# Left downward monotonicity

From Glottopedia

**Left downward monotonicity** is 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*.

### 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.