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Right downward monotonicity

SEMANTICS: a particular semantic property of some NPs, interpreted as generalized quantifiers Q. Q has the property of being right downward monotone if and only if in a domain of entities E condition (i) holds.

(i) for all X,Y subset E: if X in Q, and Y subset X, then Y in Q
Right downward monotonicity can be tested as in (ii): not every N is right downward monotone, every N is not.
(ii) Not every dog walks =>  not every dog walks rapidly
     Every dog walks    =/=> every dog walks rapidly
So, a true sentence of the form [S NP VP] with a right downward monotone NP entails the truth of [S NP VP'], where the interpretation of VP' is a subset of the interpretation of VP. Right downward monotonicity can also be defined for determiners.
LIT. Gamut, L.T.F. (1991)