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Right upward monotonicity

SEMANTICS: an NP, interpreted as a quantifier Q, has the property of being right upward monotone if and only if for all subsets X and Y of the domain of entities E condition (i) holds.

(i)  if X in Q and X subset Y, then Y in Q
Right upward monotonicity can be tested as in (ii): all N is right upward monotone, at most two N is not.
(ii) All dogs walked rapidly          =>  all dogs walked
(iv) At most two dogs walked rapidly =/=> at most two dogs walked
So a true sentence of the form [S NP VP] with a right upward monotone NP entails the truth of [S NP VP'], where the interpretation of VP' is a superset of the interpretation of VP. Right upward monotonicity can also be defined for determiners.
LIT. Gamut, L.T.F. (1991)