Generalized Quantifiers

Introduction

This section introduces a simple way of adding generalized quantifiers to dynamic semantics.

Semantics without generalized quantifiers

We use the same notion of context as before. Here are our semantics for the fragment with the syntax of predicate logic, based loosely on Heim and DPL

$c[R(x_1 \ldots x_n)] = \begin{cases} \# \text{ if any of $x_1 \ldots x_n$ not in domain of $c$}\\ \{ { \langle{f,w}\rangle} \in c : \langle f(x_1),\ldots , f(x_n) \rangle \in R_w \} \text{ otherwise} \end{cases}$

$c [\exists x \phi] = \ldots$
$\begin{cases} \# \text{ if $x$ in domain of $c$}\\ \{ { \langle{f',w}\rangle} \in c : \exists { \langle{f,w}\rangle} \in c \exists o, f'(x) = o \text{ and } f[x]f'\}[\phi] \text{ otherwise} \end{cases}$

$c[\lnot \phi] = \{ { \langle{f,w}\rangle} \in c : \not \exists { \langle{f',w}\rangle} \in c[\phi] : {\langle{f',w}\rangle} \geq { \langle{f,w}\rangle} \}$ $c[\phi \land \psi] = c[\phi][\psi]$

Alternative existential quantification (Groenendijk, Stokhof, and Veltman 1996; Beaver 1994):

$c [\exists x \phi] = \begin{cases} \# \text{ if any of $x$ in domain of $c$}\\ \bigcup_{o \in D} (\{ { \langle{f,w}\rangle} : \exists { \langle{f',w}\rangle} \in c, f(x) = o \text{ and } f[x]f'\}[\phi]) \end{cases}$

Adding generalized quantifiers

Assuming we associated with each quantifier $Q$ a binary relation on sets, $B_Q$. Relative to a restrictor $\phi$, and matrix, $\psi$, a context $c$, an assignment-world ${ \langle{f,w}\rangle}$ and a variable $x$ we'll define the restrictor set $R(\phi,c,f,w, x)$ and matrix set $M(\phi,\psi, c,f,w, x)$:

$R(\phi,c,f,w, x) = \{o : { \langle{f,w}\rangle}$ survives in $c_{x\to o}[\phi] \}$
$M(\phi, \psi,c,f,w, x) = \{o : { \langle{f,w}\rangle}$ survives in $c_{x\to o}[\phi][\psi] \}$

Now we can give a general definition of dynamic quantifiers as follows:¹

$c[Q_x(\phi, \psi)] = \{ { \langle{f,w}\rangle} \in c : R(\phi,c,f,w, x) B_Q M(\phi,c,f,w, x) \}$

Bibliography

Beaver, David. 1994. “When Variables Don’t Vary Enough.” Semantics and Linguistic Theory (SALT) 4. https://webspace.utexas.edu/dib97/salt4.pdf.

———. 2001. Presupposition and Assertion in Dynamic Semantics. CSLI. https://webspace.utexas.edu/dib97/silli.pdf.

Groenendijk, Jeroen, Martin Stokhof, and Frank Veltman. 1996. “Corefrence and Modality.” In Handbook of Contemporary Semantic Theory, edited by Shalom Lappin. Blackwell.

Nouwen, Rick. 2003. Plural Pronominal Anaphora in Context. Netherlands Graduate School of Linguistics Dissertations 84. LOT.

This ignores their dynamic effects see Beaver (2001) and, especially, Nouwen (2003) for appropriate expansions.↩