
Is there a good reason, other than historical piety for doing things the first, Tarski, over-stocking way, rather than the Toyota way? We couldn’t think of one.