indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Could not include linguistics - contents
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 use for the semantic analysis of natural language determiners in the footsteps of Richard Montague revolutionized linguistics in the early 1970s.
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
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 $\mathfrak{m}>\mathfrak{n}\;$. Note that this uses the second variable as well, whereas $\forall$ or $Q_\alpha$ depend only on $card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})$.
Generalized quantifiers were introduced in
An early textbook account is in ch.13 of
Several chapters treat their model-theoretic role in
In the context of Martin-Löf type theory they are discussed by
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, 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
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