The Semantics of Determiners

1.3.2 Montagovian approaches
The proper treatment of DPs has played a crucial role in the seminal Montagovian tradition in formal semantics, whose founding stone is Montague (1973). The major force driving the enterprise is the wish to stay as close as possible to the Fregean notion of compositionality. For Frege, the interpretation of a complex expression  is a function of the interpretation of its immediate constituents and the way they are combined. The beauty and power of the theory lie in how narrowly the syntax of an expression is assumed to determine its interpretation, which in turn depends on what modes of compositions one accepts and how narrowly one defines the semantics of each syntactic category. (Frege allowed a single mode of combination, namely function argument application, a restriction that has proven too strict.) At the same time, the approach made possible a shift to ‘direct’ interpretation, whereby expressions of natural language are assigned semantic values directly, without the need of going through a process of translation into a formal language.

In this tradition, the syntactic category of an expression is tightly connected to the nature of its semantic value. Leaving modal issues aside, the interpretation of a sentence is a truth value; for other linguistic expressions, we can assume their interpretation to be particular entities in D, the domain of the model, or functions of various degrees of complexity involving entities and truth values (as well as additional atomic types in more complex systems). Natural language expressions are assigned semantic types depending on the nature of the denotation they have. In simplest systems, there are two basic denotation types: e, for expressions referring to entities, and t for expressions referring to truth values (1 for true, 0 for false). Complex types, written as < a, b > are functions from entities of type a to entities of type b, where a and b are simple types or complex ones.⁷ Sentences are expressions of type t; predicates such as leave or see Mary are expressions of type < e, t >, i.e., they denote functions from entities to truth values. Expressions whose type is < e, t > are called properties. In set theoretic terms, we can think of them as the set of entities for which the predicate returns the value 1, i.e., the set of elements in D who left or saw Mary respectively. We will refer to such sets as the denotation of expressions of type < e, t >.

When two syntactic expressions, s₁ and s₂ combine to form a complex expression s₃, the interpretation of s₃ must be the result of combining the interpretations of its two daughters. Adhering to strict Fregean principles, the interpretation of one of these daughters must be a function and the interpretation of the other must be such as to fit the argument of that function. The interpretation of s₃ will then be the result of applying the function daughter to the argument daughter (function argument application). To illustrate with a simple example, assume that the semantic value of Proper Names is an entity in D, and therefore that their semantic type is e. Assume also that intransitive verbs are interpreted as functions from entities to truth values, i.e., they are of type

< e, t >, giving the value 1 whenever the entity that serves as their argument has the property denoted by the verb. The semantic types associated to the constituents of the sentence in (13a) are as in (13b), where the type of an expression is written under it:

(13) a. Ion pleacă. ‘Ion is leaving.’

b. Ion pleacă
t
Ion pleacă
e < e, t >

Here the subject functions as the argument of the predicate. Under the above assumptions we get the following intuitively pleasing result: the denotation of (13a) is 1 just in case the entity denoted by the subject is in the denotation of the property denoted by the intransitive verb. In other words, (13a) is predicted to be true in a model M just in case the entity denoted by Ion is among the leavers in M.

Montague’s aim was to capture what is common to all DPs. He aimed at giving them a uniform treatment in order to account for their relatively uniform syntactic distribution, as well as for the fact that one can conjoin DPs across subtypes in examples such as Jane and every student in her class/Jane and no student in her class, already mentioned above. Treating Proper Names as simply being of type e becomes problematic given that DPs such as every student and no student cannot be treated as denoting an entity, and therefore cannot be of type e.

The strategy Montague used was to reduce all DPs to the most complex case, that of DPs such as fiecare student ‘every student’ or nici un student ‘no student’, exemplified in Romanian and English in (14).

(14) a. Fiecare student a plecat.