• Dynamics of Conversation
• Notes

  Dynamic Contexts Heim and Dynamic Predicate Logic Veltman's Epistemic Modals Veltman/Yalcin comparison Generalized Quantifiers Quantifiers and Modals

• Quantifiers and Modals
• Daniel Rothschild

Introduction

This section introduces a fragment that combines the treatment of anaphora in Heim and DPL with Veltman's treatment of epistemic modality. The semantics given is a simplified version of Groenendijk, Stokhof, and Veltman (1996).¹

GSV's epistemic modal

We take our notion of file as before and define a version of Veltman's epistemic modal with this notion. Here is the most straightforward way of doing so, due to Groenendijk, Stokhof, and Veltman (1996):

\(c[\Diamond \phi] = \{ { \langle{f,w}\rangle} \in c : c[\phi] \neq \emptyset \}\)

On this definition we simply test to see if \(c\) survives the update with \(\phi\), if so \(c\) remains unchanged, otherwise we get the emptyset.

Modals and quantifiers

Note first that the two definitions for existential quantifiers canvassed here lead to different results with epistemic modals. The first definition calculates \(c[\exists x \phi]\) by first creating a new context with \(x\) free and then applying \(\phi\). Call this lumping:

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

The second definition, takes each assignment of \(x\) in \(c\) and applies \(\phi\) to that new context and then unions the result:

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

Call that slicing (following Aloni 2001).

Consider a file, \(d\), in which there 2 individuals \(a\) and \(b\), and one world where \(a\) is an \(F\) and \(b\) is not an \(F\). Note that \(d[\exists_l x Fx] = d\) while \(\exists_s Fx = \emptyset\).

GSV's boy hiding case:

GSV give this nice (if subtle) example which supports the latter treatment of exisetntial quantification in combination with their epistemic modal:

Imagine the following situation. You and your spouse have three sons. One of them broke a vase. Your spouse is very anxious to find out who did it. Both you and your spouse know that your eldest didn’t do it, he was playing outside when it must have happened. Actually, you are not interested in the question who broke the vase. But you are looking for your eldest son to help you do the dishes. He might be hiding somewhere. In search for the culprit, your spouse has gone upstairs. Suppose your spouse hears a noise coming from the closet. If it is the shuffling of feet, your spouse will know that someone is hiding in there, but will not be able to exclude any of your three sons. In that case your spouse could utter: 
(16) There is someone hiding in the closet. He might be guilty.
\(\exists Qx \land \Diamond Px\)  
But the information state of your spouse would not support:
(17) There is someone hiding in the closet who might be guilty.
\(\exists x (Qx \land \Diamond Px)\)  
If the situation is slightly changed, and it is imagined that the noise your spouse hears is a high-pitched voice, things are different. Now, your spouse knows it can not be your eldest, he already has a frog in his throat. In that case your spouse can say (17).
This also means that if your spouse yells (17) from upstairs, you can stay were you are, but if it is (16), you might run upstairs to check whether it is perhaps your aid that is hiding there.

Contradictions

Yalcin (2015) notes that sentences like this sound contradictory:

1. An infected man might not be infected.

This is easily predicted with various differen logical forms. One example is \(\exists x Ix \& \Diamond \lnot Ix\).

Similarly if we treat the generalized quantifiers as suggested we can derive the deviance of the following sentences:

2. Most people who are infected might not be.

However we do derive that the following sentence is fine:

3. Someone who might be infected is not infected.

Bibliography