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.
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.
- Barwise, J. & R. Cooper 1981. Generalized Quantifiers and Natural Language, Linguistics and Philosophy 4, pp. 159-219
- Gamut, L.T.F. 1991. Logic, language, and meaning, Univ. of Chicago Press, Chicago.