Do Logic and Existence Go Together?

It's said by some (or most) logicians that “logic must handle every possible state of affairs” and hence that it “can't imply the existence of anything” (Dale Jacquette). That almost sounds like a non sequitur. Yes, logic must handle “every possible state of affairs”. Nonetheless, how does it follow from this statement that logic can't imply the existence of anything? Why can't logic be able to handle every state of affairs and imply the existence of something (or one thing)?

Is it because if logic is applicable to everything, then implying the existence of something would pollute its ability to handle all states of affairs (note the jungle of quantifiers here)? Or is is it that the case that something (or these things) would somehow make logic contingent (or empirical) in nature? Nonetheless, implying (or allowing) the existence of something that's contingent (or empirical) isn't the same as arguing that logic itself is contingent (or empirical). Logic can still be applied to the the contingent (or empirical) even if isn't itself contingent (or empirical).

Does it mean, instead, that if logic implies the existence of anything (or even something) that it would somehow depend on that something? And, if logic did imply the existence of anything, then its logical purity would somehow be sullied?

In that sense, quantificational logic (or first-order logic) is far from being pure. Quantifiers in logic have existential import (or have ontological commitment). That is, a quantificational proposition is about the existence (or non-existence) of something (or of many things). Even free logic accepts abstract objects of various kinds. It can also be said that logical statements about self-identity have existential import. That is, the proposition (x) (x = x) has existential implications. And, more obviously, so too does, (∃y) (y = y).

It seems to follow from the acceptance of quantificational logic that an empty universe is excluded – nay, it's logically impossible. However, do these facts about quantificational logic necessarily apply to the more generic “logic” we began discussing? Perhaps quantificational logic is actually a deviant logic!