6.1 Overview
We have seen in the previous chapter that the GBA does not have a semantic notion of fragments as such: in the case of NP-short-answers for instance, the semantics of the NP is directly combined with contextual material (in this case the question they are meant to answer) to yield a propositional content.
At no point is there a representation of the semantics of the fragment independent from its context. In our set-up, we want to separate grammatical and contextual analysis, while allowing the two sources of information to interact during discourse update. Hence, we will have to provide a compositional semantics for fragments. This semantics will have to represent both the semantic information that comes from the phrase as well as the fact that there is information “missing” that has to be filled in by the context. The compositional semantics should also ensure that this ‘missing’ information is such that, were it resolved, the result would be a message—i.e., either a proposition, a question or a request.
This chapter is structured as follows. In Section 6.2 we motivate the use of semantic underspecification.
In Section 6.3 we describe a particular underspecification formalism, LL (after (Asher & Lascarides 2003)). This formalism is introduced with scope-ambiguities as examples; in 6.4 we present our ex-tensions that allow for the expression of the underspecification arising from the use of fragments. In a nutshell, our extension is a constraint that ensures that all specifications of the representation of the fragment use the semantic material of the fragment phrase in a certain way. E.g., if the fragment is
“Peter.”, then our representation demands that all resolutions denote an event which involves Peter.
Readers who are familiar with underspecification formalisms can skip directly to Section 6.4, possibly after scanning the definitions given in 6.3. To only get an idea of our approach to underspecifying the compositional semantics of fragments reading Section 6.4.1 is sufficient; further examples are discussed extensively in Section 6.4.2. In Section 6.5 some final revisions to the formalism are introduced.
will be of propositional type.1That means that once we decide that a certain utterance is a fragment (and not just a phrase), we have more semantic information than what is represented in just the translation of the phrase it consists of.
We have to answer a possible objection here. Why should we not let the pragmatic module work directly on the ‘normal’ meanings of the fragment-phrases? There are several reasons why this is not the optimal strategy, which will be discussed in detail later on, so we only highlight some of the arguments here.
Firstly, working on meanings (which will be represented at least as first-order formulae) directly rather than on descriptions means that a logic with an intractable notion of validity has to be used for pragmatic processing; this should be avoided. Secondly, it would mean that we cannot use the same definitions for rhetorical relations as for full sentences, since they expect propositions, questions and requests as arguments, and so we would have to introduce relations specifically for fragments. Moreover, since the same speech act can be performed with many different syntactic fragments (e.g., short-answers can be PP-fragments, NP-fragments, VP-fragments etc.) whose meanings all have different semantic types, we would need several relations just for fragments realising one speech act. Lastly, there is also data that suggests that fragments have a different syntactic status than phrases—this at least suggests giving them a different semantics as well.
As we said, we will discuss these points again later on, and so we just conclude for now that for our example “Peter.”, the LF(210)—the translation of the NP—isn’t sufficient as a representation of the semantics of the fragment.2
(210) λP.∃x(named(x,“Peter”)∧P(x))
What we have to express in our semantic representation, then, is that the translation of the phrase—in our case, a quantified variable that is in a “named”-relation—is an argument of some (perhaps complex) event-predicate; which predicate, the grammar can’t say. (211) shows two attempts at expressing this in the same language of predicate logic we have used in (209-b).
(211) a. ∃P∃e∃x(P(e,x)∧named(x,“Peter”)) b. ∃e∃x(P(e,x)∧named(x,“Peter”))
The formulae above simply introduce a second order variable P into the LF (informally speaking, in place of the verb-phrase translation in (209-b)). This variable is bound by a quantifier in the first formula, and free in the second. Now, does this express what we want? Let’s first look at (211-a).
1That is, if we make the simplifying assumption here that there is an un-ambiguous ‘declarative’ intonation. We will later see how this decision about the type of message can be left underspecified in the compositional semantics as well.
2Note that, as is customary in Montagovian approaches to formal semantics, we assume the use of higher-order logic (λ-calculus) during construction ofLFs here, even when the final representation for the sentence might still be first order. However, for (211) we will need higher-order variables as well, and so we can assume that the language for meaning representation used here is higher-order as well.
The formula (211-a) is true if there exists a predicate P, an eventuality e and an individual x named
“Peter”, and P is predicated of both. One now could imagine a strategy, following the abduction-based approach to discourse representation of (Hobbs et al. 1993), in which interpreting the fragment
“Peter.” consisted in finding a proof of the formula (211-a), possibly drawing on additional information.
Finding a proof involves instantiating the variable P with a ‘concrete’ instance—a predicate constant.
In the context of a question like “who saw Mary?”, P could accordingly be resolved to something like
‘λz∃y(named(y,“Mary”)∧see(e,z,y))’,3so that the fragment would be resolved to (209-b).
We have some objections to such an approach, however.4 First, this would entail that all the reasoning necessary to resolve underspecification would have to be done within the higher-order logic used for meaning representation; a logic with an intractable notion of validity. We will later see that we can use a much less expressive logic for this contextual reasoning if we use descriptions of (logical forms representing) the content. Second, such an approach also fails to distinguish situations where a coherent interpretation requires one to identify what instantiates the existentially quantified variables and where it doesn’t. Put differently, it does not distinguish between values that are unspecified for linguistic reasons and those that are for other reasons. Last but not least, such an approach would force a certain order of quantifiers (in cases where P includes a quantifier). We have seen in the discussion of theGBAthat this is problematic, given the observation that fragments can still exhibit scope ambiguities.
For the same reasons, we will not use representations like (211-b), where the predicate-variable is left free. One could imagine an approach using such representations where resolving fragments consists in updating an assignment function that provides a value for the free variable. But again, such an approach would not make it clear in the representation where there is information that is underspecified by the syntax.
We will instead use a device that is independently motivated, well studied, and can easily be integrated into modern grammar formalisms (Copestake, Flickinger, Sag & Pollard 1999, Dalrymple 1999): un-derspecification. Rather than trying to express directly in the logical form that information is “missing”, we climb to a ‘meta-level’ and let the grammar produce a description of logical forms. This description only has to be as precise as the linguistic information affords; if there are semantic differentiations the grammar can’t make, the description simply describes a set of logical forms containing representations for all different ways of making the distinction. In other words, the description produced by the grammar is a partial one. So what we want to express for our fragment example “Peter.” is that it will denote an eventuality, but one about which we only have very partial information (namely that the entity denoted by “Peter” is somehow involved in it). In (212), this is glossed as ‘P=?’ with the x being an argument to that unknown relation.
(212) ∃e∃xP(e,x)∧named(x,“Peter”)∧P=?
3Glossing over how the event variable should be dealt with.
4For a detailed argument contra such a strategy see also (Asher & Lascarides 2003).
Note that (212) is not intended to be a formula of the language of LFs; rather, it is a description of the form of such formulae. More precisely, the description will be satisfied by any ‘real’ logical form in the language we choose for meaning representation that contains “named(x,“Peter”)” and a (perhaps complex) predicate relating e and x. In other words, these normal logical forms will be models of the description.5
To give a visual impression of this approach, (213-a) shows a graphical representation of the composi-tional semantics of “Peter.” that we will eventually adopt, while (213-b) shows a graphical representation of one of the logical forms that are described by (213-a) (corresponding to “Peter walks.”). Roughly, the role of P=? in (212) is played here by the constraint unknown. The various styles of lines express certain scopal or structural relations between the various parts of the formula—this will be explained in detail presently.
(213) a. l•pr pstn
l1•
l6•de f np
l13•x l8•
l10•named
l14•x l15•Peter
l9•
l2•unknown
l11•e l12•x
b. `•pr pstn
`6•de f np
`13•x `8•named
`14•x `15•Peter
`2•walk
`11•e `12•x
5Recall the relation betweenHPSG-constraints and the feature structures they describe.
In Section 6.4, we will make formally precise what these representations express, but first we have to introduce the formalism in which they are framed. The strategy for expressing underspecification that is followed in the formalism, namely labelling bits of formulae (see the labels l1to l12in (213-a) for instance), was originally invented to deal with scopal ambiguities (see e.g. (Reyle 1993, Bos 1996)), but has since been extended to other sources of semantic underspecification (plural NPs, lexical ambiguity, anaphora, VP-ellipsis; see for example the papers collected in (Deemter & Peters 1996), also (Egg, Koller & Niehren 2001)). The particular formalisation of the labelling-idea we describe here is from (Asher & Lascarides 2003), which in turn is an extension of (Asher & Fernando 1999), combined with ideas fromCLLS(Egg et al. 2001); we have made some minor changes to it that will be indicated below.
The background section in which this approach is described is a bit longer than previous ones, because later we will need to make fairly detailed extensions which rely on the definitions of the basic approach.