The interpretation above is considerably simpler than the equivalent one using universal and gereralised quantifiers.
Non-classical quantifiers may also be captured in this way. For example, the discontinuous determiner more...than denotes the set of sets whose intersection with the extension of one common noun has more members than than their extension with a second common noun:
More students than lecturers
[more_than'(lecturer') (student')]M,g. = {X Í A | | X _ [student']M,g.| _ {X Í A | | X _ [lecturer']M,g.|}.
Set theory enables the notion of context dependence to be incorporated more directly into the semantics by letting a variable represent some pragmatically determined proportion. Thus the truth values of most, more_than_half, many and few can be determined pragmatically. For example, Many civil servants have knighthoods can be true even though less than half of such people are knighted, since on average more civil servants than members of other professions are knighted. Thus we can represent many civil servants, where c is some pragmatically determined proportion, as below:
[many' (civil-servant')]M,g.= {X Í A | | X _ [civil-servant']M,g. | _ c. | [civil-servant']M,g. | }
GTQ also allows the statement of constraints on the possible denotations of natural language quantifiers. The conservativity universal is a general property for quantifiers. It ensures that in the interpretation of a noun phrase containing a common noun, for example flower, we need not consider sets of entities in the domain of discourse which are not in the extension of flower, for example tree or bush.
Attempts have also been made to characterize the properties of subsets of quantifiers as shown below:
Monotone increasing quantifiers are unaffected by an increase in their common noun or verb phrase extension.
Monotone decreasing quantifiers are unaffected by a decrease in their common noun or verb phrase extension.
Subject monotone quantifiers are unaffected by an increase or decrease in their common noun extension.
Predicate monotone quantifiers are unaffected by an increase or decrease in their verb phrase extension.
The above properties allow interesting linguistic universals to be stated about noun phrases and enable further linguistic phenomena to be isolated and explained.
Let us now consider which types of noun-phrase can be classed as generalized quantifers. According to Barwise and Cooper (1981, p177) "the noun-phrases of a language are all and only the quantifiers over the domain of discourse". However, this statement is problematic. In addition to NPs, adverbials may have a quantificational meaning. Conversely it can be shown that not all NPs are quantifiers. Following Loebner (1987) I would like to consider definites, indefinites and quantifiers in a restricted sense in turn.
Definites
Russell's analysis of singular definites as denoting individuals and plural definites as denoting classes prevents a uniform analysis of definites, which behave the same syntactically and pragmatically. This can be remedied by treating all definites as terms.
As will be demonstrated, from within GQT itself there is evidence against the quantifier approach to definites.
Under negation definites can be shown to behave as terms. If definites are treated as terms then definite NPs should denote single individuals rather rather than classes of individuals. The definite NPs John and I correspond to single individuals. If a predicate P is false for John/I, then its negation is true, i.e. there is a global truth-value. However, for the NP the children, which, e.g., denotes three girls and two boys, there will be predicates which do not hold for all of the children, e.g. the children are girls, i.e. P is partly true and partly false, meaning that no truth value can be assigned. Compare this to the quantified proposition all children are girls, which can be assigned a truth value, namely false. If we accept truth-value gaps of this kind definite NPs in general can be treated as terms. So definite NPs denote individuals whose domain is just the true predicates together with the corresponding false predicates; whereas quantifiers with the same head noun have larger domains and also yield truth-values for all properties which hold for only parts of the head noun denotation.
Since definites are terms they cannot be negated. Quantifiers, however, can in general be negated either directly or lexically:
*not the students
not all the students
either/neither student
The last point leads us on to the differences in the type of modification quantificational and definite NPs can undergo. Quantifiers are second order predicates and can thus be modified, whereas the definite article cannot:
Not
Almost *the
[ Absolutely all students] are fast asleep
Already
Possibly
However, sentences consisting of a definite NP and a predicate can be modified using quantifiying adverbs, whereas sentences containing a quantificational NP cannot, since it is impossible to quantify the same variable twice:
*Two all
*all students are partly fast asleep
The mostly
to 70 percent
The head of a quantifying NP is definite. The quantificational determiner specifies to what extent the predicate applies to the domain under consideration.
Indefinites
Indefinite NPs can occur in the grammatical subject position of existential there-sentences. Loebner (1987) characterizes indefinite NPs as having no determiner or a determiner of a certain class. He adds that indefinite determiners all contain a cardinality specification. Barwise and Cooper (1981) distinguish indefinites on the basis of several common properties, which can however be at least partly explained by their semantic nature, which is not quantificational.
Indefinite determiners can be taken as cardinality predicates or as quantifiers proper depending on the context:
Some salesmen walked in
(i) Something walked in, namely salesmen, in fact some.
(ii) Some out of a certain set of salesmen under consideration walked in.
In (i) the indefinite introduces a new referent. This non-quantificational reading fits the discourse-semantic analysis of indefinite determiners as cardinality attributes, i.e. one-place first order predicates. The reading in (ii) is quantificational in the sense of GQT, i.e. determiners are second order predicates, and cannot be handled by the discourse-semantic approach. A set of salesmen is already introduced in the context or independently determined, otherwise the partative reading would not be available. This matches the idea of non-generic, quasi-partative quantificational NPs having partative heads.
The important question is how both uses of indefinites can be united under a single theoretical approach. Lexical ambiguity is inappropriate as a solution, since the ambiguities are not random. Arguing that the discourse-semantic interpretation applies in both cases is unsatisfactory for various reasons. The GQT approach can only handle the uses labelled quantificational. However, if quantificational interpretations of sentences containing indefinites are attributed to other semantic properties of the sentence, rather than to the indefinite determiner, then indefinites are another subcategory of NPs which should not be treated as generalized quantifiers.
If this is the case then we could posit there be in existential there-sentences as an existential quantifier. In support of this claim, there be can take neither ordinary terms nor a quantifier proper.
The cardinality attributes of indefinite NPs suggest a mental counting process which always involves two predicates. For the non-quantificational use the matrix predicate provides the counting domain, and the head noun is the counted predicate. For the quantificational interpretation these roles are reversed. This analysis can account for the common properties of indefinites observed by Barwise and Cooper (1981).
Cardinality predications involve two arguments at a conceptual level. Indefinite determiners in English are used as one-place first order predicates. The conceptual level hints at how indefinite pronouns can enter a quantificational scheme.
Quantifiers in Loebners Narrower Sense
Discluding definite and indefinite determiners we are left with classical strong determiners and proportional determiners. Both, either and the proportional determiners (excepting most) have only non-generic uses; whereas all, every, each and most, as well as the secondary determiners many, few, some, no have in addition generic uses.
Differences can be drawn between generic and non-generic, i.e. referential quantification. Referentially quantifying sentences refer to a limited domain of quantification which has previously been established. So referential quantifiers take the denotation set of the head noun as part of a limited universe of discourse.
e.g. the apples are sour
Generically quantifying sentences refer to kinds of objects on a conceptual level.
e.g. apples are sour
Generic quantification requires virtually unlimited and inherently fuzzy denotation sets. It can also be used with respect to real classes rather than sets.
The GQT format is designed to capture referential quantification, but is not appropriate for generic quantification. It appears that a notion of quantification should be prefered which involves properties instead of sets. Such a treatment would be adequate for the referential cases too.
The above arguments suggest that it is not justified to treat all NPs as genuine quantifiers in the sense of GQT. Neither definites nor indefinites are quantifiers and not all uses of the remaining quantificational NPs fit the generalized quantifier scheme. A distinction between referential and generic quantification is also needed.
Barwise and Cooper's (1981) approach following on from Montague (1973) in PTQ results in a broad and unspecific concept of natural language quantification, since it reads a quantificational interpretation into all NP meanings which in certain contexts can be interpreted as quantifiers. The inverse approach can be taken by trying to describe only those means of expression which are specific for quantification. Quantifiers must also be sought in ither syntactic categories. They can be expressed in various ways. The result is a conception of natural language quantification which is more specific, but at the same time applies to substantially more linguistic data.
Bibliography
Cann, R., 1993, Formal Semantics: an introduction, Cambridge University Press.
Loebener, S., Natural Language and Generalized Quantifier Theory,
in Gardenfoers, P. (ed), 1987, Generalized Quantifiers: Linguistic and Logical Approaches, Reidel
