_{Semantics}

**Quantification**

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 quantifier | Description | Example |
---|---|---|

existential | the predicate is valid for at least one object in the set | some, a/an |

universal | the predicate is assigned to all elements of the set | every, each, all |

**b) Propositional vs. cardinal quantifiers**

Type of quantifier | Description | Example |
---|---|---|

propositional | an asymmetric relation is expressed | all, most, few |

cardinal | a symmetric relation is expressed | no, a(n), some, many, several, (a) few |

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

**Exercises on quantification**

- Source of the picture: plato.stanford.edu