I think op is delving into a very old subject, the nature of subjective v objective reality, and how we define the parameters of this. The problem is, without care, the postulation devolves into circular reasoning, i.e. "We experience this, in a shared manner, to be real, therefore it's real", but the people sharing the observation, for a large part, share the same mechanisms of observation, so that shared mechanism may be flawed.
It may be nothing's real, it may be in some strange way everything we imagine becomes real, but this is incidental to the immediate argument, "how do we work with this?". Pragmatism, inevitably, always works against the rest of observation to mitigate the present. Sometimes we don't even, entirely, understand how something works, but it works so we use it, do it, have it, assume it.
So, with logic, as we expand horizons, we're constantly "working away" from pragmatism. We're constantly evolving, I suppose, to create more and more difficult scenarios to explain as we paint this picture through the way we manifest. I mean, really, could we revert to walking on all fours, eating grass and raw rabbits off the ground, everything would be much simpler, we wouldn't have industrial pollution, threat of nuclear catastrophe, or trolling of weak love poetry. Instead we have microwaves on our counters, jets in our skies, satellites in orbit. We didn't get those things by arguing "logic might not be applicable".
We get very real, tangible results from logical processes, so something, somewhere in there is real. As we endeavor to work with what is "real" in this fashion we develop time-tested rules, or axioms, and these tend to hold sway, regardless if we're "seeing" the truth or a mirror reflection of truth. There are many poor results, many rabbit holes, and it's best, probably, to avoid these when we're trying to garner more concrete results.
So while we can be aware of results like "empty set", or a statement with zero qualifying answers but which exists as a statement in and of itself, like a bag with no contents, we have to be aware that this doesn't really help us. Whether or not "two plus two can equal five" does not help us work in the practical world. Mathematical proofs which end in "infinity" are garbage. Whether or not "empathy is as real as a hammer that's about to crush our skulls" does not save us in the instant, we must be aware of the hammer, we must deflect or otherwise change the effect of the hammer, or none of it is going to matter. We're not out of the woods of this yet. Maybe someday. That would be nice.
