#### Heim

##### Target phenomena

Heim's target phenomena is uses of pronouns and indefinites in sentences such as these:

1. A man walked in. He wore no socks.
2. Every man who owned a donkey beat it.
##### Syntactic assumptions

Heim assumes that both indefinite descriptions, e.g, 'a man', definite descriptions, e.g. 'the man', and pronouns are syntactically free variables. So the logical form of the above sentence (1) would be:

1. $$M(x) \& W(x). N(x).$$

Sentence (2) looks like this, as quantifiers are treated syntactically as binary operators.

1. $$\text{every}(M(x)\& D(y) \& O(x,y) , B(x,y))$$

Indefinites and definites have different features which are not semantically evaluated, but rather lead to presuppositions.

##### Semantics

As described before, we will treat context as sets of pairs of partial assignment functions and worlds. This makes Heim's (1982a) system closer to that which later developed in the dynamic semantics literature such as Groenendijk, Stokhof, and Veltman (1996).1

When new variables are used, the context needs to be altered to allow the assignment function to be defined for these varialbes. We will define the following operation for 'freeing' the varaibles $$x_1 \ldots x_n$$ in a context $$c$$:

$$c[x_1 \ldots x_n] = \{ \langle f, w \rangle : f' \in c,$$ $$\exists \langle o_1\ldots o_n \rangle \in D \times \ldots \times D$$ and $$f$$ is like $$f'$$ except for each $$i$$, $$1 \leq i \leq n, f(x_i) = o_n \}$$

We also define a binary relation of assignment function as follows:

$$f \geq f'$$ if $$f$$ agrees with $$f'$$ in the domain of $$f'$$ (but $$f'$$ may have a greater domain)

Our semantic rules are as follows:

$$c[R(x\ldots x_n)] = \{ \langle w.f \rangle \in c': R(f(x) \ldots f(x_n))$$ at $$w\}$$
where $$c'$$ = $$c[y_1 \ldots y_j]$$ where $$y_1 \ldots y_j$$ are the variables in $$x_1 \ldots x_n$$ that are not defined in $$c$$.

$$c[\phi \& \psi] = c[\phi][\psi]$$

$$c[\lnot \phi] = \{ \langle f,w \rangle \in c : \not \exists \langle f' ,w \rangle \in c[\phi]\text{ s.t. } f' \geq f \}$$

$$c[\text{every}(\phi, \psi)] = \{ \langle f,w \rangle \in c :\text{ for every least f' with respect to}\geq,$$ s.t. $$f' \geq f, \langle f',w \rangle \in c[\phi]$$ there is some $$f'' \geq f', \langle f'', w \rangle \in c[\phi][\psi] \}$$

#### Dynamic Predicate Logic

##### Compositionality

Groenendijk and Stokhof present their dynamic predicate logic as the first compositional dynamic semantics. They write that no previous semantics in this vein (including Heim’s FCS) “makes compositionally its starting point”, and then they qualify this with the parenthetic remark that “it seems that Heim [1982, Ch.3], does attach some value to compositionality.” This is uncharitable, as Heim’s semantic system is clearly and straightforwardly compositional. One would think would think given the importance they accord to compositionality they would explain what lack of compositionality they found in Heim’s work. In fact, Groenendijk and Stokhof do not discuss Heim’s semantics at all, saying that they “feel justified in restricting comparison to just [Kamp’s DRT ]” since it is the most “formally explicit” theory in this area. As any careful reader of Heim ((1982b), Ch 3, (1983)) is aware, the basic semantics there is precise and explicit. The suggestion that Heim’s work need not be discussed in this context due to any lack of formal rigor is bizarre. The truth is quite simple: Heim gave a compositional dynamic semantics in her 1982 dissertation (though the term dynamic semantics was not yet in currency) and, seven years later, Groenendijk and Stokhof published a different compositional, dynamic semantics. The interest and importance of their work is clear: it does not, however, rest in its claim to being the first compositional dynamic semantics.2

##### Syntax

The syntax of DPL is simply that of first-order logic. The main difference from Heim (besides the absence of any two place quantifiers) is thus the presence of existential quantifier $$\exists x$$. In DPL then something like this:

1. A man walked in. He ordered a drink.

is understood as having the following logical form:

1. $$\exists x (Mx \& Wx). Dx$$
##### Semantics

Here is DPL, we will relativize to a world $$w$$ to make clear the relationship between DPL and FCS:

$${ {[\hspace{-.02in}[}{P(x)}{]\hspace{-.02in}]}}^w = \{ {\langle g, g \rangle} : g(x)$$ is a $$P$$ in $$w \}$$
$${ {[\hspace{-.02in}[}{\lnot \phi}{]\hspace{-.02in}]}}^w = \{ {\langle g, g \rangle}: \not \exists h : {\langle g,h \rangle} \in { {[\hspace{-.02in}[}{\phi}{]\hspace{-.02in}]}}^w \}$$
$${ {[\hspace{-.02in}[}{\exists x \phi}{]\hspace{-.02in}]}}^w = \{ {\langle g, h \rangle}: \exists k: k[x]g \ \& \ {\langle k,h \rangle} \in { {[\hspace{-.02in}[}{\phi}{]\hspace{-.02in}]}}^w \}$$
$${ {[\hspace{-.02in}[}{\phi \land \psi}{]\hspace{-.02in}]}}^w = \{ {\langle g,h \rangle} : \exists k : {\langle g,k \rangle} \in { {[\hspace{-.02in}[}{\phi}{]\hspace{-.02in}]}}^w \ \& \ {\langle k,h \rangle} \in { {[\hspace{-.02in}[}{\psi}{]\hspace{-.02in}]}}^w\}$$

Let us define DPL-contexts as sets of assignment worlds (not allowing partiality).
The empty context here could be defined as follows (though we can consider other options as well): $$c_\emptyset = \{ {\langle g,w \rangle} : g$$ is an assignment function and $$w \in W\}$$. We can then define DPL-update as follows. For a DPL-context $$c$$ and a formula in DPL $$\phi$$:

$$c[\phi] = \{ {\langle g,w \rangle} : \exists {\langle h,w \rangle} \in c \ \& \ {\langle h,g \rangle} \in { {[\hspace{-.02in}[}{\phi}{]\hspace{-.02in}]}}^w \}$$

##### Bibliography

Dekker, Paul. 1996. “The Values of Variables in Dynamic Semantics.” Linguistics and Philosophy 19: 211–57.

Geurts, Bart, and David I. Beaver. 2011. “Discourse Representation Theory.” In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, Fall 2011.

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

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

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

———. 1983. “File Change Semantics and the Familiarity Theory of Definiteness.” In Meaning, Use, and Interpretation of Language, edited by Christoph Schwarze Rainer Bäuerle and Arnim von Stechow. Walter de Gruyter.

Jäger, Gerhard. 1996. “Topics in Dynamic Semantics.” PhD thesis, Humbolt University Berlin.

Yalcin, Seth. 2012. “Dynamic Semantics.” In Routledge Companion to Philosophy of Language, edited by D. Fara and G Russell. Routledge.

1. See also Yalcin (2012) for a similar presentation of Heim, but using her representations for files.

2. I emphasize this point since some later commentators have followed Groenendijk and Stokhof in suggesting Heim’s account is somehow not overtly compositional: Geurts and Beaver (2011) label Heim’s semantics as “intended to be compositional” with no further explanation, Dekker (1996), while noting that Heim anticipates Groenendijk and Stokhof, describes her as only doing so “implicitly”, saying that the compositional presentation is “wrapped up in the representational metaphor of changing files and file cards”—as if presenting a helpful metaphor obscures the technical account. Jäger (1996) makes some interesting, and accurate, comments about how Heim’s semantics lacks certain forms of compositionality, but these relate to her felicity conditions on definiteness and the particular level of syntactic representation she works at, and so seem orthogonal to any concerns Groenendijk and Stokhof might have had.