The Semantics of Determiners

1.3 Theoretical perspectives
1.3.1 Predicate Calculus without and with restricted variables
Matters concerning the semantics of DPs in general and the semantics of determiners in particular have been crucial in shaping formal approaches to natural language semantics. Following early twentieth century proposals in the philosophy of language, formal approaches to natural language semantics relied initially on associating natural language sentences with formulas of a Predicate Calculus language for which truth conditions are well defined. In this indirect manner then, truth conditions are associated to natural language sentences as well. In the tradition stemming from Russell (1905), the sentences in (8) are associated with the formulas in (9), ignoring matters pertaining to tense and aspect.

(8) a. A plecat un student.

has left a student
‘A student left.’
b. Studentul a plecat.
student.Def has left
‘The student left.’
c. Fiecare student a plecat.
every student has left
‘Every student left.’
d. Nici un student nu a plecat.
Neg a student not has left
‘No student left.’

(9) a. ∃xstudent(x)  leave(x))

b. ∃xstudent(x)  leave(x))
c. ∀x (student(x)  leave(x))
d. ∃xstudent(x)  leave(x))

In (9) bold-faced expressions stand for the Predicate Calculus predicates corresponding to natural language nouns, verbs and adjectives. The ! mark is short for the uniqueness requirement associated by Russell with the definite article.

Formulas in a Predicate Calculus language (PC) are interpreted relative to a model M consisting, minimally, of a set of entities D and a valuation function V that assigns denotations to predicates and other constants of the language. Predicates are typed for the number of arguments they take. The truth conditions of the formulas in (9) ensure that (9a) is true in a model M if and only if there is an entity d in D, such that d is a student in M and d left in M. The formula in (9b) adds an extra requirement of uniqueness: there should be no d’  d such that d’ is a student in M and d’ left in M. The truth conditions for the universal formula in (9c) require it to be the case that for every entity d in D, if d is a student, d left. The negative formula in (9d) is true just in case the expression within the scope of the negative operator  is false, and therefore it will be true just in case there is no entity d in D that is a student and that left. The formal statement of these conditions involves either quantifying over elements in D (as our paraphrases have done) and using modified assignment functions or quantifying over modified assignment functions directly. (See Gamut (1991), Volume 1 for a particularly clear discussion of the former approach.)

In this system, the domain from which variables bound by quantifiers get their values is D, the set of entities in the model M. There are two unwelcome consequences of this assumption: (i) The correspondence between the constituents of the PC formulas and the syntactic make-up of the sentences they translate is problematic. In particular, there is no systematic correlation between the DPs of interest here, underlined in (8), and their contribution to the formulas that translates these sentences, underlined in (9). (ii) The approach just outlined cannot generalize to other quantifier-like expressions such as most, more than half, or finite cardinals.

In order to remedy this situation, McCawley (1981) advocates reverting to a pre-Russellian predicate calculus logic that uses restricted variables. In such PC languages, variables bound by quantifiers are restricted, i.e., their values are taken from subsets of D, subsets determined by Restrictor expressions. The formulas translating the sentences in (8) in a Predicate Calculus language with restricted variables are given in (10):

(10) a. ∃xstudent(x) leave(x)

b. ∃x:student(x) leave(x)
c. ∀x: student(x) leave(x)
d. ∃xstudent(x) leave(x)

Importing terminology from Heim (1982), the italicized expressions in (10) are the Restrictor part of the quantificational formula, while the part left ununderlined and unitalicized is the N(uclear) S(cope). The truth conditions for (10a) require there to be some entity among those entities in D of which the Restrictor is true such that the Nuclear Scope is true of it as well; the extra requirement imposed by (10b) is that there be a single entity of which the Restrictor is true. For (10c) to be true, it must be the case that every entity of which the Restrictor is true is such that the NS is true of it as well, and for (10d) to be true, (10a) must not be true. These truth conditions turn out to be equivalent to those given for (9) except that it is now possible to assume that particular determiners place particular restrictions on the set denoted by their Restrictor, restrictions that are ‘presupposed’ in the sense that they have to be met in order for the sentence to have a truth value at all. Thus, for fiecare ‘every’ one could assume that this set is presupposed to be non-empty, and therefore get no truth value for (10c) in a model without students, while (9c) would be true in such a model. A major advantage of this approach is that further quantifiers such as most or more than half can be added without any difficulty.

We can now generalize over DPs that serve as arguments of predicates. Their D contributes a quantifier and a variable, while their NP contributes the Restrictor. Expressions that form the Restrictor as well as those that form the NS may be complex and may contain quantificational formulas in their turn leading to more complex truth conditions. In this chapter we adopt the restricted variable version of Predicate Calculus as the lingua franca of formal semantics.

Proper Names such as Ion, on the other hand, are translated as individual constants whose semantic value is an individual. The connection between constants and their values is done at the level of the model by the valuation function V. Definite pronouns can be treated as contributing variables whose semantic value is determined either by the extra-linguistic or the linguistic context. There are many complex issues raised by the many ways in which pronouns are interpreted, which we will leave out of the current discussion.

Turning now back to articles, note that so far, the difference between definite and indefinite determiners (un vs. -(u)l) appears to be a matter of the latter bringing an extra requirement relative to the former. Understanding this requirement and its status is crucial to anyone interested in the semantics of determiners, since it involves understanding the definite/indefinite dichotomy. There are two issues that have been raised in connection with Russell’s uniqueness proposal from the start: (i) In order to apply the uniqueness treatment of the definite article to ordinary examples in ordinary language one has to be able to relativize it since (10b) is judged as true in many contexts in the absence of an assumption that would require the set of students in the model to be singleton.⁶ The uniqueness requirement therefore must be contextually restrictable. (ii) An independent problem, raised by Strawson (1950), is whether or not the uniqueness condition together with the existence requirements packed into the definite article have the same status as the NS requirements. In a Russellian approach, which makes no distinctions between these conditions, an expression such as (10b) is rendered false under any of the following circumstances: (a) the model/context contains no students; (b) the set of students in the model/context is not a singleton; (c) there is a singleton set of students in the model/context but the student in this set did not in fact leave. Strawson argued that the conditions under (a) and (b) should be seen as having a different status from the condition under (c): (a) and (b) appear to be pre-conditions that have to be met in order for the use of the definite article felicitous. This view led to treating the existence and uniqueness requirements that make up the contribution of the definite article as presuppositions rather than as being part of what the sentence containing the DP asserts.

Under a Strawsonean approach, we are led to the following view of the four Romanian determiners bold-faced in (8): (a) the unmarked indefinite is a simple existential; (b) the definite is an existential that presupposes that its Restrictor denotes a singleton set (relative to the context); (c) fiecare is a universal possibly presupposing that its Restrictor is not empty; (d) nici un is an existential that must occur in the immediate scope of the negative operator . This latter constraint is encoded in the negative morpheme nici. The negative operator is introduced by the verbal negation nu as well as by other morphemes, such as the complementizer fără ‘without’:

(11) Maria a plecat fără să ia nici o carte.

Maria has left without Subj. take Neg a book
‘Mari left without taking any book.’

The requirement that nici indefinites occur within the immediate scope of negation means that in order for nici indefinites to be grammatical the smallest clause in which they occur must be negated. Nici indefinites then are a strict type of what is know in semantics as a N(egative) P(olarity) I(tem).

What we have seen so far is that determiners decide what quantifier is involved in the interpretation of their DP, i.e., they determine the quantificational force of the DP. In addition, they may impose various restrictions on the properties of the Restrictor or on the syntactic/semantic environment of the DP. The former is exemplified by the hypothesized uniqueness presupposition associated with the definite determiner, the latter, with the special requirement carried by the negative morpheme nici.