# Dual (semantics)

**Dual** is the dual Q^{*} of a generalized quantifier Q can be made by taking both the external negation and the internal negation of Q, i.e.:

(i) Q^{*}= Neg Q Neg

This can be written out as:

(ii) Q^{*}= { X subset E : (E - X) not_in Q }

*All N* and *some N*, for instance are pairs of quantifiers which are each other's duals:

(iii) a All dogs bark <-> b It is not the case that some dogs do not bark

*Some dogs* is the dual of *all dogs* because every set X that belongs to the interpretation of *some dogs* contains at least one dog; so there is no set (E - X) that belongs to the interpretation of *all dogs*. If Q = Q^{*}, then Q is called *self-dual*. Proper names, for instance, are self-dual.

### Link

Utrecht Lexicon of Linguistics

### References

- Gamut, L.T.F. 1991.
*Logic, language, and meaning,*Univ. of Chicago Press, Chicago.