Summary and Keywords
Nominal reference is central to both linguistic semantics and philosophy of language. On the theoretical side, both philosophers and linguists wrestle with the problem of how the link between nominal expressions and their referents is to be characterized, and what formal tools are most appropriate to deal with this issue. The problem is complex because nominal expression come in a large variety of forms, from simple proper names, pronouns, or bare nouns (Jennifer, they, books) to complex expressions involving determiners and various quantifiers (the/every/no/their answer). While the reference of such expressions is varied, their basic syntactic distribution as subjects or objects of various types, for instance, is homogeneous. Important advances in understanding this tension were made with the advent of the work of R. Montague and that of his successors. The problems involved in understanding the relationship between pronouns and their antecedents in discourse have led to another fundamental theoretical development, namely that of dynamic semantics. On the empirical side, issues at the center of both linguistic and philosophical investigations concern how to best characterize the difference between definite and indefinite nominals, and, more generally, how to understand the large variety of determiner types found both within a language and cross-linguistically. These considerations led to refining the definite/indefinite contrast to include fine-grained specificity distinctions that have been shown to be relevant to various morphosyntactic phenomena across the world’s languages. Considerations concerning nominal reference are thus relevant not only to semantics but also to morphology and syntax. Some questions within the domain of nominal reference have grown into rich subfields of inquiry. This is the case with generic reference, the study of pronominal reference, the study of quantifiers, and the study of the semantics of nominal number marking.
Nominal reference has been at the center of work in philosophy of language and linguistics at least since Aristotle, and therefore an overview of the main threads of scholarship on this topic would barely fit into a thick tome. This article is by necessity selective. Left outside its scope are important topics such as the semantics of pronouns, bare nominals, nominal number, numerals, and genericity, which have grown into semi-independent subfields of their own.
The starting point of the discussion is a nominal typology drawn along both formal and semantic lines in Section 2; a review of the main theoretical tools and approaches follows in Section 3. Section 4 is devoted to theoretical issues raised by definite and indefinite DPs, and Section 5 concludes.
In what follows, the maximal syntactic category whose lexical head is a noun is called a D(eterminer) P(hrase); D stands for ‘determiners,’ a category that includes articles, and quantifiers; the term ‘nominal’ is used as an umbrella term for nominal projections, maximal or not.
2. Semantic Typology of Nominal Expressions
Under the assumption that the basic building blocks of a core sentence are a predicate, denoting an n-ary relation, and n arguments, the canonical role of nominals is to contribute information about arguments, as in (1a). There are, however, predicative uses of nominals as well, exemplified in (1b).
The focus of the discussion below is on argumental nominals, which is the subject of the next section.
2.1 Types of DPs
Figure 1 gives a traditional taxonomy of DPs based on both formal and semantic characteristics.
Leaving pronouns and bare nominals aside, the first category to be reviewed is that of P(roper) N(ames). Among non-pronominal DPs, they are the only category that lacks descriptive content. The issue of how PNs refer has been a subject of debate in the philosophy of language for a long time. One school of thought, associated with G. Frege and B. Russell, and known as descriptivism, holds that the semantics of PNs is a set of descriptions that uniquely identify the entity that bares that name. In this account, then, PNs are a special, shortened form of definite DPs, i.e., DPs whose D is the.
An opposing theory, pioneered by J. S. Mill and more recently developed in Kripke (1980), known as the causal theory of PNs, holds that the connection between PNs and their referent is direct, unmediated by any semantic content. For Kripke, this connection involves a causal chain that leads from an initial baptism event to the user of the name. The causal theory has been dominant in both the philosophy of language and linguistics, but see Bach (2015) for an overview of recent defenses of various descriptive approaches.
Languages differ with respect to whether PNs are used without an article (English) or with one (Greek uses a definite article with personal proper names), but they are always functionally definite, in the sense that their reference is (contextually) unique, a fact captured in both approaches. A semantic property of PNs that follows from the causal theory, and can be stipulated in a descriptive approach, is the fact that the connection between a PN and the entity it names is, typically at least, constant across contexts of evaluation.
The rest of this article focuses on DPs whose functional head is a D, the largest and most heterogeneous group. Their semantic properties follow to a large extent from their determiner. Semantically, the immediate building blocks of these DPs are the D and its nominal sister, whose semantic value is a set (or a property), which will be called the domain of the DP. The contribution of the D is to provide information about the elements of the domain that are relevant to the truth conditions of the clause in which the DP appears.
The nominals in Figure 1 should be treated as belonging to the same syntactic category for two main reasons: (a) they can be conjoined with one another (Jane and every one of her friends, a neighbor but none of the prominent officials in town), and (b) they can occupy main argumental positions in clausal structure, i.e., they can be Subjects, Direct Objects, Indirect Objects, and Objects of a preposition.
On the semantic side, it is harder to find a common denominator for these DPs aside from the fact that they can function as the arguments of the main predicate of the sentence in which they occur. Some of them—pronouns, proper names, definite and indefinite DPs—can refer to entities. Other DPs, such as every/each girl, no girl, any girl, cannot.
The brief discussion above treated articles (such as a or the) and quantifiers (such as every or most) as one class. There are many languages, however, that not only lack a definite or an indefinite article but do not employ articles at all. In Chinese, for instance, bare nominals are freely used in argumental position, with context guiding the interpreter in deciding whether the intended reference is definite or not. Such languages use classifiers when the reference of the nominal involves particular quantities.
Note that even in a language that makes heavy use of articles, such as English, there is a class of nouns, called mass nouns, that are used in argumental position without an article, and which use a classifier when particular quantities are referred to:
The basic semantic distinction between mass nouns (sugar, gold, water) and count nouns (book, child, idea) is that the reference of the latter is intrinsically individuated, i.e., it is made up of discrete parts. The denotation of these nouns can be thought of as a set of entities, some of which are atomic (singular) and some of which are not (plural). The reference of mass nouns, on the other hand, is non-individuated, i.e., not structured into separate parts. Thus, not any subpart of a book counts as a book, while any reasonably coarse subpart of sugar counts as sugar.
In order to account for the mass/count distinction, one has to enrich the ontological domain within which nominals refer, and distinguish between the individualized count domain and the non-individualized mass domain while at the same time positing linguistic means whereby nominals can switch their reference from one domain to the other. Further ontological distinctions within the individualized domain have to be drawn in order to account for genericity and the semantics of number. For a typology of languages based on how the mass/count distinction, number, and individuation are marked, see Chierchia (1998) and (2010).
2.2 Types of Ds
This subsection expands the Determiner-headed node of Figure 1, concentrating on distinctions within the class of articles and quantifiers. In work rooted in the philosophical literature, the emphasis has been on theoretical considerations based primarily on English data. The more recent linguistically rooted research has focused on the large cross-linguistic variation uncovered in this area.
The starting point is provided by a taxonomy of Ds based on English, given in Figure 2:
Following terminology introduced in Heim (1982), the constituent that is the sister of the D will be called the Restrictor of the determiner, and the rest of the sentence will be called the N(uclear) S(cope). The semantic value of the Restrictor is a set of entities, i.e., the domain of the determiner. The semantic value of the NS is also a set of entities. In (3) below,
the semantic value of the Restrictor is a set of students, and the semantic value of the NS is the set of entities that left.
The division between strictly quantificational and existential determiners will be motivated shortly. Strictly quantificational determiners fall into two subclasses, universal Ds (every, each) and proportional Ds (most, more than half). The truth conditions imposed by universal DPs require every entity in the domain of the determiner (in our case, the set of relevant students) to have the property denoted by the NS, in our case, to have left. Proportional quantifiers impose a proportional restriction on their domain. In the case of most, more than half of the entities in its domain have to have the property denoted by the NS.
In the case of existential determiners, the truth conditions require the existence of an entity in the semantic value of the Restrictor which also has the property denoted by the NS. All that is needed for (4) to be true,
is for there to be an entity within the relevant set of students of whom the property denoted by the VP is true, i.e., who left early.
Within the existential class, a major division is drawn between definite and indefinite DPs, to be discussed in Section 4. Membership in the indefinite category is large and contested, even if we restrict our attention to English. First, the ‘free choice’ determiner any, exemplified in (5),
could either be analyzed as a universal with a modal in its scope or as an existential within the scope of a modal. The many issues that arise in connection with free choice items remain outside the scope of this article.
Another contested class of determiners classified in Figure 2 as indefinite are negative determiners, exemplified in (6):
The issue is that negative determiners can be treated either as negative quantifiers or as indefinite articles signaling that the DP they head is within the scope of sentential negation. Sentential negation in such cases may remain implicit, as in (6), or be expressed, as in languages said to exhibit ‘negative concord.’ Note that languages (or even dialects of the same language) may differ as to whether and when the sentential negation operator may overtly co-occur with negative indefinites. See Zeijlstra (2011) and Penka (2011) for a generalization of the negative indefinite approach to English, German, and Dutch.
A common property of all the Ds discussed above, with the possible exception of the free choice determiner in (5), is that they allow their Restrictor to be contextually narrowed down to a subset of relevant or salient entities. Thus, in the appropriate context, (6) may be interpreted as claiming that no student in a particular class complained. Whether free choice determiners allow such contextual restrictions is a matter of dispute. Kadmon and Landman (1993) propose that the contribution of any is to widen the interpretation of the common noun along a contextual dimension which results in limiting domain restriction. Zamparelli (2007) has argued for treating ‘domain wideners’ as explicitly ruling out such restrictions. The topic of indefinite DPs is picked up again in 4.3.
3. Theoretical Tools
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. The main formal approaches to the semantics of nominal expressions are reviewed in the rest of this section, going as little as possible into matters that are independent of them.
3.1 Predicate Calculus With Restricted Variables
Early formal approaches to the semantics of natural language relied on associating natural language sentences with formulas of a Predicate Calculus language for which truth conditions are well defined. In this indirect manner, truth conditions are associated to natural language sentences as well. Following McCawley (1981), it is customary to use a predicate calculus language with restricted variables. In this system, ignoring matters pertaining to tense and aspect, and treating no as a negative indefinite, the sentences in (7) are translated as in (8).
Bold-faced expressions are used for Predicate Calculus predicates corresponding to natural language nouns, verbs, and adjectives. The symbol ! 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 and an assignment function f. M consists, minimally, of a set of entities D and a valuation function V that assigns denotations to the constants of the language. Assignment functions associate variables to elements in D. The italicized expressions in (10) are the Restrictor part of the quantificational formulas, while the parts left unitalicized are the NS. The truth conditions for (8a) require there to be some entity among those entities in D of which the Restrictor is true such that the NS is true of it as well. This amounts to requiring that there be a student who left. The extra requirement imposed by (8b) is that there be a single entity of which the Restrictor is true. For (8c) to be true, it must be the case that every entity of which the Restrictor is true, i.e., every student in M, is such that the NS is true of it as well, and for (8d) to be true, (8a) must be false. An advantage of this approach over a predicate calculus version that uses unrestricted variables is that it generalizes to other quantifier-like expressions such as most. Thus, for (9) to be true, more than half of the entities of which the Restrictor is true have to be such that the NS is true of them as well.
In this approach, then, DPs headed by determiners are translated as a quantifier and a Restrictor, with the D deciding what the quantifier is.
3.2 Generalized Quantifiers
A different, though related, view of strictly quantificational and existential DPs that is quite influential was developed in Barwise and Cooper (1981). The proposal is within a ‘direct’ approach to natural language semantics in which linguistic expressions are connected to their semantic values directly, skipping the intermediary logic language step. The gist of Barwise and Cooper’s proposal is to treat determiners as expressing relations between two sets A and B, where the set A is given by the interpretation of the expression in the Restrictor and the set B is given by the interpretation of the expression in the NS. For (7) and (9), A is the set of students, and B is the set of people who left. The relation denoted by the indefinite article a is that which obtains between two sets, A and B, when their intersection is non-null, i.e., A ∩ B ≠ ∅.The extra requirement imposed by the definite article the is that A be a singleton. The universal determiner every, on the other hand, requires A to be a subset of B, i.e., A ∩ B = A. The negative determiner no, if treated like a quantifier rather than a negative indefinite, denotes a relation that holds between A and B just in case A ∩ B = ∅. Note that this approach, like the restricted quantifier view, generalizes to proportional determiners such as most. Such determiners express a relation between A and B that involves relating A ∩ B to A\B.
This proposal led to a better understanding of the formal properties of natural language determiners. This, in turn, led to distinguishing between universal properties of all natural language determiners and linguistically relevant properties that are determiner-specific. An example of the former is conservativity. The relation imposed on the Restrictor set A and the NS set B by a conservative determiner holds of A and B if and only if it holds of A and A ∩ B. Thus, the properties of elements of B that are not in A are immaterial. There is an immediate connection between requiring natural language determiners to be conservative and the claim that all quantification in natural language is restricted.
An example of a formal property of generalized quantifiers that is relevant to natural language determiners though it is not universal concerns their monotonicity properties. Assume that the relation denoted by a determiner, understood as a generalized quantifier, holds between two sets, S and S′. Monotonicity concerns the issue of whether the relation in question still holds if we consider supersets or subsets of each of these sets. Taking one of these sets, say S, if the relation in question holds of supersets of S, the determiner is monotone increasing on S. If the relation holds of subsets of S, the determiner is monotone decreasing on S. Monotone decreasing environments have been shown to be relevant to the characterization of various types of special expressions called Negative Polarity Items (NPIs). The discussion of this class of nominals remains outside the scope of this article.
The generalized quantifier approach allows one to capture the commonalities across a wide range of DPs, from simple indefinites, such as a chair; universals, such as every chair; and proportional DPs, such as most chairs to complex DPs, such as at least 15 but not more than 20 chairs. This is in line with Montagovian approaches to the semantics of nominals, which is the subject of 3.3.
3.3 Montagovian Approaches
The semantic 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, while also covering as much as possible of the complexities of natural language. In Frege’s view, the interpretation of a complex expression α 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 composition, namely function argument application, a restriction that has proven too strict when faced with the vagaries of natural language.
Leaving modal issues aside, under this approach the interpretation of a sentence is a truth value, i.e., 0 or 1; 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. 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 a truth value. Complex types, written as < a, b >, where both a and b are types, are functions from entities of type a to entities of type b. 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. Such expressions are referred to as properties. They are functions that apply to entities and yield the truth value 1 (true) just in case the entity has the property in question. Alternatively, we can think of the denotation of a predicate as the set of entities whose characteristic function is the property in question.
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 allow that expression to play the role of 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 PNs 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 relative to a model M whenever the entity that serves as their argument has the property denoted by the verb in M. The semantic types associated to the constituents of the sentence in (10a) are as in (10b), where the type of an expression is written next to it:
The denotation of the sentence will be 1 just in case the entity denoted by the subject has the property denoted by the intransitive verb.
Montague’s aim was to capture what is common to all argumental 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. Treating PNs as simply being of type e becomes problematic given that DPs such as every student and no student cannot be treated as referring to entities.
The strategy Montague used was to reduce all DPs to the most complex case, namely quantificational DPs, exemplified in (11).
In Montague’s treatment, in such examples, the italicized DPs are of type < < e, t >, t >: they denote a function that takes a property as its argument and it yields a truth value as its value. The subjects in (11) denote functions that take the denotation of the VP as their arguments. 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 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 takes a property P and yields a function that takes a property as an argument and yields the value 1 just in case P ∩ Q = ∅. It is easy to see that these truth conditions are parallel to Barwise and Cooper’s approach (where P is the characteristic function of the Restrictor set and Q is the characteristic function of the Nuclear Scope set). The interpretation of (11a) is exemplified in (12), where the denotation of student plays the role of the property P, and the denotation of leave plays the role of the property Q:
The function denoted by every student is the characteristic function of the set of properties that every student has. The sentence Every student left then denotes 1 just in case the property of leaving is an element of this set, i.e., just in case for every student in D, that student left.
The type < < e, t >, t >, called generalized quantifier type, is the most complex type a DP may have. Now if the aim is to give all argumental DPs a uniform type, all the other argumental DPs, including PNs, must be generalized quantifiers 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 ≠ ∅. As before, the definite would impose an additional special restriction concerning the cardinality of P. Extending the analysis to PNs amounts to treating their denotation as a function from properties to truth values as well. As before, one can think of this function as the characteristic function of a set of properties. In the case of John, this set will be the set of properties that a particular entity (the one called 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.
It is customary in semantics to characterize the denotation of generalized quantifiers by means of lambda expressions, as in (13), where double brackets stand for a function from expressions to their interpretation, and where the model and assignment function parameters are omitted (the notation used in (13) is taken from Heim & Kratzer, 1998):
The lambda expression in (13) denote a function from properties to a truth value, where the value is 1 just in case the condition after the dot in the expression is met.
Note that in each case, one arrives at truth conditions close or equivalent to those of the restricted predicate logic 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 just sketched, that assigns syntactic categories a single uniform type and 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: (i) allow a richer inventory of modes of combination, and (ii) allow a single expression to be associated with multiple types. Both routes have been taken in the literature. Which one to follow, as a general strategy or as the solution to a particular problem, should depend on how well the solution fits the problem and how principled the departure is from the simple Fregean view.
To illustrate the need for further distinctions, note that while Montague’s unifying sweep explains the similarities found across various DP types, it is unable to explain the equally important syntactic and semantic differences encountered. As noted in Kamp (1981) and Heim (1982), for instance, PNs, pronouns, 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:
Note also that while indefinite DPs are quite natural as predicative nominals (exemplified earlier in (1) and repeated below), DPs with universal Ds are not:
These contrasts concern the distinction between universal and existential determiners in Figure 2 above. Ignoring negative and free choice determiners, one arrives at a basic distinction between purely quantificational and non-quantificational DPs, as in Figure 3:
A further distinction within the non-quantificational DP category is needed to separate those that may function as predicative DPs from those that may not. PNs and definite pronouns belong to the latter, while DPs with definite and indefinite articles and possibly indefinite pronouns belong to the former.
One way of addressing the variety of DPs found in language is to assume that a syntactic constituent can denote in several semantic types. In a multiple type approach to DPs one can assume 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 of quantificational DPs, 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 is to allow the category of 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).
3.4 Dynamic Approaches
The need to differentiate strictly quantificational DPs from all the other DPs in Figure 2 was the driving force behind the dynamic approaches to semantics pioneered in Kamp (1981) and Heim (1982). The former is the founding stone for the framework known as D(iscourse) R(epresentation) T(heory); the latter is the source of dynamic approaches to semantics known under the name of C(ontext) C(hange) S(emantics). The insights in dynamic semantics have led to the formulation of a dynamic predicate logic language, DPL.
Common to work in dynamic semantics is the assumption that the interpretation of a linguistic expression amounts to specifying how the expression in question affects and is affected by its context. The dynamic nature of these approaches consists in treating linguistic utterances as occurring against the background of an input D(iscourse)R(epresentation)S(tructure) or context. The primary task of the interpretation of a linguistic expression is to specify how that expression changes its input context.
Work on dynamic semantics of nominals has concentrated on three basic issues: (i) how to provide a uniform account of all types of definite pronouns, a problem beyond the scope of this article; (ii) how to account for the similarities and differences between existential and purely quantificational DPs; and (iii) how to account for the definite/indefinite contrast. This subsection concentrates on (ii), while (iii) will be the focus of Section 4.
While there are important differences between DRT and CCS, this discussion will concentrate on the aspects that are shared, using DRT terminology. Both approaches assume an intermediate level of representation that mediates between linguistic expressions and the model. A crucial element of this level is the notion of discourse referent (or variable), a discourse element that receives values from the domain of the model. The notion of discourse referent was first introduced in semantics in Karttunen (1976). For recent discussion, and connections to the Montagovian tradition, see Muskens (1996) and Brasoveanu (2007).
Crucial here is the idea that at the DRS level, nominals that are not quantificational simply introduce a discourse referent (or free variable), while quantificational DPs, besides doing this, also introduce a complex structure that ends up determining the quantificational force of the DP. The discourse referents introduced by non-quantificational DPs acquire existential force by default, as it were, via the truth conditions associated with the representations in which they appear.
Simple discourse representations (those that do not involve structures introduced by quantification, for instance) are made up of a set of discourse referents. Specific ‘construction rules’ specify the way DRSs are constructed based on particular linguistic expressions.
To exemplify, assume that (16) is uttered relative to an empty context.
The DRS resulting after the processing of (16) is of the form in (17), where the contribution of the DP is bold-faced:
Interpretation rules connect DRSs to their truth conditions relative to a model M with a domain D. By general rules, a DRS K is true in a model M just in case one finds a function from the set of discourse referents in K to the entities in D such that this function meets all the conditions imposed by K. For (19), such a function has to map x to an element in D that is a student and that came in M. A function f that meets the conditions imposed by a DRS K relative to a model M is an embedding function for K in M. Thus, a DRS K is true in M if and only if there is an embedding function for K in M. The contents of the DRS are, in essence, constraints on the embedding function. As a result, existential force is conferred on discourse referents by general truth conditions. PNs and definite pronouns, like (in)definite DPs, simply introduce a discourse referent and a condition on it without creating extra structure.
A quantificational determiner is responsible for the introduction of a complex structure consisting of two DRSs linked by an operator. The first DRS (in our case, K′) is contributed by the Restrictor, the second (K′′), is contributed by the NS, and the connecting operator is determined by the nature of the determiner. The DRS resulting after interpreting (18) relative to an empty input DRS is given in (19):
The complex conditions associated with such complex DRSs say that a function f is an embedding function for K in M just in case every f′ that extends f and is an embedding function for K′ can be extended to a function f′′ that is an embedding function for K′′ in M. Embedding functions are partial functions from discourse referents to D. A function f′ extends a function f iff f and f′ agree on all values for which f is defined. Note that the quantificational DP does not contribute a discourse referent in the main DRS K, a property that is responsible for the fact that such DPs cannot normally act as antecedents for singular pronouns.
Within this general approach, the next step is to further distinguish between types of non-quantificational nominals. The distinctions can be made at the level of the properties of the discourse referent introduced (such as whether its value is an ordinary entity or a kind-level entity, an atomic entity, a group, or a sum), as well as at the level of the conditions the nominal brings along. In addition, the constraints accompanying a particular determiner, whether quantificational or not, may involve the properties of the discourse structure that serves as its input (i.e., presuppositions) or properties of the output structure (postsuppositions). Section 4 concentrates on definite and indefinite DP, i.e., DPs whose D is on the existential branch in Figure 2.
4. Definite Versus Indefinite DPs
This section first lays out the basic issues in this area and then goes into the properties of different groups of definite and indefinite DPs.
4.1 Basic Questions
The philosophical and early linguistic approaches to definiteness focused on the issue of differentiating DPs involving the unmarked definite article the (definite DPs) from DPs involving the unmarked indefinite article a (indefinite DPs). The issue is what property distinguishes the two.
Another important issue that has been somewhat neglected is how to generalize the notion of definiteness to a larger class of DPs, involving no article or different articles from the two unmarked cases. An intuitive grouping is given in (22), where quantificational DPs have been set aside.
This division is justified beyond the intuitive level inasmuch as there are grammatical phenomena that treat the DPs in one group differently from the DPs in the other. Such a phenomenon is Differential Object Marking (DOM). Languages that exhibit DOM (Turkish, Persian, Romanian, Spanish, Hindi, to name just a few) mark a subclass of direct objects (DOs) by special morphosyntactic means. The question of interest for the semanticist is what semantic parameters are relevant to characterizing the class of DOs that are specially marked. The vast literature on this topic has shown that definiteness and animacy are the main factors involved. The data within a language as well across languages point to hierarchies of DPs depending on how likely they are to trigger DOM.
With respect to definiteness, Aissen (2003) proposes the hierarchy in (21):
The further on the left of this hierarchy a DP is, the stronger DOM trigger it will be relative to the definiteness parameter. Note that the placement of PNs and definite pronouns to the left of definite DPs justifies treating these two types of DPs as definite, as in (20).
The scale in (21) raises the issue of how to make narrower semantic distinctions within each DP type and how to understand specificity distinctions within the context of definiteness. Some of these matters are addressed in the rest of this section, organizing the discussion around the basic definite/indefinite contrast.
4.2 Definite DPs
Starting with the basic contrast between the unmarked definite and the unmarked indefinite articles (the vs. a in English), a preliminary question that needs to be settled is whether the distinction is presuppositional or whether it is part of ‘at issue’ content. In the philosophical literature of the early 20th century, the former view was taken in Russell (1905) and the latter in Strawson (1950). The presuppositional view has prevailed in the linguistic literature and will be assumed below without further discussion.
When it comes to the issue of what distinguishes these two articles, the two most prominent contenders are uniqueness versus non-uniqueness and familiarity versus novelty.
According to the uniqueness approach to definiteness, the definite determiner presuppose that the DP it heads refers uniquely relative to the context in which it is uttered.
According to the familiarity approach, introduced into formal semantics by Karttunen (1976) and Heim (1982), the definite determiner presupposes that the discourse referent associated to the DP it heads is already present in the input context. While this view has the immediate advantage of extending straightforwardly to definite pronouns, it runs into serious empirical challenges.
For instance, Poesio and Vieira (1998), among many others, have documented numerous cases of novel definite DPs. Two instances are exemplified in (22):
For further arguments, see Abbott (1999).
Swayed by the empirical worries associated with the familiarity approach, the consensus has shifted back to an account based on uniqueness. Note, however, that a Russell-style absolute uniqueness view fares well with only a very small number of DPs, such as those in (22) or DPs like the current Queen of England, or the number of words in this article. In order for such accounts to become empirically respectable, uniqueness has to be made sensitive to the input context. This means that the uniqueness condition on definite DPs has to require unique reference relative to the context against which the DP is interpreted. Once context is properly taken into account, uniqueness characterizations of definiteness become dynamic as well, since what is required is unique reference relative to the input context of the DP.
Several versions of dynamic uniqueness approaches to definiteness have been proposed in the literature—see Kadmon (1990), Farkas (2002a), and Roberts (2003). The upshot of these proposals is that dynamic uniqueness amounts to having a unique appropriate value for the relevant discourse referent relative to its input context. Thus, relative to the immediate input context of a definite DP, all output functions have to assign the relevant discourse referent the same value.
Note that the characterization of definiteness in dynamic uniqueness terms extends to PNs and definite pronouns. In the case of definite DPs like the present Queen of England or this/the proposal, dynamic uniqueness is achieved because the description denotes a singleton set relative to the model in the first case, and relative to what is salient in the context in the second. Cases of anaphoric definite descriptions, exemplified in (23), can be treated along similar lines:
Definite pronouns achieve dynamic uniqueness because they have to have a unique antecedent in the discourse. PNs are dynamically unique because of the special connection between the name and its referent. See Farkas (2002a) for more discussion of these distinctions and on how they relate to the hierarchy in (21).
The dynamic uniqueness approach can deal with definite DPs that rely on familiarity because being anaphorically linked to a unique salient familiar discourse referent is one of the ways in which dynamic uniqueness can be achieved. It is therefore not surprising to see familiarity and definiteness connected in some principled way. For further discussion of how uniqueness and familiarity are connected, see Hawkins (1991) and Farkas and de Swart (2007).
When it comes to narrowing the scope of the discussion to the distinction between the ordinary definite article (the and its cross-linguistic relatives) and demonstrative determiners, note first that they pattern with ordinary definites with respect to phenomena sensitive to definiteness. Note also that in many languages demonstrative DPs are often formally marked relative to their ordinary definite counterparts in that they may involve both the definite and the demonstrative determiner. In terms of interpretation, they appear special as well in that they achieve unique reference relative to a subpart of the context made salient either by gesture or by the fact that the value of the relevant discourse referent has been introduced in the recent discourse.
The dynamic uniqueness view of definiteness is not without its empirical challenges. They involve cases where a definite DP is acceptable even though the context does not provide a unique salient value but rather provides a small number of such values, with the choice between them being of no consequence for the purposes of the discourse. Such examples are given in (24):
On balance, a contextual uniqueness account of definiteness, if it can be adequately weekend to deal with the recalcitrant cases in (24), appears the most promising approach that has been proposed so far.
An independent question that arises is whether the definite/indefinite contrast, as exemplified in the definite/indefinite article contrast, is symmetrical or not. The symmetrical view holds that the is marked for a particular property (uniqueness or familiarity), while a is marked for the complement of that property (non-uniqueness or novelty). Under the assumption that the contrast involves a DEF feature, in the symmetric view this feature is equipollent. The asymmetrical view, which is simpler and therefore in principle preferable, holds that the is marked for a particular property, while a remains unmarked or underspecified for this property. Under this assumption, DEF is a privative feature. For arguments for the asymmetrical view, which will be adopted here, see Hawkins (1991) and Abbott and Horn (2012).
4.3 Indefinite DPs
The class of indefinite determiners within a language as well as cross-linguistically is large and heterogeneous. Consequently, indefinites have been the focus of much theoretical and empirical research, which cannot be fully reviewed within the confines of this article. The discussion will concentrate on some aspects of the interpretation of DPs with indefinite determiners, leaving indefinite pronouns and bare singulars outside its scope.
4.3.1 Unmarked Indefinite Ds
In languages like English, with a motley class of indefinite determiners, it is useful to distinguish the unmarked element of the group from the other, marked indefinite articles. The unmarked indefinite, which in English is a, comes with no further requirements on the interpretation of the discourse referent it introduces. The other indefinite articles are marked in that they involve additional interpretive requirements. The particular interpretive requirements to which marked indefinites are subject vary across languages, though there are recurring similarities. In many languages, interpretive markedness distinctions are paralleled by morphological markedness. Thus in Hungarian, for instance, the ordinary, unmarked indefinite determiner is egy, while marked indefinites come with special morphology added to this form: egy-ik is a partitive indefinite, and egy-egy is a dependent indefinite, i.e., it is marked for co-variation with a previously introduced discourse referent. While there are languages that lack unmarked indefinite articles altogether, or fail to mark various types of special indefinites, there are no languages that have a morphologically simple indefinite article coming with a complex interpretive condition and a morphologically complex article best analyzed as simply being underspecified for dynamic uniqueness or any other property associated with indefinites.
As a consequence of their semantic blandness, unmarked indefinites have remarkable freedom of distribution and interpretation. It is precisely because of their interpretive versatility that unmarked indefinites have attracted attention in the literature. For instance, a much-discussed property they exhibit, which separates them from purely quantificational Ds, is that DPs headed by unmarked indefinites have unbounded upward scope (for discussion, see Farkas, 1981a and Abusch, 1993). A DP i occurring in a sentence S has unbounded upward scope iff its interpretation may or may not be affected by the interpretation of any operator or quantifier Q in S occupying a structurally higher position relative to i. If the interpretation of i is affected by Q, i is said to be in the semantic scope of Q, while if the interpretation of i remains unaffected by Q, i is said to be outside the scope of Q.
To illustrate, the indefinite in (25),
may be interpreted either within or outside the scope of the universal. In the latter interpretation, the sentence is true in case there is a story by Alice Munro such that every student read that story; in the former, the sentence is true in case for every student there is a story by Alice Munro such that the student read that story, with possible co-variation between students and stories. In predicate logic, these two readings are represented as in (26), where sbAM abbreviates story by Alice Munro, and where square brackets mark the NSs of quantifiers:
A more complex example is given in (27):
Here, the indefinite may be interpreted as having widest scope, represented in (28a), where the story does not co-vary with either professors or students. A second possible interpretation, represented in (28b), is one where the indefinite has narrowest scope, in which case the story may co-vary with the students. Under this interpretation all that the professors cared about is whether students had read a story by Alice Munro. Finally, there is an intermediate scope reading as well, represented in (28c), in which case the story co-varies with the professors but not with the students. Under this interpretation, for each professor there is a particular story by Alice Munro (though not necessarily the same for all professors) and the professor talked to every student who had read that story.
In contrast, the upward scope of strictly quantificational DPs is clause bound, i.e., they may scope over a structurally higher indefinite only if the indefinite and the universal are clause-mates. Thus, in (29), the universal may take scope over the indefinite (in which case the student may co-vary with the professor), or not, in which case co-variation is ruled out:
In (30), however, the universal cannot be interpreted as scoping over the indefinite, i.e., there is no possible interpretation of (30) in which professors co-vary with students:
The issue of how to account for this scope contrast has been hotly debated in the literature. Fodor and Sag (1982) attributed it to an inherent ambiguity in unmarked indefinites between an existential and a referential interpretation. According to this view, existential indefinites obey the same scopal constraints as universals, while referential DPs, like deictic demonstratives, have fixed reference, and therefore give the illusion of having only widest scope. As pointed out by Farkas (1981a), the existence of intermediate scope readings presents an empirical problem for this account.
Reinhart (1997) and Winter (1997) propose a different solution by introducing a new formal tool, namely choice functions, deployed solely for the purpose of accounting for the scope contrast between indefinites and purely quantificational DPs. In this account, indefinites are interpreted as choice functions whose domain is given by the NP sister of the determiner. The choice function variable is bound by an existential quantifier that is freely inserted at any point in the structure, thus assuring the desired freedom of scope. Under this account, indefinites have exceptionally free scope because their interpretation involves choice functions, and choice functions are bound by an exceptionally free mechanism. The interpretation of quantificational DPs does not involve choice functions, and because of this, their scope is fixed to the clause in which they occur.
Matthewson (1999) and Kratzer (1998) propose to do away with the necessity of the exceptional choice function binding mechanism by suggesting that choice functional variables are contextually bound, resulting in interpretations equivalent to widest scope readings. To account for intermediate scope readings, the mechanism is enriched with optional indices on choice functions, which result in co-variation interpretations. For problems with this view, see Chierchia (2001) and Schwarz (2011). See also Schwarzschild (2002) for an argument that exceptional wide scope is an illusion due to contextual uniqueness.
The approaches in Farkas (1997) and Brasoveanu and Farkas (2011) pursue a different solution altogether. The essential novel assumption is that the issue of co-variation or lack thereof that underlies scopal phenomena is only partially determined by structural considerations. The value of an indefinite is allowed, but not required, to co-vary with structurally higher operators. Such operators render co-variation possible for an indefinite that is lower in the structure but do not enforce it. Under these approaches, the account of existentials in general and of unmarked indefinites in particular does not involve any special mechanism. Marked indefinites, on the other hand, are special precisely because they contribute special conditions, at least some of which require or rule out variation of values assigned to the discourse referent introduced by the indefinite relative to various parameters of evaluation.
The interpretive versatility of unmarked indefinites is responsible not only for their freedom of upward scope but also for their ability to be interpreted within the scope of a variety of operators or other quantifiers. To exemplify, note first that the italicized DPs in (31) can be interpreted within or outside the scope of the bold-faced expressions:
The referential properties of the narrow scope indefinites in (31) depend on the semantic properties of the operators they have narrow scope relative to. Thus, in (31a), it is possible to list the different professors that the candidates talked to, while in (31b–d) such a specification is not possible. Finally, note that the truth of (31d) requires the denotation of the nominal to be empty in the world of evaluation, while the opposite is arguably the case in (31a) and (31c); finally, the truth of (31b) requires Susanna to assume (correctly or not) that the denotation of the indefinite is non-empty. Making sense of these differences is the task of the semantics of the operators the indefinite interacts with. The semantics of the unmarked indefinite itself should be sufficiently bland to allow this interaction. Whether a DP is interpreted within or outside the scope of certain operators turns out to have morphological consequences in some languages. For instance, it has been shown that in Romance languages and elsewhere, being interpreted within the scope of certain intensional predicates such as want, wish, or look for allows a DP to host subjunctive relative clauses. (See Rivero, 1975 and Farkas, 1981b).
Unmarked indefinites also exhibit interpretive contrasts that are independent of scope. Thus, in (34), taken from Fodor and Sag (1982), the indefinite can be interpreted either as epistemically specific, in which case the speaker has a particular student in mind, or as epistemically non-specific, in which case (32) is interpreted as a mere existential statement:
The distinction between epistemically specific and epistemically non-specific indefinites can be modeled as involving a constant value assigned to the discourse referent relative to the epistemic state of the speaker in the case of the former, and varying values in the case of the latter.
Another contrast that is independent of scope involves the possibility of unmarked indefinites to be interpreted partitively in the right context, as exemplified in (33):
The italicized DP can be interpreted either as a simple existential or as implicitly partitive, i.e., as referring to one of the several men who came in. In the case of the partitive indefinite, the set from which values for the discourse referent introduced by the indefinite are to be chosen is contextually restricted to a previously introduced set.
There are morphosyntactic properties that are sensitive to epistemic and partitive specificity. Thus, it has been shown that the specificity rung in Aissen’s hierarchy in (21) concerns epistemic and partitive specificity. In many DOM languages, partitive and/or epistemically specific indefinite DPs are better DOM triggers than non-specific indefinites. An important issue which will be discussed shortly is why this should be the case.
4.3.2 Marked Indefinites
The focus of research on nominal reference in the 21st century has been on morphologically marked indefinite DPs, i.e., DPs whose determiner is a special indefinite article signaling a special interpretive property. The main challenge in this area is to account for the subtle interpretive and distributional constraints of each of these special determiners while at the same time proposing a predictive cross-linguistic taxonomy. A useful perspective, suggested in Farkas (2002a), is to see marked indefinite determiners as falling into two large classes, namely pro-variation and anti-variation determiners. Pro-variation special determiners impose interpretive restrictions that result in enforcing variation across the values assigned to the relevant discourse referent across particular evaluation parameters. Anti-variation determiners impose interpretive restrictions that result in enforcing constancy of values for the relevant discourse referent across particular parameters.
Under this perspective, anti-variation determiners form a natural class with definite determiners, because the contextual uniqueness condition imposed by definite DPs is a strong anti-variation requirement.
Partitive DPs (discussed above) form a natural class with definite DPs, because the variation of values they permit is restricted to a contextually salient set. Partitivity, then, is a restriction on allowable variation in values for the relevant discourse referent. In many languages, there are special indefinite determiners that mark this restriction. An example already mentioned is the complex determiner egy-ik in Hungarian, which requires the value of the discourse referent it introduces to be chosen from a contextually salient set.
Epistemically specific DPs are like definites in that they require constancy of values assigned to their discourse referent. The essential difference is that for definites, constancy is required relative to the function that gives them values relative to their input context, while for epistemically specific indefinites, constancy is required relative to the worlds in the epistemic base of the speaker.
The marked indefinite a certain in English has been claimed to signal epistemic specificity. Farkas (2002b) argues that in fact the interpretive constraint relevant to this particular anti-variation indefinite is weaker: it requires the discourse referent it introduces to be in principle identifiable. This means, roughly, that there must be a world in the context set relative to which the value assigned to the discourse referent in question is unique, i.e., constant. As a consequence, a certain DPs are correctly predicted not to occur within the scope of intensional predicates like want or negation, but to be acceptable within the scope of universals or epistemic predicates. This is in line with the general goal put forth in Farkas (2002b), which is to have the scopal properties of special indefinites follow from the interpretive constraint they impose on the discourse referent introduced.
Within the class of anti-variation DPs, the two groups that have received most attention are dependent (or distributive) indefinites and epistemic indefinites.
Concerning dependent indefinites, it has been noted that many languages—Hungarian, Romanian, Hindi, Albanian, and Kaqchikel among them—use special morphological marking on indefinite articles or numerals that requires the DP to be interpreted as co-varying with another variable, which is assigned a range of values. This characterization is due to Farkas (1997), who discusses Hungarian, which has both dependent indefinites and dependent numerals, with subtle differences among them. Balusu (2006) and Henderson (2014) consider simpler data from Telugu and Kaqchikel respectively, and propose simpler constraints, which, however, also result in enforced variation of values.
The class of marked indefinites called epistemic are associated with a constraint that results in an interpretation akin to epistemic non-specificity, often, though not always associated with implications of speaker ignorance or indifference relative to the value chosen for the relevant discourse referent. Special indefinites that belong to this group are singular some DPs in English, discussed in Farkas (2002b); algún indefinites in Spanish, discussed in Alonso-Ovalle and Menéndez-Benito (2010); vreun indefinites in Romanian, discussed in Farkas (2007) and Fălăuş (2011); and, arguably, German irgend, discussed in Kratzer and Shimoyama (2002), Kratzer (2005), and Aloni and Port (2013, 2015), among others. The various conditions proposed to capture the special nature of these indefinites result in enforced variation of values assigned to the relevant discourse referent across different types of evaluation parameters. A common theme across many of these accounts is the notion of alternative values for the relevant discourse referent; what differs is how these alternatives are defined.
5. Some Open Issues
The aim of this article is to survey some of the main research strands in the realm of nominal reference, attempting to bring together theoretical and empirical perspectives. Despite the venerable tradition of work in this area, many important questions are still open. The overall issue that the field faces now is to characterize the theoretical tools that are most appropriate for dealing with the vast array of problems nominal reference raises. In particular, the cross-linguistically rich variety of special indefinites just discussed has shown that the many subtle distinctions drawn across various indefinites show not only variety but also common themes. It is, however, still left to future work to capture these differences and similarities in a way that not only describes the facts but also explains why they are the way we find them.
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