Semantic answers to syntactic questions

A preliminary issue: partially defined determiners.

(9)

The cat slept.

(10)

The cats slept.

(11)

Neither cat slept.

Question 1: Which NPs can appear in there sentences and why?

(12)

There is/are a cat/some cat/no cat/three cats/less/more than ten cats/between five and ten cats/many cats/few cats in the garden.

(13)

*There is/are every cat/most cats/all cats/the cat(s)/neither cat in the garden.

Barwise and Cooper: The answer has to do with the semantics of there sentences. A determiner is allowed to appear in there sentences if and only if it does not make them tautological or contradictory.

(14)

A sentence there is/are NP is true iff E is in the quantifier denoted by the NP.

For instance:

(15)

There are three cats.

(16)

A determiner D is called positive strong iff for every : D(A)(E) is true whenever it is defined.
A determiner D is called negative strong iff for every : D(A)(E) is false whenever it is defined.
A determiner that is neither positive strong nor negative strong is called weak.

The determiners in (12) are weak. Those in (13) are strong. Generally: strong determiners are those that make there sentences semantically trivial (=tautological or contradictory whenever defined). Consequently they are ruled out.
Question 2: Which NPs can appear in partitives and why?

(17)

All of the book/the books/the five books/these books \ is/are interesting.

(18)

*All of a book/no book/five books/most books is/are interesting.

Barwise and Cooper/Ladusaw: The answer is again semantic. The denotation of of retrieves a set from the denotation of the complement NP. If this set cannot be retrieved, the partitive becomes ill-formed.

(19)

The constituent of NP is defined and denotes the set A iff NP denotes the quantifier .

For instance:

(20)

The NP the cats denotes, whenever defined, i.e. whenever , the set: . Thus, the constituent of the cats denotes the set in these cases.
Consequently, all of the cats, whenever defined, denotes the same quantifier as every cat/all cats.

(21)

A determiner is called definite iff for every : whenever the quantifier D(A) is defined A is non-empty and .

The determiners in (17) are definite. Those in (18) are not. Generally, only definite determiners guarantee, whenever defined, that the of in the partitive can extract from them a non-empty set. Hence, only these NPs are allowed in partitive constructions.
A general strategy: Reduction of syntactic anomaly to semantic anomaly that naturally arises from the semantics of the construction.
Question 3: What licenses negative polarity items?
Examples (22)-(28) from [Keenan (1996].
The Ladusaw-Fauconnier Generalization: Negative polarity items occur within arguments of monotonic decreasing functions but not within arguments of monotonic increasing functions.
Exercises

1. Write down the quantifiers denoting the following NPs: every woman or every man, Mary and most students, John or five girls, neither many women nor many men, John and the girls, more than five girls but less than five boys.
2. Show that given these denotations you can account for the corresponding equivalences with sentential coordinations as in (2)-(4).
3. Derive the semantics of the determiner not every using the boolean complement in the determiner domain. Hint: recall that determiners are relations between sets, and hence subsets of .
   Can you think of other possibly grammatical constructions with boolean operations in the determiner domain?
4. Consider the familiar sentence: Tina is not tall and thin. What readings are expected by a rule of CR? Compare these predictions with those of the boolean treatment.
5. Consider the sentence there are cats in the garden, as well as the questionable sentence ?there is John in the garden. What problem do these cases pose to the proposal made in class? Can you think of a modification that would improve the situation?
6. Consider sentences like both/the two children are smiling. Define both and the two as determiners. Hint: see the definitions of the and neither. Contrast now the cases in all of the two/*both books are interesting. What is the problem these two examples pose for Barwise and Cooper's proposal? Can you improve the situation?
7. Find more contexts that allow negative polarity items and check whether they agree with the Ladusaw-Fauconnier Generalization. Hint: check determiners, or consult Keenan's paper for a more general definition of monotonicity.