Downward monotonicity

Downward monotonicity is a property of a determiner D(A,B). A determiner D can be downward monotone with respect to its left argument (A) or its right argument (B). It is left downward monotone (or left monotone decreasing or antipersistent) if a true sentence of the form [S [NP D CN] VP] entails the truth of [S [NP D CN'] VP] where CN' denotes a subset of the set denoted by CN.

Example
the D at most two is left downward monotone:

(i) If at most two animals walked, then at most two dogs walked.

A determiner is right downward monotone (or right monotone decreasing) if a true sentence of the form [S [NP D CN] VP] entails the truth of [S [NP D CN] VP'] where VP' denotes a subset of the set denoted by VP. EXAMPLE: the D at most two is also right downward monotone:

(ii) If at most two dogs walked, then at most two dogs walked in the garden.

If a determiner D is right downward monotone, then the generalized quantifier D(A) is often called downward monotone or monotone decreasing. The opposite of downward monotonicity is upward monotonicity.

Link
Utrecht Lexicon of Linguistics