Quantifiers are linguistic expressions that specify or quantify a set. In everyday language, they are expressed by indefinite adjectives or pronouns (e.g. all, some, several), numerals (e.g. one, two, three), the definite and indefinite article (the, a/an) or indefinite plurals (books).

In formal logic, quantification specifies for how many objects in a certain set a predicate are valid. Quantification is determined by the quantifier used, which connects freely occuring variables in a sentence. Thus, quantifiers are analysed as expressing a relation between two sets: A sentence like Most dogs are domestic can be interpreted in terms of the intersection of the set of dogs with the set of domestic things is greater than the remainder of the set of dogs.

If the predicate in question is valid for at least one object in the given set (e.g. some, a/an), the quantifier is referred to as an existential quantifier. This contrasts to the universal quantifier, through which the predicate in question is assigned to all elements of the underlying set of individuals (e.g. every, each, all).

A further distinction is made between propositional quantifiers (e.g. all, most, few), which express asymmetric relations, and cardinal quantifiers (e.g. no, a/an, some, many, several, few, a few), which refer to a symmetric intersection of two sets. Thus, the sentence Most leaves are green is not equivalent to the sentence Most green things are leaves (propositional quantifier). In contrast, we can say that the two sentences No rose is black and No black thing is a rose are equivalent (cardinal quantifier).

a) Existential vs. universal quantifiers

Type of quantifierDescriptionExample
existentialthe predicate is valid for at least one object in the setsome, a/an
universalthe predicate is assigned to all elements of the setevery, each, all

b) Propositional vs. cardinal quantifiers

Type of quantifierDescriptionExample
propositionalan asymmetric relation is expressedall, most, few
cardinala symmetric relation is expressedno, a(n), some, many, several, (a) few

The examples are taken from Kearns, D. (2000). Semantics. Basingstoke: Macmillan. Chap. 4

Exercises on quantification

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