Quantifiers in Natural Language

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Quantifiers in natural language correspond to words such as every, some, most, few, and many others.

The Semantics of Determiners

What is the semantics of expressions like every and most ? An answer to this question emerged in the early 1980s, in work of Jon Barwise and Robin Cooper (1981), James Higginbotham and Robert May (1981), Edward L. Keenan and Jonathan Stavi (1986), Johan van Benthem (1986), Dag Westerståhl (1985), and many others.

The basic idea of how to interpret quantified expressions comes from Gottlob Frege (1879). Frege observed that the familiar quantifiers (everything ) and (something ) can be thought of, in Frege's terms, as second-level concepts. Let us call whatever gives the interpretation of an expression its semantic value. Assuming an extensional and set-theoretic framework, we my assign predicates sets of individuals as their semantic values. Frege's idea can then be recast as saying that the semantic values of and are sets of sets. xFx (something is F ) is true if the semantic value of F is in the interpretation of , which happens just in case the semantic value of F is nonempty. More generally, quantifiers have as semantic values sets of the values of predicates which result in true sentences when the quantifiers are applied.

In logic, this idea was later investigated by Andrzej Mostowski (1957) and then Per Lindström (1966). But it does not apply to natural language without an important modification. Consider:

(1) Most students attended the party.

In this, most does not tell us something about a single predicate. Rather, it compares the students with the people attending the party. In particular, it compares the size of the set of students with the size of the set of people attending the party.

This binary or relational character of quantification in natural language is extremely widespread (as is demonstrated by the extensive list of examples in Keenan and Stavi 1986). It is also no accident. Rather, it reflects a fundamental feature of the syntax of natural languages. Simplifying somewhat, sentences break down into combinations of noun phrases (NPs) and verb phrases (VPs). Noun phrases also break down, into combinations of determiners (DETs) and common nouns (CNs) (or more complex construction with adjectival modifiers like small brown dog ). Quantifier expressions occupy the determiner positions in noun phrases, as in:

(2) [S [NP [DET most ] [CN students ] ] [VP attended the party] ]

(See any current syntax text for a more thorough presentation of this material, or the handbook discussions of Bernstein [2001] and Longobardi [2001]. For some interesting cross-linguistic work, see Matthewson [2001] and the papers in Bach et al. [1995].)

Quantifier expressions, such as every and most, are determiners. Their semantic values must be relations between sets of individuals, representing the semantic values of CNs and VPs in simple syntactic configurations like (2). Using some set theory, we may give examples of the semantic values of determiners explicitly. For instance, for a universe of discourse M and sets of individuals X, Y M :

(3) a. every M (X,Y ) Ξ Y
b. most M (X,Y ) |X Y | > |X \ Y |

(Here the boldface every M is the semantic value of the expression every.) This characterization of the semantic values of determiners as relations between sets is often called the relational theory of determiner denotations.

As the semantic values of determiners are relations between sets, the semantic values of noun phrases built out of determiners (or most determiners) are interpreted as sets of sets, along Fregean lines. For instance, the semantic value of most boys is most M boys = {Y M : |boys Y | > |boys \ Y |}. We may use the term 'quantifier' for either sort of semantic value. The latter are often called unary or simple quantifiers. Quantifiers taking more than two arguments are well documented in natural language, and have been investigated by a number of authors, including Filippo Beghelli (1994) and Edward L. Keenan and Lawrence S. Moss (1984). Quantifiers taking as inputs relations rather than sets, called polyadic quantifiers, have also been investigated, by authors including Higginbotham and May (1981), May (1989) and van Benthem (1909), though their place in natural language remains controversial. The survey by Keenan and Westerståhl (1997) is a good place to look for an introduction to these issues.

Properties of Quantifiers

The relational theory of determiner denotations has been applied to a number of issues in logic, philosophy of language, and linguistics. Many of these are discussed in the surveys by Keenan (2002), Keenan and Westerståhl (1997), and Westerståhl (1989). These applications rely on some important properties of quantifiers, of which two examples are given here.

restricted quantifiers

Quantifiers in natural language appear to be restricted quantifiers. Whereas and range over the entire universe, a quantifier like most M ranges over its first input, corresponding to the CN position in an NP. Most boys are happy expresses Most M (boys, happy ). Whether this holds or not depends on the properties of the boys , and not anything about the rest of the universe.

The mere presence of the CN argument is not enough to show that it functions as the domain of quantification. But the CN does play an important role, which is brought out by the following pattern:

(4) a. i. Every student attended the party.
ii. Every student is a student who attended the party.
b. i. Most students attended the party.
ii. Most students are students who attended the party.

In each of these, (i) and (ii) are equivalent.

The pattern we see in (4) is called conservativity:

(5) (CONS) Q M (X,Y ) is conservative if and only if for all X,Y M, Q M (X,Y ) Q M (X,X Y ).

Conservativity expresses the idea that the interpretation of a sentence with a quantified noun phrase only looks as far as the CN, so the CN restricts the domain of quantification.

One of the striking facts about natural languages, observed in Barwise and Cooper (1981) and Keenan and Stavi (1986), is that the semantic values of all natural language determiners satisfy CONS. This is a proposed linguistic universals: a non-trivial empirical restrictions on natural languages.

Conservativity has proved an extremely important property. The space of conservative quantifiers is much more orderly than the full range of relations between sets. This is brought out most vividly by the conservativity theorem due initially to Keenan and Stavi (1986), further investigated by van Benthem (1983, 1986) and Keenan (1993). The key insight is that the class of conservative quantifiers can be build up inductively, from a base stock of quantifiers and some closure conditions. Let M be a fixed finite universe and let CONS M be the collection of conservative quantifiers on M. We will build up a class of quantifiers D-GEN M on M as follows. D-GEN M contains every M and some M . We also assume that each set of members of M is definable by a predicate, and that D-GEN M is closed under Boolean combination and predicate restrictions. The latter assumes that if Q M (X,Y ) is in D-GEN M , so is Q M (X C,Y ) for C M. This amounts to closure under (intersective) adjectival restriction in an NP.

The conservativity theorem says that for each M :

(6) CONS M = D-GEN M

This tells us that the domain of natural language determiners is far more orderly than it might have appeared. Some logical properties extending CONS have been studied, by van Benthem (1983, 1986) and Westerståhl (1985, 1989). These appear to strengthen the proposed universal as well.


Quantified NPs are often described as expressions of generality. One way to articulate the relevant notion of generality is that it requires the truth of a sentence to be independent of exactly which individuals are involved in interpreting a given quantifier. This can be captured formally by the constraint of permutation invariance. A permutation π of M is a 1-1 onto mapping of M to itself, which can be thought of as a rearranging of the elements of M. The constraint of permutation invariance then says:

(7) (PERM) Let π be a permutation of M. Then Q M (X,Y ) QM (π[X ],π[Y ]).

(Here π[X ] = {π(x ): x X }.) PERM, or some strengthening of it, is commonly assumed in the mathematical literature, and is built into the definitions of quantifier in Lindström (1966) and Mostowski (1957). The semantic values of most natural language determiners satisfy PERM. (At least, the values of most syntactically simple determiners do.) There remain some hard cases, such as possessive constructions (as well as proper names, which can be interpreted as unary quantifiers not satisfying PERM). As these may not be examples of genuine determiners, the hypothesis that all natural language quantifiers satisfy both CONS and PERM is commonplace.

Semantic Composition

The relational theory of determiner denotations does not explain how quantifiers interact with the rest of syntax and semantics. The way the values of determiners combine with other semantic values provides an example of such interaction.

According to the relational theory, the semantic values of quantified NPs are sets of sets, while the values of VPs are sets. How do these combine? When we have a quantified NP in subject position, the semantics of composition is given by set membership. For a quantified NP value α:

(8)[S [NP α ] [VP β ] ] is true if and only if β α.

This simple story does not always work. Transitive verbs with quantified NPs in object position provide one sort of problem. A transitive verb will be interpreted as a relation between individuals. Now, consider an example like:

(9) a. John offended every student.
b. [S [NP John ] [VP [V offended ] [NP every student] ] ]

The value of offended is a relation between individuals, while the value of every student is a set of sets. We have no way to combine these to give us a set of individuals, which the value of the VP must be.

The theory of determiner denotations does not help solve this problem. Instead, some more apparatus is needed, either in the semantics or in the syntax. One approach is to posit underlying logical forms for sentences which are in some ways closer to the ones used in the standard formalisms of logic.

The goal is to replace the quantified NP every student with a variable that can occupy the argument position of a VP, that is, a variable over individuals. This variable is then bound by the quantifier. We thus want a structure that looks something like:

(10) [ [NP every studentx ] [S John offended x ] ]

In fact, many theories (following May 1977, 1985) argue that a structure like (10) is the underlying logical form of a quantified sentence. This is a substantial empirical claim about natural language, which holds that syntactic structures like (10) provide the input to semantic interpretation. Typically, such theories also hold that a syntactic process of movement produces a syntactic structure with initial quantifiers, and variables in the argument positions those quantifiers originally occupied. (For a survey of ideas about logical form in syntactic theory, see Huang 1995.)

Providing a structure like (10) does not by itself explain the semantics of binding: It does not explain semantically how the quantified NP binds the variable in the VP. The theory of the semantic values of determiners does not explain this either. Some separate account is needed.

The semantic operation that corresponds to binding is one of forming the right set to be the input of the semantic value of the determiner. Hence, even though we think of the syntactic structure John offended x as sentence-like (with the variable functioning like a pronoun), its interpretation needs to wind up being {x : John offended x }. Once we have this, we can say the sentence is true if this set is in the semantic value of the quantified NP every student. Hence, binding is carried out by the appropriate form of set abstraction (as in Barwise and Cooper 1981). Many current presentations are embedded in the framework of the typed lambda-calculus, which treat sets as functions from individuals to truth values. In such a framework (Büring 2005, Heim and Kratzer 1989), set abstraction is replaced by lambda-abstraction. Other approaches use similar syntactic structures to (10), but offer a more Tarskian account of binding (Higginbotham 1985; Larson and Segal 1995). Finally, there are approaches that avoid positing syntactic structures like we see in (10), including early work of Cooper (1983), and type shifting approaches (Hendriks 1993, Jacobson 1999, Steedman 2000, van Benthem 1991). There is also an approach that seeks to explain semantic composition via a generalized account of the semantic values of determiners (Keenan 1992).

See also Artificial and Natural Languages; Frege, Gottlob; Semantics.


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Michael Glanzberg (2005)