# Strength

Strength is a property of determiners and generalized quantifiers in Generalized Quantifier Theory. An NP is positive strong if and only if its denotation (a set of sets) always contains the denotation of the CN (common noun). An NP is negative strong if and only if its denotation never contains the denotation of the CN. An NP which is neither positive nor negative strong is called weak. Sentences of the form in (i) provide a test for strength of a determiner D:

```(i)   [S [NP DET CN] is a CN/are CN's]
```

If the sentence is true in every model, D is positive strong (Every dog is a dog); if it is false in every model, D is negative strong (Neither dog is a dog); and if it is true depending on the domain, D is weak (At least two dogs are dogs is only true if there are at least two dogs in the domain E). The distinction between strong and weak determiners can be used to account for the contrast in (ii)-(iv) (due to the definiteness restriction).

```(ii)   *There is every dog in the garden.
(iii)  *There is neither dog in the garden.
(iv)    There are at least two dogs in the garden.
```