##### Propositional common grounds

What formal object should we use to represent contexts in semantics (whether dynamic or not). In the propositional case, following Stalnaker (1974; 1978), we can treat the context as a set of propositions in the common ground, i.e. those that are mutually accepted for the purposes of conversation. When modelling contexts, however, we do not always keep track individually of every proposition commonly accepted. Rather we often use the context set, instead. The context set is just the set of all worlds that are compatible with every proposition in the common ground. If propositions themselves are identified with sets of worlds (the worlds they are true at) then the context set is just the intersection of all the worlds in the common ground.

Given this, propositional, worldly, notion of context we can easily define the update assocated with an assertion as follows (again following Stalnaker). If in a context $$c$$ a sentence $$s$$ is asserted and $$s$$ expresses the proposition $$p$$, then the new common ground after $$s$$ has been accepted is just the old common ground $$c$$ intersected with $$p$$, $$c\cap p$$. This is a static notion of conversational dynamics. That is we assume a static semantics in which sentences are assocated with propositions.

Once the context is adjusted to accommodate the information that the particular utterance has produced, how does the CONTENT of an assertion alter the context? My suggestion is a very simple one: To make an assertion is to reduce the context set in a particular way, provided that there are no objections from the other participants in the conversation. The particular way in which the context set is reduced is that all of the possible situations incompatible with what is said are eliminated. To put it a slightly different way, the essential effect of an assertion is to change the presuppositions of the participants in the conversation by adding the content of what is asserted to what is presupposed. This effect is avoided only if the assertion is rejected. (Stalnaker 1978, 323)

##### Contexts with file structure

Heim (1982) enriches Stalnaker's notion of content with information about what she calls, following Karttunen (1976), discourse referents. The file metaphor suggest a system in which we keep dossiers of information about a series of unspecified individuals. Heim differentiates dossiers by number, but we will rather assume that each dossier is associated with an individual. A sample dossier, then, might be something like this:

Individuals $$x$$ and $$y$$: $$x$$ is a man, $$y$$ is a dog, $$x$$ owns $$y$$.

What sort of formal object should we use to keep track of such information? This is actually a somewhat complex question to ask as it is not clear what kind of information is in here. If we just want to keep track of the information that some individuals $$x$$ and $$y$$ are such that $$x$$ is a man $$y$$ is dog, and $$x$$ owns $$y$$, then we might just take the sets of worlds where that is so:

$$P = \{ w: \exists x \exists y:$$ $$x$$ is a man, $$y$$ is a dog and $$x$$ owns $$y$$ in $$w \}$$

This representation, however, loses the structure of the file system it represents. Heim's notion is more fine-grained than this, as she wants to keep information about individual discourse referents. Her way of doing this is to equate contexts with what she calls satisfaction sets. A satisfaction set is set of ordered pairs the first member of which is an assignment function (a function from the set of variables $$V$$ to the domain $$D$$) and the second member of which is a world $$W$$.1 The satisfaction set corresponding to the file described above is as follows:

$$S = \{ \langle f,w \rangle : f(x)$$ is a man, $$f(y)$$ is a dog and $$x$$ owns $$y$$ in $$w \}$$

Satisfaction sets encode both informationa about worlds and discourse referents at once. For we can extract a propositional context set $$P$$ from a satisfaction set $$S$$, by simply applying the following function $$E$$ to $$S$$:

$$E(S) = \{w : \exists f : \langle f,w \rangle \in S\}$$

It should be clear that $$E$$ maps many different satisfaction sets to the same propositional context set. Here are two examples of files that map to $$P$$.

$$S_1 = \{ \langle f,w \rangle : f(y)$$ is a man, $$f(x)$$ is a dog and $$y$$ owns $$x$$ in $$w \}$$
$$S_2 = \{ \langle f,w \rangle : f(x)$$ is a man, $$f(y)$$ is a dog and $$x$$ owns $$y$$, $$f(z)$$ is a man in $$w \}$$

Satisfaction sets, as fine-grained as they are, do not themselves carry sufficient information for some purposes. Consider, first, the most uninformative satifaction set, $$T$$, that simply includes all worlds and assignment functions:

$$T = \{ \langle f,w \rangle : f$$ is an assignment function and $$w$$ is a world$$\}$$

$$T$$ is also equivalent to this file:

$$\{ \langle f,w \rangle : f(x) = f(x) \}$$

But we might want to be able to talk of a file in which a discourse referent $$x$$ is introduced even if the file contains no non-trivial information about $$x$$. To do this Heim proposes to identify files with pairs of satisfaction sets and domains where domains are simply the set of variables that are active in the file.

##### Our representation: sets of pairs of partial assignment functions and worlds

A more perspicuous representation of files is as sets of pairs of partial assignment functions and worlds. A file is a set $$F$$ such that each element of $$F$$ is an ordered pair $$\langle f, w \rangle$$ where and for any two elements $$x$$ and $$y$$ the assignment functions associated with $$f$$ and $$y$$ have the same domain (i.e. they are defined for the same variables, so have the same degree of partiality).

Now we treat the uniformative file, $$\top$$, as follows, where $$\emptyset$$ is understood to be the assignment function that is completely undefined.

$$\top = \{ \langle \emptyset, w \rangle : w$$ is a world $$\}$$

On the other hand the file corresponding to our example would be as follows:

$$S = \{\langle f,w \rangle :$$ the domain of $$f$$ is $$\{ x,y\},$$ $$f(x)$$ is a man, $$f(y)$$ is a dog and $$x$$ owns $$y$$ in $$w \}$$

One extra useful piece of notation: $$f \geq f'$$ if $$f$$ is the same as $$f'(x) = f(x)$$ for any $$x$$ in the domain of $$f'$$ (but $$f$$ may have a larger domain).\$

##### Freeness and translating between Heim's files and the system here

For Heim a file is a set of pairs of total assignment functions and worlds (a satisfaction set), along with a set of variables, the domain. The file must also meet the constraint that every variable in the domain is free in the satisfaction set. A variable $$x$$ is free in the satisfactions set $$S$$ iff for all objects $$o$$ in the domain for each $$\langle f,w \rangle$$ in $$S$$, $$\langle f_{x\to o} w\rangle$$ is in $$S$$. For a file $$c$$ we call the domain of $$c$$ the set of variables for which each assignment function in $$c$$ is defined.

The 1-1 mapping from Heim's files to ours is as follows: for any Heimian file with satisfaction set $$S$$ and domain $$V$$, the file corresponding to it is just the $$\{ \langle f,w \rangle :$$ there exists $$\langle f',w \rangle \in S$$ s.t. $$f$$ is undefined for each variable not in $$D$$, and for all $$x$$ in $$D$$, $$f(x) = f'(x)\}$$.

##### Bibliography

Heim, Irene. 1982. “The Semantics of Definite and Indefinite Noun Phrases.” PhD thesis, Amherst: University of Massachusetts. http://semanticsarchive.net/Archive/Tk0ZmYyY/.

Karttunen, Lauri. 1976. “Discourse Referents.” In Syntax and Semantics, edited by J. McCawley. Vol. 7. Academic Press.

Stalnaker, Robert. 1974. “Pragmatic Presuppositions.” In Semantics and Philosophy, edited by M. K. Munitz and D. K. Unger, 197–213. NYU.

———. 1978. “Assertion.” In Syntax and Semantics, edited by P. Cole. Academic Press.

1. We assume a constant domain across worlds.