My thesis is that the concepts of "all" and "no" have an unspoken "zero" built in; specifically, the thesis is that "all" literally means "all with zero exception" and "no" means "none with zero exception." But once the zeroes are made explicit it is clear that "all but zero" and "none but zero" stand at the beginning of two infinitely long numerical series, viz.,
| All but 0 | None but 0 |
| All but 1 | None but 1 |
| All but 2 | None but 2 |
| All but 3 | None but 3 |
| Etc. | Etc. |
Here "All but 0" of a category refers to the category's full membership, and as the exceptions increase to "All but 1," "All but 2," "All but 3," etc., reference is to increasingly fewer of its members. On the other hand, "None but 0" of a category does not refer to any of the category's members, but as the exceptions increase to "None but 1," "None but 2," "None but 3," etc., reference is made to an ever greater number of its members. So "All" and "None" are the extreme "poles" of the membership continuum, and the exceptions "back away" from these extremes in a minus or plus direction. That is, "All but x" means "All minus x" while "None but x" means "None plus x."
Letting "∀" symbolize "all" and "∅" symbolize "none," this can be pictured on a vertical continuum as follows:
| All Pole | All but 0 | = ∀-0 |
| All but 1 | = ∀-1 | |
| All but 2 | = ∀-2 | |
| All but 3 | = ∀-3 | |
| None but 3 | = ∅+3 | |
| None but 2 | =∅+2 | |
| None but 1 | = ∅+1 | |
| None Pole | None but 0 | = ∅+0 |
(Incidentally, for categories with finite membership each "All but x" claim will in fact refer to the same number of members as some "None but y" claim. For example, if the category has exactly 20 members, then "All but 7" of them is the same number as "None but 13" of them; but this will not be relevant in what follows.)
Now to complete the general statement forms—of which the traditional statement forms are the limiting instances—we need to qualify these quantities with "at least" or "at most" to indicate the affirmative and negative character, respectively, and we also must affix the terms and copula for each. The result is:
| Universal Affirmative | At least all but x A are B | At least ∀-x A are B |
| Universal Negative | At most none but x A are B | At most ∅+x A are B |
| Particular Affirmative | At least none but x A are B | At least ∅+x A are B |
| Particular Negative | At most all but x A are B | At most ∀-x A are B |
Note that I continue to use the term universal even though there will be a deviation from full universality (by the amount of x) in these quantifiers; also, I continue to use the term particular even though x may be greater than one.