\(\exists x[\phi]\) in DPL has two parts in its semantics:
\(g[\![ \exists x [\phi] ]\!]g'\) iff for some \(f\), \(g[x]f\) and \(f[\![ \phi ]\!]g'\)
We could actually change the syntax a little bit and define \(\exists x\) as a formula.
\(g[\![ \exists x ]\!]g'\) iff for some \(f\), \(g[x]f\)
The operation encoded in \(\exists x\) is called random assignment.
The semantics of DPL is relational, rather than functional, and a given input assignment \(g\) can have multiple output assignments, e.g.
There is a linguist in Göttingen.
\(\exists x[Lx \land Gx]\)
Suppose there are exactly three linguists in Göttingen, \(a\), \(b\), and \(c\). Then for any input assignment \(g\), the output can be any of the following three: \[ g[x\mapsto a] \quad g[x\mapsto b] \quad g[x\mapsto c] \] Notation: \(g[\xi\mapsto e]\) is the assignment that differs from \(g\) at most in that \(g[\xi\mapsto e](\xi) = e\).
In fact, random assignment is what’s responsible for the indeterminism. When there’s no \(\exists x\) in the formula, the output can be determined uniquely. Specifically, either the output is the same as the input, or there’s no output.
Exercise: Prove this. You can also prove a stronger version of this using the notion of active quantifier occurrences defined in Groenendijk and Stokhof (1991).
It’s sometimes convenient to talk about meaning more deterministically. We could do so by defining functions over sets of assignments, based on the relational semantics of DPL.
Let’s call this transformation lifting.
New notation. Let \(G\) be a set of assignments.
\(G[\phi] = \{g' \,|\, \text{for some }g \in G, g[\![\phi]\!]g'\}\)
(Another common way of writing this is \(G+\phi\) but we’ll use the more compact notation.)
Using the above general recipe, we can re-write the semantics as follows.
Note that as before, \(\exists \xi\) can be seen as a formula on its own, \(G[\exists \xi] = G[\xi]\).
In this lifted version of DPL, the output set of assignments is uniquely determined.
The meaning of each formula determines a way to update \(G\).
We can call a system like this an update semantics.
This particular update semantic system has the property of distributivity: \[G[\phi] = \bigcup_{g\in G} \{g\}[\phi]\]
If we are only interested in anaphora, a conversational context can be modelled as a set of assignments, \(G\).
\(G\) represents different ways of assigning values to discourse referents, e.g.
Let’s write \(G(x)\) for \(\{e |\) for some \(g\in G, g(x) = e\}\). \(G(x)\) is the possible values of \(x\) with respect to \(G\), and represents what kind of information you have about the discourse referent \(x\) in the context represented by \(G\).
An utterance of a sentence updates \(G\).
Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof 1991) is a dynamic version of Predicate Logic (PL).
We can also construct a dynamic version of Propositional Logic (PropL).
Let’s momentarily forget about anaphora. There is another aspect of conversational context that is important, the assumptions/knowledge commonly shared by the conversational participants.
We can think of the shared assumptions in terms of a set of possible worlds, often called the context set (after Stalnaker).
Robert Stalnaker has developed a theory of pragmatics based on the context set in a series of works. See Stalnaker (1978), and other papers reprinted in Stalnaker (1999). See also Stalnaker (1998), Stalnaker (2002), Stalnaker (2004), Stalnaker (2014).
Stalnaker defines the context set in terms of the conversational agent’s mutual beliefs.
The iterative part of the definition of mutual beliefs is important. That I know \(p\) and you now \(p\) doesn’t necessarily mean that \(p\) is a mutual belief, e.g.:
In Stalnakerian pragmatics, an assertion of a sentence that denotes a proposition \(p\) is seen as a proposal to update the context set. And the update amounts to:
\(C[p] = \{w \in C | w\in p\} = C \cap p\)
This framework has been employed to account for a number of linguistic phenomena. In particular:
Many extensions of this framework have also been proposed.
The syntax of DPropL is the same as the syntax of PropL. The language \(\mathscr{L}\) is recursively defined as follows.
A model for PropL is an interpretation function that assigns a truth-value to each atomic proposition.
We can see such a model as a possible world, so let’s write \(w\) for a model. (Note: in proper intensional semantics, possible worlds are part of the model and you have operators on possible worlds; the system here is extensional).
Let’s first define relational semantics.
DPropL seems underwhelming. The language contains no operator that changes the input possible world, so whenever \(w[\![ \phi ]\!]w'\), \(w = w'\). (Note: In DPL \(\exists x\) is the operator that manipulates input; were it not for it, its semantics would be similarly boring)
It becomes a little more interesting when we lift it so that formulae operate on sets of possible worlds. The strategy is the same as before: We define update rules, in terms of the relational semantics according to the following recipe.
\(C[\phi] = \{w' \,|\,\) for some \(w\in C, w[\![\phi]\!]w'\} = \{w\in C \,|\, w[\![\phi]\!]w\}\)
To put it differently, \([\![\phi]\!]\) characterises the set of possible worlds in which \(\phi\) is true: \(\{w \,|\, w[\![\phi]\!]w\}\). The update process returns its intersection with \(C\).
Equivalently, the update rules can be stated as follows.
Remarks (Rothschild and Yalcin 2016)
This update semantics is distributive in the same sense as before, i.e. \(C[\phi] = \bigcup_{w\in C}\{w\}[\phi]\)
It is also eliminative, i.e. for any formula \(\phi\in\mathscr{L}\), \(C[\phi]\subseteq C\)
One of the first dynamic semantic theories for natural language was developed by Heim (1982). She called it File Change Semantics (FCS).
In FCS, we operate on possible worlds and an assignment at the same time. In the unlifted version, a formula denotes a relation between world-assignment pairs, \(\langle w, g\rangle\).
Heim (1982) gives a direct translation of English, but for now, let’s stick to the formal language for the sake of clarity. We’ll develope a fragment of English in Lecture 3.
The language is the same as (D)PL, \(\mathcal{L}\). The relational semantics looks as follows.
If we see \(\exists x\) as a formula: \(\langle w, g\rangle [\![\exists x ]\!]\langle w', g'\rangle\) iff \(w = w'\) and \(g[x]g'\)
This system is still extensional, so whenever \(\langle w,g\rangle[\![\phi]\!]\langle w', g'\rangle\), \(w = w'\), and there’s no operator that changes it in the compositional interpretation. If you decide to include an intensional operator (not done here), this will not be the case, of course.
Let’s lift this system using the same recipe as before. We write \(c\) for a set of world-assignment pairs.
\(c[\phi] = \{\langle w', g'\rangle \,|\,\)for some \(\langle w, g\rangle\in c, \langle w, g\rangle[\![\phi]\!]\langle w', g'\rangle\}\)
The update rules will just follow from this, but let’s restate them in a more understanding way. We write \(c[\xi]\) for \(\{\langle w, g'\rangle \,|\,\) for some \(\langle w,g\rangle\in c, g[\xi]g'\}\).
Some examples:
A cat is sleeping.
A cat is sleeping. It is black.
Every farmer is happy.
Every farmer who owns a donkey beats it.
If a farmer owns a donkey, he beats it.
We’ve seen some versions of update semantics, but they are all of the following form:
\(\{p_1, \dots, p_n\}[\phi] = \{p'_1, \dots, p'_m\}\)
Lifted DPL: \(p_i\) is an assignment \(g\)
Lifted DPropL: \(p_i\) is a possible world \(w\)
FCS: \(p_i\) is a world-assignment pair\(\langle w, g\rangle\)
(Propositional) inquisitive semantics (Ciardelli, Groenendijk, and Roelofsen 2013, 2018): a possibility is a set of possible worlds.
Depending on which aspect of context you are trying to analyse, you choose your possibilities and information states (Nouwen et al. 2016).
Our version of is very close to the version of FCS given in Heim (1982), but there are some differences.
One of the main proposals of Heim (1982) is that indefinites are not quantifiers, but simply variables. The theory we have developed is similar in this respect, but not completely the same as hers.
There is a donkey.
Heim’s underlying intuition was that indefinites and definites in natural language are actually the same thing, except that they have indefinites and definites have different felicity conditions on their usage.
Our formal language has no ‘definites,’ but ultimately we want to have a formal semantic system for natural languages.
Heim’s felicity conditions on indefinites and definites:
However, Heim decided to develop a different theory later on, in part because this analysis of definites has some issues. One of them is that definites can contain bound variables. But if the definite is interpreted as a single discourse referent that is not bound itself, it should have a fixed denotation, e.g. the following sentence should presuppose that there’s a unique paper written by every student.
Every student sent me the paper they wrote last semester.
Heim also later proposed to drop the Novelty Condition for indefinites, in favor of so-called Maximize Presupposition! (Heim 1991, 2011). But she never fully worked out this view, as far as I know. To do so, one would need a reasonably good theory of definites, which we don’t really have (but see Köpping 2018, 2019).
Another issue of FCS is so-called unselective binding.
Fortunately, solutions to this issue were developed in the 1990s (Kanazawa 1993, 1994; Chierchia 1995; Berg 1996; Nouwen 2003; Brasoveanu 2007). Essentially, these authors selective quantifiers that quantify over individuals, rather than assignments, to the system.
It is agreed (with very few exceptions) that natural language conditionals cannot be analysed in terms of material implication \(\to\) (as Heim 1982 remarks), and a proper analysis of conditionals requires intensional semantics. This literature is huge.
But the mechanism of unselective binding might not be too bad for conditional donkeys after all. Unlike quantificational donkeys, conditional donkeys don’t run into the Proportion Problem.
We won’t be able to talk about a serious analysis of conditionals in this course, but we can hopefully discuss this point later.
Disjunction is a problem. Heim (1982) actually doesn’t really talk about disjunction herself. The above meaning of \(\lor\) is based on DPL as presented in Groenendijk and Stokhof (1991).
As mentioned in Lecture 1, there are some recalcitrant issues with disjunction.
As in DPL, \(\lor\) in our version of FCS is both internally and externally static, so neither of these examples is accounted for.
The first type of examples is called Stone disjunction after Stone (1992; but it was already discussed in Groenendijk and Stokhof 1991).
The second example is due to Barbara Partee and is called bathroom disjunction.
To account for Stone disjunction, we could have an externally dynamic disjunction.
\(c[(\phi\dot{\lor}\psi)] = c[\phi]\cup c[\psi]\)
However, this gives rise to another issue.
Giorgos has a cat, or he has no pets. #
\((\exists x[Cx \land Hgx] \lor \neg \exists y[Py \land Hgy]) \land [Wx]\)
This anaphora is wrongly predicted to be fine.
You might think that the infelicity comes from some other conditions on the use of it. It normally cannot refer to a human (or a plurality of entities), and if some values of \(x\) are non-humans (and/or pluralities), the infelicity is expected.
But the above sentence is infelicitous even with anaphora that doesn’t have such conditions, e.g.
A related issue is that if assignments are total, that means that all discourse referents are already there from the beginning. The vast majority of them carry no information, but total assignments are defined for all of them.
For a logical language, this is not an issue, but for natural language, a discourse referent needs to be introduced somehow (linguistically or not), before it can be anaphorically referred to.
For this reason, the theory in Heim (1982) has a way to keep track of a set of discourse referents that have been mentioned.
Heim (1982) mentions an argument of the following sort for this, but it’s perhaps not very cogent.
Exercise: Discuss this argument, and possible ways to improve it.
Another way to deal with this issue is by introducing partiality.
Yet another way of implementing this idea is by making use of a dummy individual (Rothschild 2017).
Partiality makes everything formally very complex (see e.g., Berg 1996; Krahmer 1998; Nouwen 2003). But partiality could potentially solve the above issue of externally dynamic disjunction by assuming that the second disjunct does not fully resolve undefinedness.
Exercise: Formulate a constraint on partial assignments that explains the above data with disjunction (There are probably several potential ways).
Our fragment for English will have partiality.
The bathroom anaphora is still an issue.
Related to it is the issue of double negation.
It’s not true that Flora doesn’t have a cat. I’ve seen him.
These issues have been recently attracting attention in the literature (again), e.g. Rothschild (2017), Gotham (2019), Hofmann (2019), as well as Patrick D. Elliott’s and Matt Mandelkern’s recent work.
Another well-known issue is quantificational subordination (Berg 1996; Nouwen 2003, 2007; Brasoveanu 2007, 2008, 2010).
Every linguist has a cat, and they all post pictures of
\((\forall x[Lx \to \exists y[Cy\land Hxy]])\)
There is another way of combining DPL and DPropL, namely, by defining the relational semantics between pairs consisting of a possible world \(w\) and a set \(G\) of assignments.
Each \(\langle w, G\rangle\) in context \(c\) will be given the following pragmatic interpretation.
Exercise: Reformulate the semantics.
Discourse Representation Theory (DRT) is another well-known dynamic theory developed for natural language (Kamp 1981; Kamp and Reyle 1993, among others).
It has been popular (in certain circles, at least) and many different versions have been developed and being developed.
I will not covered DRT, but see Brasoveanu (2007) for a way to think about it similarly to FCS. See also the comparison between DRT and DPL in Groenendijk and Stokhof (1991).