Quantification

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Routledge Dictionary of Language and Linguistics

In formal logic, quantification refers to the specification of for how many objects in a certain set a predicate is valid. Quantification is determined by quantifiers ( operator) which connect freely occurring variables in a sentence. A distinction is made between the existential quantifier, which says that the predicate in question is valid for at least one object in the given set, and the universal operator, through which the predicate in question is assigned to all elements of the underlying set of individuals. In quantification, the logical analysis is abstracted from the many colloquial interpretations, which may appear as the expressions several, some, many, by rendering these expressions non-distinctive through the existential operator. On the other hand, ambiguities such as those found in the colloquial statement Everybody loves somebody can be specified in formal logic by illuminating the different scopes of the quantifying expressions. Such specifications constitute an important area of investigation for linguistic descriptions. Compare the approach of generative semantics (Lakoff 1971; Partee 1970) as well as the corresponding proposals of categorial grammar and Montague grammar, specifically Montague’s milestone essay of 1973, The proper treatment of quantification in ordinary English.’ (abbrev. PTQ)

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formal logic

1 In predicate logic, a frequently used synonym for operator in the narrower sense, that is, an umbrella term or synonym for the universal quantifier and the existential quantifier.

2 Linguistic term taken from predicate logic that designates operators that specify or quantify a set and are expressed in everyday language by indefinite adjectives or pronouns (all, some, several, and others), numerals (one, two, three, etc.), the definite article (The books are expensive), or indefinite plurals (Books are expensive). In transformational grammar quantifiers are derived from noun phrases in the deep structure, in generative semantics they are introduced as higher-order predicates. In Montague grammar quantifying phrases like all humans denote sets of properties such that a universal proposition like All humans are mortal can be analyzed as simple predication: ‘mortal’ is a property that belongs to the set of properties that apply to all humans. This analysis corresponds to the syntactic structure of natural-language sentences and presents an important example of the methodological principle of compositionality in grammar theory and semantics ( principle of compositionality). It is a point of departure for more recent research on the semantics of natural-language quantifiers (see Barwise and Cooper 1981; Benthem and Meulen 1985).

Bartsch, R. 1973. The semantics and syntax of number and numbers. In P.Kimball (ed), Syntax and semantics. New York. Vol. 2, 51–93.

Barwise, J. and R.Cooper, 1981. Generalized quantifiers and natural language. Ling&P 4. 159–219.

Lakoff, G. 1971. On generative semantics. In D.D. Steinberg and L.A.Jakobovits (eds), Semantics, Cambridge, MA. 232–96.

Van Benthem, J. and A.ter Meulen (eds) 1985. Generalized quantifiers in natural language. Dordrecht.

Van der Auwera, J. (ed.) 1980. Determiners. London.

Westerstähl, D. 1989. Quantifiers in formal and natural language. In D.Gabbay and F.Guenthner (eds), Handbook of philosophical logic. Dordrecht. Vol. 4, 1–131.

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