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Left upward monotonicity

SEMANTICS: a property of a determiner D in Generalized Quantifier Theory. A determiner D has the property of being left upward 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 upward monotonicity can be tested as in (ii); as shown there, some and at least two are left upward monotone, but all and exactly two are not.
(ii) a  If some/at least two dogs walked, then 
        some/at least two animals walked.
     b  If all/exactly two dogs walked, then 
        all/exactly two animals walked.
Other terms are persistent and left monotone increasing.
LIT. Gamut, L.T.F. (1991)