nLab generalized quantifier

Idea

Generalized quantifiers were introduced into model theory by the Polish logician A. Mostowski in order to supplement the usual existential and universal quantifiers of first-order predicate logic with the aim to circumvent shortcomings concerning expressivity and categoricity.

Their subsequent use for the semantic analysis of natural language determiners in the footsteps of Richard Montague revolutionized linguistics in the early 1970s.

Definition

Definition

A generalized quantifier or an interpretation of a quantifier symbol $Q$ is a mapping $\mu_Q$ from triples of cardinal numbers $\langle \mathfrak{m},\mathfrak{n},\mathfrak{p}\rangle$ such that $\mathfrak{m}+\mathfrak{n}=\mathfrak{p}$ to $\Omega=\{0,1\}$. The satisfaction relation for $Q$ is defined for structures $\mathfrak{A}$ with carrier $A$ and valuation $x$ by

$$\begin{aligned} \mathfrak{A}\models_x(Qv_n)\phi\quad&{iff}\quad \mu_Q(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1\;,\; with \\ \mathfrak{m}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\}) \\ \mathfrak{n}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\not\phi\}) \\ \mathfrak{p}&=card(A)\;. \end{aligned}$$

Examples

• The universal quantifier $\forall$ has $\mu_\forall(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{n}=0$.

• The quantifier $Q_\alpha$ “there exist at least $\aleph_\alpha$” is given by $\mu_{Q_\alpha}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}\geq\aleph_{\alpha}\;$.

• The quantifier $\mathsf{W}$ “most” is given by $\mu_{\mathsf{W}}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}\gt\mathfrak{n}\;$. Note that this uses the second variable as well, whereas $\forall$ or $Q_\alpha$ depend only on $card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})$.

Related entries

• counting quantifier

• model theory

• second-order logic

• cardinal number

• Lawvere distribution

• Richard Montague

References

Generalized quantifiers were introduced in

• A. Mostowski, On a Generalization of Quantifiers , Fund. Math. 44 (1957). (pdf)

An early textbook account is in ch.13 of

• J. L. Bell, A. B. Slomson, Models and Ultraproducts: An Introduction , North-Holland Amsterdam 1969. (Dover reprint)

Several chapters treat their model-theoretic role in

• J. Barwise, S. Feferman (eds.), Model-theoretic Logics , Springer Heidelberg 1985 (freely available online: toc) .

In the context of Martin-Löf type theory they are discussed by

• G. Sundholm, Constructive Generalized Quantifiers , Synthese 79 no.1 (1989) pp.1-12.

For the use of generalized quantifiers in natural language semantics see

• J. Barwise, R. Cooper, Generalized Quantifiers and Natural Language , Linguistics and Philosophy 4 no.2 (1981) pp.159-219.

• E. Keenan, D. Paperno (eds.), Handbook of Quantifiers in Natural Language , Springer 2012.

• E. Keenan, D. Paperno (eds.), Handbook of Quantifiers in Natural Language: vol.2 , Springer 2017.

• E. Keenan, J. Stavi, A semantic characterization of natural language determiners , Linguistics & Philosophy 9 (1986) pp.253–326.

• D. Westerståhl, Generalized quantifiers: linguistics meets model theory , in Aloni, Dekker (eds.), The Cambridge Handbook of Semantics , Cambridge UP 2014. (draft)

Some problems with the standard approach in linguistics are discussed in

• H. Ben-Yami, Generalized Quantifiers, and Beyond , Logique Et Analyse no.208 (2009) pp.309-326.

• M. Hackl, On the Grammar and Processing of Proportional Quantifiers: Most Versus More Than Half , Natural Language Semantics 17 no.1 (2009) pp.63-98.

A dependent type theoretic analysis of natural language generalized quantifiers is in

• T. Fernando, Conservative generalized quantifiers and presupposition , Semantics and Linguistic Theory XI (2001) pp.172–91. (draft)

They are modelled in compact closed categories in

• J. Hedges, M. Sadrzadeh, A Generalised Quantifier Theory of Natural Language in Categorical Compositional Distributional Semantics with Bialgebras , Math. Struc. Theor. Comp. Sci. 2019. (link)