6.4 The Logical Form of Fragments
6.4.1 A Constraint for Fragments
solutions).17 Finally, and for the same purpose of restricting where quantifiers can ‘float’ to, we adopt the convention from (Copestake et al. 1999) to explicitly denote`0, the supremum of the partial order Succ, with l>. Again, for the example above this is not necessary, but we will need it in cases where we want to ensure that certain material outscopes all quantifiers.
This concludes our description of how scope-underspecification can be represented in a logical language.
In the next section we will extend this language to deal with the underspecification needed to represent the compositional semantics of fragments. We are only concerned with the semantics here; we deal with details of the syntax/semantics interface in the next chapter.
into theLLwhich encapsulates these demands on the subformula. We will call this constraint unknown, because it stands in for an unknown subformula.
We should add a word of caution here. Unlike formulae ofLLthat only contain underspecification of scope, those that contain unknown rel-constraints will describe, if they are satisfiable at all, a countably infinite number ofBLFformulae. This is what we want, since it reflects the fact that viewed in isolation from the context there is an infinite number of propositions (or, depending on the sentence mood of the fragment, questions, or requests) that can be conveyed with such fragments. The compositional semantics specified here simply constrains the readings to make use of the material from the fragment in some way. But this of course makes it impossible to specify a procedure that actually generates the set of all readings in finite time; something which, as we have said, is possible for the fragment ofLL without unknown rel. However, as we mentioned above, we are not actually interested in all readings anyway. What we are interested in is a different relation, namely the one that relates to theULFthat particular reading that is pragmatically preferred in the given context. We will define this relation in Chapter 8.
We give a definition of the semantics of the constraint unknown below in Definition 7, but first we show here what the representation of the compositional semantics of our example fragment “Peter.”
now finally looks like. (232) give this representation in all three notations we have introduced (the relation notation; the infix notation; and the graphical notation, which we already have seen as (213-a) above).18,19
(232) a. Rpr pstn(l1,l>) ∧unknown rel(l11,l12,l2)∧
Re(l11) ∧Rx(l12) ∧
Rdef np(l13,l8,l9,l6)∧Rx(l13) ∧ Rnamed(l14,l15,l10) ∧Rx(l14) ∧ RPeter(l15) ∧
outscopes(l8,l10) ∧
b. l>: pr pstn(l1)∧l2: unknown rel(e,x)∧ l6: def np(x,named(x,Peter),l9)∧ outscopes(l8,l10)
18We use as base language here a predicate language that is inspired by the predicates in the “English Resource Grammar”
(ERG, (Copestake & Flickinger 2000)). We will say more about this grammar below, so for now just note it uses in itsLFs a predicate prpstn that outscopes all others and that signals that the formula denotes a proposition (rather than a question or a request), and also that there is a quantifier for definite NPs, aptly named def np.
19Recall that, given the additional assumption about scoping possibilities made above, we do not have to explicitly state out-scope constraints between l1and l6or l2. For clarity, however, we will draw such relations in the constraint-trees.
c. l>•pr pstn
l1•
l6•de f np
l13•x l8•
l10•named
l14•x l15•Peter l9•
l2•unknown
l11•e l12•x
Note that unknown is a constraint more like outscopes than like for example def np, in that it constrains the configuration of the base language formula and does not get translated into a base language predicate.
We draw it in the graphical representation in the same way as the translations of the base-language predicates, with the exception that its arguments are connected with a dotted line. This expresses that the arguments of unknown do not have to be immediately outscoped by the label of unknown (as arguments connected with solid lines have to), and on the other hand have to satisfy a stricter constraint than simply outscopes (which is indicated by dashed lines).
Let us now look at an example that indicates that this indeed expresses what we want, before we formally define the semantics of this new constraint. (233) shows an LΣS corresponding to the base language representation for the sentence “Peter walks.” (this was (213-b) above). This should be one possible resolution of the fragment “Peter.”—for example if the fragment is uttered in the context of the question
“who walks?”—and so we’d expect this to be described by (232). (233) shows the LΣS corresponding to theLFof “Peter walks.”, and in colour one possible variable assignment. We also indicate the node which is the root for the unknown-subformula by marking it withuk. Since the variables x and e do occur unbound in the subformula ‘below’`2(the denotation of l2), this is indeed a solution for (232).
Examples that show that (232) also describesBLFs where “Peter” is an object (e.g. “Marty loves Peter”), or an argument in a PP (e.g. “John made a picture of Peter.”), are given below in Section 6.4.2.
(233) l,`•pr pstn
l1,l6,`6•de f np
l13,`13•x l8,l10,`8•named
l14,`14•x l15,`15•Peter
l9,l2,`2•walkuk
l11,`11•e l12,`12•x
With this motivating example behind us, we can proceed to formally define the semantics of the new constraint. For this, we need a couple of auxiliary definitions. First, labels:
Definition 5 labels
Given an LΣShU,Succ,Ii, we say that`labels a constructor R∈Σof arity n iffh`1,...,`n,`i ∈I(R) (with`,`1,...,`n∈U ).
Further, we define two relations, Free in and First Arg, that will be used in the definition of the inter-pretation of unknown rel. Roughly, Free in formalises the base-language notion of free variables on LΣSs. Formally, it is the set of all ordered pairs of labels where the former labels a variable that is free in the subformula labelled by the latter. We say that a label`is free in a subtree whose root is`0 if` is subordinate to`0 and does not label a constructor that is the first argument (i.e. the bound variable) of a quantifier which is subordinate to`0and superordinate to`. Or, to put it simpler,`is free in`0if the variable labelled by`occurs in the subformula labelled by`0, but is not bound in it. A label is in First Arg if it labels an event-variable that is the first argument of some constructor f . These definitions are illustrated by (234) (a fragment of for example theLFof “a dog bites sandy”).
(234) `1•de f np
`2•x `3•dog
`5•x
`4•bites
`6•e `7•x `8•y
`8in (234) is free relative to`1, because it is not bound by any quantifier below`1;`6is in First Arg relative to`1, since it is the first argument of bites;`5on the other hand is not free relative to`1, since it labels the constructor x which is the first argument (i.e., the bound variable) of a quantifier that is subordinate to`1(recall that subordination is reflexive).
We can now give the formal definition of these notions.
Definition 6 Free in and First Arg Given an LΣShU,Succ,Ii,
• the binary relation Free in consists of exactly the pairsh`,`0i ∈U×U such that 1. `∈Uil; and
2. `0`; and 3. `labels R; and
4. there is no R0(with arity n)∈Q (Q is the set of quantifiers inΣ), s.t.
(a) ∃`00∈U where`00labels R0,`0`00and`00`; and
(b) there are`000,`2,...,`n∈U , s.t. h`000,`2,...,`n,`00i ∈I(R0)and both`labels R and`000 labels R.
• First Arg is the binary relation consisting of exactly the pairsh`,`0i ∈U×U such that 1. `∈Uel; and
2. ∃Rn∈Σs.t.h`,`2,...,`n,`00i ∈I(R), with`,`2,...,`00∈U and`0`00. With the help of these relations we can now interpret the predicate unknown rel:
Definition 7 Interpretation of unknown rel hU,Succ,Ii |=gl unknown rel(l,l0,l00)iff
i) h[[l]],[[l00]]i ∈Free in (givenhU,Succ,Iiand g), ii) h[[l0]],[[l00]]i ∈Free in (givenhU,Succ,Iiand g), iii) h[[l]],[[l00]]i ∈First Arg (givenhU,Succ,Iiand g).
This definition makes sure that unknown rel indeed constrains any subformula ‘below’ l00 to contain the variables labelled by l and l0. Moreover, it also requires that these variables are free in that sub-formula. This is necessary, because otherwise we would also describe formulae where these variables are ‘captured’ by other quantifiers (remember that the fragment that introduces the variable will also introduce a quantifier; e.g. the fragment “Peter” will introduce a variable bound by a def np-quantifier).
Additionally, we have constrained the first argument, the event variable, to appear as the first argument of some predicate in the subformula. This makes sure that the event variable is indeed the main event of the subformula.
Let’s return to our example (232) now and see whether the LΣS and variable assignment shown in (233) does indeed satisfy it, given the semantics of unknown we have now defined. (We repeat the two graphs here as (235) and (236).)
(235) l•pr pstn
l1•
l6•de f np
l13•x l8•
l10•named
l14•x l15•Peter l9•
l2•unknown
l11•e l12•x
(236) l,`•pr pstn
l1,l6,`6•de f np
l13,`13•x l8,l10,`8•named
l14,`14•x l15,`15•Peter
l9,l2,`2•walkuk
l11,`11•e l12,`12•x
(237) shows the relations Free in and First Arg as specified by (236).
(237) {h`14,`8i,h`12,`2i,h`11,`2i,h`11,`6i,h`11,`i}=Free in {h`11,`2i}=First Arg.
With a variable assignment as indicated above in (233), that LΣS does indeed satisfy unknown(l11,l12,l2): (a)h[[l11]],[[l2]]i ∈Free in is true; (b) as ish[[l12]],[[l2]]i ∈Free in; and (c) also[[l11]]∈First Arg holds.