Quantification and Scope

Quantification studies the semantics of expressions like 'every', 'some', and 'most', and scope concerns how their relative ordering produces distinct interpretations.

Definition

Quantification is the semantics of quantificational determiners and noun phrases; scope is the relative semantic ordering of operators such as quantifiers, negation, and modals that determines a sentence's reading.

Scope

This topic covers the theory of generalized quantifiers, which treats determiner phrases as denoting relations between sets or sets of sets, and the formal properties (monotonicity, conservativity) that natural-language quantifiers exhibit. It also covers scope ambiguities, in which sentences with more than one quantifier (or a quantifier and negation) have multiple readings, and the syntactic and semantic mechanisms (such as quantifier raising) proposed to derive them.

Core questions

• What do quantificational determiners denote, and what semantic universals do they obey?
• How do sentences with multiple quantifiers come to have several readings?
• What mechanism derives scope relations from syntactic structure?
• Why are some logically possible scope readings unavailable?

Key concepts

• generalized quantifier
• conservativity
• monotonicity (upward / downward entailing)
• scope ambiguity
• quantifier raising
• logical form

Key theories

Generalized quantifier theory

Determiners denote relations between sets (or noun phrases denote sets of sets), allowing a uniform treatment of 'every', 'some', 'most', and 'few', and revealing universals such as conservativity.

Quantifier raising and logical form

Scope ambiguities are derived by covertly moving quantifier phrases to positions at a level of logical form, where their relative scope is read off the resulting structure.

History

The treatment of noun phrases as generalized quantifiers, anticipated in mathematical logic by Mostowski and Lindstrom, was applied to natural language in Barwise and Cooper's influential 1981 paper, which uncovered semantic universals such as conservativity. Work on scope, drawing on May's notion of quantifier raising, connected these semantic insights to syntactic theory and was consolidated in the Heim and Kratzer textbook.

Debates

Syntactic vs. type-shifting accounts of scope

Whether scope ambiguities are best derived by covert syntactic movement (quantifier raising) or by in-situ semantic mechanisms such as type-shifting and continuations.

Key figures

• Jon Barwise
• Robin Cooper
• Edward Keenan
• Irene Heim
• Angelika Kratzer

Seminal works

• barwisecooper1981
• heimkratzer1998

Frequently asked questions

What is a scope ambiguity?

It is a sentence with more than one reading depending on which operator takes wider scope, as in 'Every student read some book', which can mean each student read a possibly different book or that there is one book all students read.