The Semantics of Determiners

Every student left.
b. Nici un student nu a plecat.
No student left.

In Montague’s analysis, the denotation of the bold-faced subjects is the function, and the denotation of the VP is its argument. The DP then is an expression of type < < e, t >, t > : it is a function that takes a property as its argument and it yields a truth value as its value. The type of the D in each DP is a function from properties to the functions denoted by the DP, i.e., a function from expressions of type < e, t > to expressions of type < < e, t >, t >. Determiners then are expressions of type < < e, t >, < < e, t >, t > >. The function denoted by every/fiecare then takes a property P as an argument and yields as value a function that takes a property Q as an argument and yields the value 1 just in case P  Q. The function denoted by no/nici un takes a property P and yields a function that takes a property Q as an argument and yields the value 1 just in case P  Q = . These truth conditions, as you can see, are parallel to the Barwise and Cooper’s approach (where P is the Restrictor set we denoted by A above, and Q is the NS set denoted above by B).

We exemplify in (15) how the types of the constituents of (14a) combine. Here the denotation of student plays the role of the property P above, and the denotation of leave plays the role of the property Q:

(15) Every student left

t

every student left
< < e, t >, t > < e,t >

every student

< < e, t >, < < e, t >, t > > < e,t >

The function denoted by every student can be thought of as the set of properties for which it gives the value 1, i.e., the set of properties that every student has. The sentence Every student left then denotes 1 just in case the property of leaving is within the set of properties that every student has, i.e., just in case for every student in D, that student left, which is as it should be.

The type given to every student above is the most complex type a DP may have and it goes under the name of generalized quantifier type. Now if our aim is to give all DPs a uniform type, all the other DPs we looked at so far, including Proper Names must be generalized quantifiers, i.e., they must be of type < < e, t >, t >. For indefinite and definite DPs the analysis would work just like in the case of universal DPs with the difference that the relation imposed by the indefinite would require P  Q  , while the definite would impose an additional special presupposition concerning the cardinality of P. The function denoted by a student can be thought of as the set of properties such that some student or other has those properties. The sentence A student left then denotes 1 just in case leaving is among these properties, i.e., just in case you can find some student in D who left. Extending the analysis to Proper Names means treating their denotation as a function from properties to truth values as well. As before, we can think of this function as the set of properties for which the value it gives us is 1. In the case of John, this set will be the set of properties that a particular entity (the one we call John) has. The sentence John left is true just in case the property of leaving is among this set of properties, i.e., just in case the individual we call John is among the entities who left.⁸

Note that in each case we arrive at truth conditions close or equivalent to those we had for our restricted PC formulas, but via a route that computes the denotations of expressions step by step and without resorting to an intermediate level of representation.

A system like the one sketched above, that assigns syntactic categories a single uniform type and which allows a single mode of combination, namely function argument application, is, alas, not rich enough to account for the complexities of natural language. There are two possible routes toward enriching the system: allow a richer inventory of modes of combination, and allow a single expression to be associated with a set of types.

Both routes have been taken in the literature.

The multiple type route is most directly relevant to DP semantics. While Montague’s unifying sweep explains the similarities we find across various DP types, it is unable to give us an explanation for the equally important syntactic and semantic differences we encounter. As noted in Kamp (1981) and Heim (1982), for instance, Proper Names, as well as definite and indefinite DPs may serve as antecedents to definite pronouns in discourse, while DPs whose D is a universal or proportional quantifier may not:

(16) a. John/a student/the student left. He was tired.

b. Every student/most students left. *He was tired.⁹

Note also that while indefinite DPs are quite natural as predicative nominals (exemplified earlier in (2) and repeated below in English) DPs with universal Ds are not:

(17) a. John is a famous doctor.

b. *John and Mary are every famous doctor in our family.

Finally, it has been shown that syntactic positioning within a clause is sensitive in some languages, such as Hungarian, to whether the D in a DP is a universal or proportional quantifier or not (see Szabolcsi 1997).

These contrasts take us back to the distinction between universal and existential determiners we drew in Figure 2 above. We have now a basic distinction between purely quantificational DPs and non-quantificational DPs, exemplified below:

non- quantificational DPs quantificational DPs

Mary, he, somebody, the/a/this student every student, most students
Figure 3 Semantic DP classification

A further distinction within the non-quantificational DP category is needed to separate those that may function as predicative DPs in ordinary sentences such as the one exemplified in (17) from those that may not. Proper Names and definite pronouns belong to the former while DPs with definite and indefinite articles and possibly indefinite pronouns belong to the latter.¹⁰

Turning back to semantic types, it appears that nominal constituents live in the following three types: (i) e, where non-quantificational DPs are at home but quantificational DPs are not; (ii) < e, t >, the type assumed by DPs and other nominals when acting as predicative nominals, as in (17a), which only a subset of non-quantificational nominals may inhabit; (iii) < < e,t >, t >, the type we find quantificational DPs in and which other DPs may take on as well, when in conjunction with quantificational DPs. In Partee (1986), the classic on this subject, this messy situation is elegantly cleaned up. The proposal, in essence, is to indeed allow the category of DPs (or, more generally, nominals) to live in all three types mentioned above, and to define type shifting rules that change the type of a nominal to a more complex type (type lifting) or to a simpler type (type lowering). The interested reader is referred to Partee’s paper and the large literature it spawned for details and discussion. Interestingly, Partee suggests that the ‘unmarked’ type for DPs is e and < < e,t >, t >, the types we need for nominals when not in predicative position. The upshot as far as the analysis of determiners goes, is that the definite article, for instance, is given two related interpretations, both presuppositional: (a) a function (called iota) that takes a property P and returns the unique entity in the model that has that property; (b) a function (called THE) that takes a property P and returns a generalized quantifier meaning (the set of properties characterizing the unique individual in P). The former meaning is what we get in ordinary examples of definite DPs such as the student in simple examples like The student left, where the denotation of student is the property P; the latter is what we get when a definite DP is conjoined with a quantificational one. Finally, for cases where a definite DP functions as a predicate nominal (as in Sam is the treasurer), Partee proposes a function that takes the generalized quantifier meaning of a definite description and lowers it back to a predicate meaning.