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.