Ideal is a particular type of generalized quantifier Q (Q =/= 0) which obeys the condition in (i):
(i) for all X,Y subset E: (X in Q and Y in Q) <=> union(X,Y) in Q
Only N, no N and none of the N are examples of ideals; not all N and at most N are not. The condition in (i) captures the contrast in (ii).
(ii) a Only dogs bark and only dogs run <=> only dogs bark or run b Not all dogs bark and not all dogs run <=/=> not all dogs bark or run
- Zwarts, F. 1981. Negatief polaire uitdrukkingen, Glot 4, 35-132