Common relevance - situational assumptions.
The common-use ontological notions of nothing/emptiness are context dependent, with enthymematic information filling out the gaps concerning what kind of absence is under consideration, and as such relevant absence. This isn't very different from notions of universal nature, such as all/everything, where what counts as anything at all, is conditioned on the relevant context. I will argue that the empty class is a relative notion, and is conditioned by what is considered as possible/impossible in any given context, whether it be an informal one (situation) or formal (formal theory).
For example "she ate everything" (in most everyday cases) certainly doesn't refer to some paradoxical Leviathan who has devoured the entire world and would have to eat itself also, if the statement was to be considered as is, without qualification. Rather, almost always, the context dictates what the scope of quantification is restricted to - and in most cases the type of food stuff contents of, say a plate, or fridge etc.
Similarly in the cases of common expressions such as "the house is empty", or even ones like "the house is absolutely empty", or "there's nothing left" the scope of quantification is restricted to the things that count as ontologically relevant e.g. people, furniture, etc.
Relevant scope of quantification can be thought of as relevant possibility (context conditioned/indexed possibility). When I say "Put down your pencils everyone!", in a room full of students somewhere in one of the University of Queensland St. Lucia campus lecture rooms, at the end of a multivariate calculus quiz, this address does not include the security guard checking id's in the Biological Sciences Library, nor does it include my friend who at that very moment is sailing the Mediterranean Sea - they're outside the scope of relevant quantification, since they're not even enrolled in MATH 1052 (arguably, a non-controversial minimal requirement). This would still hold if by some strange circumstances my sailor friend somehow acquired a copy of the quiz and was solving the problems simultaneously with my students. Not being a relevant someone in that context, she is a relevantly impossible someone.
Of course there is a world where she arranges with the school to be allowed to take the test remotely, and as such participate in the quiz via some appropriate video-phone technology. However in most cases of common language use, such peculiar circumstances do not tend to be part of enthymematic content (common situations don't imply uncommon information about them, i.e. our reasoning in most cases is governed by a ceteris paribus set of assumptions), but rather carefully explicated instead.
Furthermore, if after 15 minutes or so, having marked "all" (the ones handed out in the classroom) the quizzes, I said "No one got full marks.", and if it turned out that my sailor friend in fact got all the answers right, I would still be telling the truth, for my friend fails to be in the scope of the relevant quantification (indexed by that informal context).
Formally, we can think of my friend as being an element of the empty set (here indexed by the above quiz scenario), as far as "someones" are concerned. But since empty sets have no elements, instead we think of the sailor friend being in some sense associated with the empty set indexed by the quiz scenario. Since such relevantly impossible objects can be thought of as a collection, we can also talk of the cardinality of that collection. But since we're interested in a particular relation such a collection has to the empty set indexed by the quiz context, rather than using the term cardinality, we use the term rarity, which captures the intuitive meaning of the cardinality in question.
Meta relevance - logical assumptions.
There seems to be no reason why the above arguments, and their form, cannot be generalized, and as such extended to formal languages, logical systems and their theories. That is, what counts now as relevantly possible in some theory (a theory being the situation analogue of the above quiz scenario), is just what is possible in the logic on which the theory is based.