# 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

### References

- 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.