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Left downward monotonicity

SEMANTICS: a property of a determiner D in Generalized Quantifier Theory. A determiner D is left downward monotone if and only if in a domain of entities E condition (i) holds.

(i)  for all A, B, A' subset E: if D(A,B) and A' subset A, then D(A',B)
Left downward monotonicity can be tested as in (ii); as shown there, all and no are left downward monotone, but some and exactly two are not.
(ii) a  If all/no animals walked, then all/no dogs walked.
     b  If some/exactly two animals walked, then some/exactly two dogs walked.
Other terms are antipersistent and left monotone decreasing.
LIT. Barwise, J. & R. Cooper (1981)
Gamut, L.T.F. (1991)