Research on Language and Computation 00: 1–18, 2006.
c 2006 Springer Science+Business Media, Inc. Manufactured in The Netherlands.
1
On Appositives and Dynamic Binding
RICK NOUWEN ([email protected])
Utrecht Institute of Linguistics (OTS),
Utrecht University, NL-3512BL/13 Utrecht, the Netherlands
July 3, 2006
Abstract. Quantified appositives have a limited distribution which is reminiscent of quantified
discourse anaphora. This article investigates whether this parallel can be fleshed out by means of a
two-dimensional dynamic semantics. The proposal works towards an explanation of why quantified
appositives are generally infelicitous. Crucial is the fact that variables bound by a strong quantifier
have a singular value, while the antecedents for discourse anaphora introduced by the quantifier are
plural.
Key words: appositives, quantifiers, dynamic binding, multi-dimensional semantics
1. Introduction
On many occasions (see del Gobbo 2003b and references therein), a parallel has
been observed between discourse anaphoric phenomena on the one hand and constraints on appositives on the other. An illustration of this is presented in (1) and
(2).
(1)
a.
b.
Jake, a famous Dutch boxer, lives in Utrecht.
Jake lives in Utrecht. He is a famous Dutch boxer.
(2)
a. #Every Dutch boxer, a famous one, took part in the event.
b. Every Dutch boxer took part in the event. #He is famous.
The example in (1-a) first and foremost informs us that Jake lives in Utrecht. The
appositive gives the supplementary comment that Jake is a famous Dutch boxer.
The two sentences in (1-b) do something similar. The first one states that Jake lives
in Utrecht and the second one adds to this that he is a famous Dutch boxer. While
there might be a difference in how we use (1-a) and (1-b), it is clear that what they
share is the expression of two propositions, both about properties of Jake.
In (2-a), the appositive fails to supply additional information about (the) Dutch
boxers. In fact, the appositive does not yield an interpretation at all, and the example is simply infelicitous. In discourse, once again something similar happens.
We cannot use the second sentence in (2-b) to express that every Dutch boxer is
famous.
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The parallel between discourse and appositive is very consistent. For instance,
if we substitute ‘every’ in (2-a) and (2-b) by the indefinite article ‘a’, then both
examples are felicitous.
(3)
a.
b.
A Dutch boxer, a famous one, took part in the event.
A Dutch boxer took part in the event. He is famous.
Nevertheless, there is an obvious problem for the comparison between the two
kinds of phenomena. In discourse, one can turn to a notion of scope to explain these
data. The supplementary nature of appositives severely complicates the notion of
scope. In (2-a), for instance, the appositive is sentence internal and, thus, in some
sense in the scope of the quantifier. However, the appositive content is invisible to
operators in the host sentence. For instance, (4) is not true in a situation where Jake
lives in Utrecht but is not a famous Dutch boxer.
(4)
It is not the case that Jake, a famous Dutch boxer, lives in Utrecht.
Still, there is an obvious way in which host sentence and appositive connect. The
appositive’s anchor provides a logical subject: in (4), it is Jake who is said to be
a famous Dutch boxer. In parenthetical clauses, moreover, the host sentence may
provide an antecedent to supplementary material containing an anaphoric pronoun.
(5)
Jake’s coaches – and, mind you, they worked very hard – were criticised
beyond belief.
In general, then, apposition and parenthesis is like discourse in that it offers limited
access to resources presented in the host/antecedent sentence. However, the fact
that appositives are semi-integrated in a sentence makes it difficult to pursue an
analysis which does justice to this parallel with discourse. The reason for this is
that as long as there is no clearly stated semantics of apposition, we cannot apply
our insights from dynamic semantics, the e-type strategy, discourse representation
theory or whatever other view we have on the interpretation of discourse anaphora.
What is needed is a way of representing the role of appositives and (other)
parenthetical material in semantics. Potts 2005 presents a thorough study of what
he calls ‘supplements’ by using a multidimensional framework. This means that
although the supplementary material is partly integrated in the host sentence, it
contributes its content to a separate dimension, thereby escaping the scope of host
sentence operators.1
In multidimensional semantics, a sentence S does not result simply in a proposition [[S ]] , but rather in an array of propositions h [[S α ]] , [[S β ]] , . . .i.2 The propositions in the array do not all have the same status. Some of them might be asserted,
some might be conventionally implicated, some might be presupposed, some might
be separated in a way we do not yet understand. For the purpose of this paper, it
does not really matter how these arrays are themselves interpreted. The central
topic here is namely to what extent there is an interaction between the dimension
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APPOSITIVES AND DYNAMIC BINDING
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associated to an appositive (or a parenthetical clause) and the dimension associated
to the host sentence. I will therefore focus on a two-dimensional semantics only
and will leave implicit what the (extra) dimensions stands for.3
In this paper, I will apply the idea of two-dimensionality to the dynamic approach to discourse anaphora. In particular, I will spell out the consequences of
combining two null assumptions. The first one involves a statement of simplicity:
there is no more to the semantics of apposition than two-dimensionality. That is, we
expect that if ψ occurs appositively in ϕ, that these two forms are teased apart and
that a 2D interpretation results which contains nothing but [[ϕ]] and [[ψ]] . In other
words, apposition has no semantic side-effects. The second assumption we start
out with is that the distribution of discourse anaphora is correctly accounted for by
dynamic semantics. In particular, I assume that a dynamic analysis of quantifiers
gives us exactly the anaphoric properties we are interested in.
Central to the account of appositives and parentheticals that I propose will be the
notion of scope in the semantics. Syntactically, scope is given by scope delimiters
such as parentheses. Standardly, parentheses in logic are syntactic instructions that
guide interpretation. Normally, there is no meaning assigned to them. Visser and
Vermeulen 1996 observe that normally ‘[g]rammar is syncategorematic in [the]
approach to semantics’ (p. 322), and that ‘no semantical objects are ascribed to the
symbols fixing grammatical structure’ (ibid.). Visser and Vermeulen subsequently
aimed at devising formalisms where the role of grammar is categorematical rather
than syncategorematical. It goes beyond the scope of this paper to review Visser
and Vermeulen’s radical appeal for syntactic flatness in detail. However, I will in
part embrace their approach by adhering to the following maxim: do not attribute
scope to scopeless actions. By stating a semantics that is fully explicit about scope
matters, I will show in detail how appositive content interacts with scope taking
operators from the host sentence.
The structure of the paper is as follows. Section 2 will present a dynamic
semantics for strong generalised quantifiers and will discuss the notion of scope
they entail. On the basis of this formalism, in section 3, I construct a simple
two-dimensional semantics, which is more expressive only in that it adds a representation of appositive content. Section 4 shows that the dynamic multidimensional
formalism readily explains a large part of the data. Finally, in section 5, I discuss
some data that is less clearly accounted for.
2. The Dynamics of Quantifiers
It is well-known that nominal appositives generally do not occur with quantificational anchors, whereas they are felicitous in combination with referential expressions (McCawley, 1998).
(6)
a. #Every Dutch boxer, a famous one, takes part in the event.
b. #No Dutch boxer, a famous one, takes part in the event.
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c.
A Dutch boxer, a famous one, takes part in the event.
As I discussed above, the contrast in (6) has a full parallel in the domain of discourse anaphora. Singular pronominal reference to a quantificational antecedent
not possible.
(7)
#Every/A Dutch boxer takes part in the event. He is famous.
In dynamic semantics, this contrast between referential and quantificational antecedents is accounted for by assuming a different kind of relational interpretation
for these expressions. For existential quantifiers, the so-called donkey equivalence
in (8) holds, while for universal quantifiers such an equivalence is invalid.
(8)
a.
b.
(9)
a.
b.
∃x[dutch(x) ∧ boxer(x) ∧ take-part(x)] ∧ famous(x)
= ∃x[dutch(x) ∧ boxer(x) ∧ take-part(x) ∧ famous(x)]
A Dutch boxer takes part in the event. He is famous.
= A Dutch boxer takes part in the event and is famous.
∀x[dutch(x) ∧ boxer(x) ∧ take-part(x)] ∧ famous(x)
, ∀x[dutch(x) ∧ boxer(x) ∧ take-part(x) ∧ famous(x)]
Every Dutch boxer takes part in the event. He is famous.
, Every Dutch boxer takes part in the event and is famous.
As is common in dynamic semantics, expressions in the language denote relations
between pairs of assignments and worlds. This allows for the simultaneous representation of changes forms make with respect to both knowledge about discourse
referents and knowledge of the world. As the discourse progresses less worlds will
be accepted and the assignments will contain less valuations to an increasing number of variables. In dynamic predicate logic (DPL, Groenendijk and Stokhof 1991),
an existential quantification ∃x[ϕ] represents random assignment to x. It relates
assignment functions f to functions that differ from f only in assigning some
value to x such that ϕ is verified. It is easy to see that there is really no reason
why one should have to define the semantics of existential quantification with
respect to some scopal form ϕ. In fact, the donkey equivalence makes clear that
a notion of scope is absent with existentials. In the style of dynamic semantics
offered by Visser and Vermeulen (Vermeulen, 1993; Visser and Vermeulen, 1996),
∃x is therefore often interpreted as a simple random assignment action:
h f, wi [[∃x]] hg, w0 i :⇔ w = w0 & ∀v , x : f (v) = g(v)
According to this, ∃x does not tell us anything about the world we are in and it
moreover says that any value for x will be accepted. In other words, ∃x resets
whatever we knew about x. Subsequent predicates over x may then specify which
values for x will be acceptable in which worlds. It is important to realise that there
can be no similarly decomposed way in which we may interpret ∀x. Universal
quantification only makes sense in combination with the quantifier’s scope. The
fact that the donkey equivalence in (9-b) does not hold is a good illustration of this.
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APPOSITIVES AND DYNAMIC BINDING
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Consequently, for ∀x we need an interpretation that tests values for x with respect
to the whole quantificational structure.
h f, wi [[∀x[ϕ][ψ]]] hg, w0 i :⇔ f = g & w = w0 &
∀ f 0 : h f, wi [[∃x ∧ ϕ]] h f 0 , wi
→ ∃g0 : h f 0 , wi [[ψ]] hg0 , wi
A definition like this does justice to the fact that universal quantifiers cannot antecede singular pronouns in discourse. However, it fails to account for the fact that
once plural anaphora is taken into consideration everything changes. Examples like
(10) show that strong quantifiers are not completely discourse intransparent.
(10)
Every Dutch boxer will take part in the event. They are all famous.
It has been known since Evans 1977 that a non-coreferential, non-bound-variable
type of anaphora exists in discourse, where a plural pronoun refers to the maximal
set of individuals that satisfy restrictor and scope of quantification. That is, in (10),
the plural pronoun refers to (all) the Dutch boxers that will take part in the event. It
is fairly straightforward to account for this in dynamic semantics. Given a form ϕ, a
world w and an assignment f , let f +w ϕ be the set of output assignments that result
from interpreting ϕ with respect to h f, wi. So, f +w ϕ := { f 0 | h f, wi [[ϕ]] h f 0 , wi}.
Let G be a set of assignment functions, then tG is the assignment function such
that for any variable v: (tG)v = {g(v) | g ∈ G}. In words, t transforms a plurality
of assignments into a single assignment to pluralities. Combining these two operations gives us a way of introducing plural entities in the context: t( f +w ϕ) returns a
single assignment function which represents all the different output valuations for ϕ
in pluralities. For instance, t( f +w (∃x∧ boxer(x))) yields a function that assigns the
set of boxers in w to x. A definition like this gives us a way to represent generalised
quantification dynamically. Given some contextual f , for a quantified statement
Qx[ϕ][ψ] the restrictor set (in w) is given by t( f +w (∃x∧ϕ)), while the intersection
between restrictor and scope of the quantifier results from t( f +w (∃x ∧ ϕ ∧ ψ)). On
the basis of this, I propose the following dynamic interpretation for the universal
quantifier. (Related proposals can be found in work like Krifka 1996, van den
Berg 1996, Elworthy 1995, Asher and Wang 2003, Wang et al. 2006, Nouwen 2003,
Nouwen 2006.)4
h f, wi [[∀x[ϕ][ψ]]] hg, wi :⇔ w = w0 &
g = t( f +w (∃x ∧ ϕ ∧ ψ)) &
(t( f +w (∃x ∧ ϕ)))(x) ⊆ g(x)
This definition is completely parallel to the standard “Every(A)(B)⇔A⊆ B”. What
is more, we may generalise this definition to all strong quantifiers. Assuming that
a quantifier symbol Q is interpreted in the model as Q0 , we get:
h f, wi [[Qx[ϕ][ψ]]] hg, wi :⇔ w = w0 &
g = t( f +w (∃x ∧ ϕ ∧ ψ)) &
Q0 (t( f +w (∃x ∧ ϕ))(x) , g(x))
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Table I. The semantics of LQDPL
h f, wi
[[ ∃x ]]
h f 0 , w0 i :⇔ w0 = w & ∀v , x : f (v) = f 0 (v)
h f, wi [[ P(x1 , . . . , xn ) ]] h f 0 , w0 i :⇔ w0 = w & f 0 = f & f, w |= P(x1 , . . . , xn )
h f, wi
[[ ϕ ∧ ψ ]]
h f 0 , w0 i :⇔ ∃w00 ∃ f 00 : h f, wi [[ ϕ ]] h f 00 , w00 i [[ ψ ]] h f 0 , w0 i
h f, wi [[ Qx[ϕ][ψ] ]] h f 0 , w0 i :⇔ w = w0 &
f 0 = t( f +w (∃x ∧ ϕ ∧ ψ)) &
Q0 (t( f +w (∃x ∧ ϕ))(x) , f 0 (x))
In terms of dynamics, the definition gives an important result. Any occurrence of x
in the restrictor or scope of a quantifier Qx is bound by the quantifier. That is, such
occurrences are given singular values only. This not only accounts for the fact that
co-indexed pronouns in the scope of a quantifier are always bound,5 but also for the
fact that strong quantifiers are distributive (see, for instance, Kamp and Reyle 1993
for discussion). Occurrences of x outside the scope of Qx have an exhaustive plural
value, which accounts for the famous observation by Evans.
To summarise, strong quantifiers differ from existential/referential ones in that
they come with a notion of scope. In dynamic semantics, this difference is reflected
by the fact that existential quantifiers are actions by themselves, whereas strongly
quantified statements have to be interpreted as complexes that include the quantifier’s full scope. As I will show below, the dynamics of quantifiers accounts for a
considerable part of the apposition data, given a simple two-dimensional semantics. In fact, the 2D semantics will be nothing more than the quantified dynamic
formalism with an added mechanism for dimensionality. That is, the dynamic semantics of quantifiers discussed above gives the bulk of what is needed to account
for quantified appositives. I end this section therefore by stating the details of a
quantificational version of dynamic predicate logic.
The language LQDPL is the language of dynamic predicate logic with generalised quantifiers. The syntax is as follows. First, I assume a set of constants Con.
I write Conn for that subset of Con that contains the n-ary constants only. I will
furthermore assume that there exists a set of variables Var and a set of terms
T erm = Con0 ∪ Var. Finally, there exists a set of (binary) quantifier symbols
Quan.
∃v ∈ LQDPL ⇔ v is a variable
P(x1 , . . . , xn ) ∈ LQDPL ⇔ P ∈ Conn & x1 , . . . , xn ∈ T erm
Qx(ϕ, ψ) ∈ LQDPL ⇔ x ∈ Var & Q ∈ Quan & ϕ , ψ ∈ LQDPL
ϕ ∧ ψ ∈ LQDPL ⇔ ϕ , ψ ∈ LQDPL
nothing else is in LQDPL
Well-formed formulae in LQDPL will be presented on top of a slightly shaded
background, this to allow for a clearer distinction between the quantified dynamic
logic defined here and the multi-dimensional formalism that will follow. This colour
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APPOSITIVES AND DYNAMIC BINDING
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coding will become useful in the semantics of the multi-dimensional logic which,
in part, is based on that of LQDPL . The semantics is summarised in table I.
3. 2D Semantics
Below, I will introduce a language that will function as a representation for the
semantics of appositives. I’ll first provide an informal glance at this language on
the basis of (11).
(11)
a.
b.
Jake, a famous boxer, lives in Utrecht.
∃x ; x = j ; h ; famous-boxer(x) ; i ; live-in-Utrecht(x)
At first sight, this form may seem rather mysterious. But it should be clear that all
the relevant information is there. The formula in (10-b) contains both the fact the
Jake lives in Utrecht and that he is a famous boxer.
The way to read the formal representation in (11-b) is to view it as a series
of instructions. That is, I replaced the conjunction symbol with ‘;’ to emphasise
the fact that (11-b) is a list of operations as in a complex program. So, ϕ ; ψ
corresponds to “first do ϕ, then ψ.” This means that everything apart from these
connectives corresponds to an action of some sort. This includes, as I discussed
above, ∃x. Moreover, nothing in (11-b) has a scope in the traditional sense of the
word. Each of the forms that are connected by semicolons is interpreted by itself
as an action on contexts.
In what follows, I will use the terms primary and secondary content in order to
be able to talk about the different dimensions without obviously adhering to some
theoretically assumption on the pragmatic status of the dimensions. The brackets h
and i direct the flow of information to the two dimensions. Just like ∃x is a primitive
action, so are h and i. The left bracket is an instruction to add future information to
the dimension that stores secondary information. The right bracket is an instruction
to change back to the dimension of primary content. The action h comes with a
presupposition that the current level of interpretation is the standard one, while i
presupposes the current level to be the one for apposition. The bracketing in the
logical language is therefore not always interpretable. For instance, h; ϕ; h will not
receive a defined interpretation in case ϕ does not contain a i, but note that h; ϕ may
be defined even if the level is not changed in ϕ. That is, non-matching bracketing
is not interpretable, but incomplete bracketing is.
Let us now turn to the semantics of the multidimensional language. In dynamic
semantics, forms express relations between world-assignment pairs. The simplest
way of viewing an example like (11) in terms of relations, is to have the twodimensional language express operations on two completely separated information
states. In such an approach, as the discourse unfolds, a doubled context is continuously updated, representing two different ways in which information may be
conveyed. Such a view is too simple, however. It leads to a well-known problem
with multidimensional formalisms, which was first noticed in Karttunen and Pe-
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ters 1979. For an example like (11), a doubled-context approach would generate
two separated updates, one saying that Jake lives in Utrecht, the other saying that
there is a famous Dutch boxer. Since the two information dimensions have no
access to each other’s referential resources, there is no possibility to do justice
to the fact that the appositive’s meaning is referentially related to the meaning of
its anchor.
The solution is to have dimensions share their referential resources. So, instead
of two distinct information states, I take there to be states that consist of one assignment function and two possible worlds. Informally, the interpretation of (11)
in some given context, results in triples h f, w, w0 i such that:
- f (x) lives in Utrecht in w and f (x) = j
- f (x) is a famous boxer in w0
(primary content)
(secondary content)
In other words, (11) results in triples containing, first of all, a world in which
Jake lives in Utrecht and, second, a world in which Jake is a famous boxer. The
triple-based semantics for two-dimensionality supports two intuitions: (i) referential resources are shared, and (ii) an appositive expresses a proposition that is
independent from the one expressed by the host sentence. As we will see below the
sharing of resources has important consequences for the interpretation of quantification into appositives. Before turning to quantification, however, let us see how
the current proposal accounts for the second intuition. At each point in discourse,
the information that has been conveyed so far is captured in the set of triples that
are related to some initial context. Given such a set S , two propositions may be
extracted: S 2 = {w | ∃ f, w0 : h f, w, w0 i ∈ S } and S 3 = {w0 | ∃ f, w : h f, w, w0 i ∈ S }.
So, for the above example S 2 amounts to Jake lives in Utrecht and S 3 is Jake
is a famous boxer. These two propositions represent the information conveyed
by host sentences and by supplementary material respectively. However, an agent
participating in the discourse might assume that both the proposition expressed by
the appositive and the one expressed by the host sentence are true in the actual
world. In such cases, the information state needs to collapse. What I mean by this
is that when both the appositive and the host are assumed to be true, then there
ought to be no difference in the worlds that occur in the two dimensions. There is a
straightforward operation that can represent this. Let S be a set of triples h f, w, w0 i:
(collapse)
!S = {h f, w, w0 i|w = w0 & h f, w, w0 i ∈ S }
In what remains, I will ignore the option of information state collapse, however,
and assume that for the purpose of studying conditions on appositives one should
study the effects of separating out information.
One of the merits of the account presented here is that the widest-scope behaviour of appositives follows immediately. For instance, assuming that none of
ϕ, ψ and γ contain brackets, then ¬(ϕ; h; ψ; i; γ) will neatly separate the parenthetical meaning ψ (viz. the proposition in S 3 ) from the negative conventional meaning
¬(ϕ; γ) (in S 2 ). (Cf. the discussion of example (4) in the introduction.)
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I now turn to the details of the 2D semantics. What follows is basically an
application of Visser’s logic for polarity switching (Visser, 2002). That is, the
existing language of DPL with generalised quantifiers is augmented with control
features that allow for the switching between different kinds of content (here,
supplementary versus ordinary meaning).
Let a possibility be a quadruple hπ, f, ws , wp i, where π ∈ {s, p}, f is an assignment function from variables to (possibly plural) entities and ws , wp possible
worlds. The worlds ws will represent the ordinary meaning, while wp represents
the meaning that is contributed appositively. The slot π keeps track of what kind of
meaning is being processed.
Together with the multiplier ×, the meaning types {s, p} form a monoid Π =
{{s, p}, ×, s}, where s is the identity symbol and:
s×p = p×s
s×s = p×p
= p
= s
This monoid is isomorphic to the monoid consisting of 1 and -1 with multiplication (where 1 is the identity symbol and corresponds to s and -1 is the switcher
corresponding to p.) In what follows, we will omit the × symbol from our notation
and write e.g. ps instead of p × s for p.
Let α = hπ, f, ws , wp i be a possibility. We write α p for π, αa for f , αs for ws
and αp for wp .
The language of LQDPL−2D is that of LQDPL except that we additionally have h
and i as atomic formulae. As mentioned before, we will use a background colour
to keep the languages apart. The one-dimensional formalism is presented with a
shaded background. The first step in interpreting the forms of our multi-dimensional
formalism is to provide the link to the quantified dynamic semantics.
Let ϕ be an atomic form in LQDPL .
q[{ϕ}]q0 :⇔ q0p = q p & q0pq p = qpq p & hqa , qq p i [[ ϕ ]] hq0a , q0q p i
What this says is that ϕ is interpreted in LQDPL−2D almost exactly as it is in
LQDPL . The only difference is that it depends on the value of q p whether the
information conveyed by ϕ ends up in the s or in the p-slot. If q p = s than
the q0s slot might differ from that of qs , whereas q0p will remain the same as qp . If,
on the other hand, q p = p than the p-information will change, while the s part will
be left untouched.
For the extension LQDPL−2D offers to LQDPL , we will need the following additional semantics.
q {[h}] q0 :⇔ q0p = p & q0a = qa & q0s = qs & q0p = qp
q {[i}] q0 :⇔ q0p = s & q0a = qa & q0s = qs & q0p = qp
q[{ϕ; ψ}]q0 :⇔ ∃q00 : q[{ϕ}]q00 {[ψ}]q0
The interpretation of the brackets is straightforward. They change the value of the
slot that keeps track of what kind of information is being processed. Brackets come
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with the presupposition that they actually do change something. So, h cannot be
performed in a context where the p is already the current mode. Similarly, i is
unsuitable in context q should q p be s. So, with ↓ expressing definedness:
q {[h}] q0 =↓ :⇔ q p = s
q {[i}] q0 =↓ :⇔ q p = p
q[{ϕ; ψ}]q0 =↓ :⇔ ∃q00 : q[{ϕ}]q00 =↓ & q00 {[ψ}]q0 =↓
Here are some simple examples. Assume ϕ , ψ etc. are well-formed formulae
from the language LQDPL (so they do not contain brackets). Assume moreover that
ϕ and ψ do not contain quantifiers over the same variable. Let q be an arbitrary
information state with q p = s: q = hs, f, ws , wp i. The following holds (I use ↑ to
express undefinedness):
(12)
q[{ϕ; h; ψ}]q0 ⇔ q0p = p & h f, ws i [[ ϕ ]] hq0a , q0s i & h f, wp i [[ ψ ]] hq0a , q0p i
(13)
q[{ϕ; h; ψ; h}]q0 = ↑ for every q and q0
(14)
q[{ϕ; h; ψ; i; ϕ0 ]}q0 ⇔ q0p = p & h f, ws i [[ ϕ; ϕ0 ]] hq0a , q0s i &
h f, wp i [[ ψ ]] hq0a , q0p i
Since quantified sentences are interpreted as syncategoremata, we need to restate
their semantics in the extended language. Before we can do this, however, we first
of all need to redefine the ‘+0 -operation introduced above for the summation of
assignment functions, so that it can deal with multidimensional information states.
We write +q ϕ for the set of assignment functions that result in an information state
after interpreting ϕ w.r.t. q. Formally: +q ϕ := {q0a | q[{ϕ}]q0 }
I furthermore define an operation which tests whether the content of a nonactual dimension is compatible with a possibility. This will be useful to test whether
the appositive content (if any) within the scope of a quantifier is compatible with
an output possibility created by quantification.

∃w : hq p , qa , w, qp i[{ϕ}]hq0p , q0a , w, q0p i & qp = q0p




! 0
q[{ϕ}] q :⇔ 


 ∃w : hq , q , q , wi[{ϕ}]hq0 , q0 , q0 , wi & q = q0
p
a
s
p
a
s
s
s
if q = s
if q = p
Given a possibility q with q p = s, an action {[(∃x; ψ; h; ϕ; i; ψ0 )}]! is successful
only if the secondary world (qp ) is such that the value for x provided by qa is
compatible with ϕ. Moreover, any variable-introductions embedded in ϕ end up in
the output assignment. With respect to the primary content dimension, the definition merely states the condition that there exists a world for which the relation in
question holds, which is a triviality.
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.10
11
APPOSITIVES AND DYNAMIC BINDING
The semantics of a quantificational sentence is now as follows.
q[{Qx[ϕ][ψ]}]q0 :⇔ q p = q0p & qs = q0s & qp = q0p & ∃ f, g :
f = t(+q (∃x; ϕ; ψ)) &
g = t(+q (∃x; ϕ)) &
Q0 (g(x) , f (x)) &
hq p , f, qs , qp i[{ϕ; ψ}]! q0
This is completely parallel to the dynamic semantics of generalised quantifiers
given above. The only difference is that it incorporates forms from the 2D language
instead of forms from the original dynamic logic and that it tests whether its output
is compatible with any content for the non-actual dimension which might be embedded in the quantifier’s scope.6 Informally, this definition yields the following
separation of conditions:
- The set of atomic values for x that would yield an output for interpreting ϕ is
in the Q0 relation to the set of atomic values for x that would yield an output
for interpreting ϕ; ψ. The value for x is this latter set.
(primary content)
- Any appositive conditions that exist within the form ϕ; ψ are compatible
with the output assignment and qp .
(secondary content)
The assignment function resulting from interpreting a quantificational statement
is the result from combining possible atomic extensions of the input assignment.
However, such extensions might involve appositive conditions on the value of the
variable. Since the two dimensions share their referential resources, the quantifier
counts value assignments not only constrained by the actual, but also by the other
dimension. So, importantly, the resulting value for x is restricted by conditions
from all dimensions. As we will see below, this has important consequences for the
distribution of quantified appositives.
4. Multidimensionality and Quantified Appositives
I start with the case of existential anchors.7
(15)
a.
b.
A Dutch boxer, a famous one, won the tournament.
∃x; dutchboxer(x); h; famous(x); dutchboxer(x); i; won(x)
The relation expressed by the form in (15-b) contains pairs of quadruples hq, q0 i
where: q p = q0p = s and where q0a is such that it assigns to x a value that is a Dutch
boxer who won the tournament in the world q0s and who is a famous Dutch boxer
in world q0p . In less formal terms:
- the value for x is a Dutch boxer who won
- the value for x is a famous Dutch boxer
(primary content)
(secondary content)
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12
RICK NOUWEN
Note that (15-b) already yields an important result. Existentially anchored appositives are interpretable and result in separated conditions on the same variable
assignment.
Turning now to a quantificational variation on (15-a):8
(16)
a. #Every Dutch boxer, a famous one, took part in the tournament.
b. Every x[dutchboxer(x)] [h; famous(x); dutchboxer(x); i; tookpart(x)]
Informally, this results in the following:
- the value for x is the set of (famous) Dutch boxers who took part in the
tournament; this set has the set of Dutch boxers as a subset
(primary content)
- the value for x is a famous boxer
(secondary content)
There can be no such value assignment, since one dimension demands the value
to be a plurality, whereas the other demands it to be singular. Consequently, the
form in (16-b) denotes the empty relation.
Strong quantifiers compare possible valuations for variables. So, (16-b) yields
an output possibility if and only if all values d for x, such that d is a Dutch boxer
in the s-world are such that d is famous in the p-world and took part in the tournament in the s-world. Because of this comparison of valuations for the restrictor
and scope, any supplementary information will be taken into account as well. In
other words, appositive occurrences of x in the scope of a strong quantifier Qx are
bound. The quantifier summates the singular values for its running variable into one
maximal antecedent for future anaphora. This is in conflict, however, with the fact
that the appositive content is supposed to hold independent of the quantificational
relation. In other words, the plural value for x that results from quantification is
incompatible with the appositive condition that x be a single famous boxer.
The above suggests that quantified appositives are only infelicitous because the
quantifier wants to set up (plural) exhaustive reference for the quantified variable.
This implies that when the appositive’s content predicates over a variable other
than the quantifier’s running variable, the apposition can be quantified without
problems. In natural language, this prediction is borne out by examples like (17),
first observed by Wang et al. (2005).
(17)
If a professor, a famous one, writes a book, he will make a lot of money.
The most salient reading for (17) is one that says that famous book-writing professors make a lot of money. In discourse, a conditional like that in (17) does not give
rise to exhaustive reference to the set of professors that wrote a book and made a
lot of money:
(18)
If a professor writes a book, he will make a lot of money. #They are
famous.
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.12
APPOSITIVES AND DYNAMIC BINDING
13
Such a conditional, represented here as (ϕ) ⇒ (ψ), ought to receive a simple static
interpretation that tests whether all cases in which ϕ holds are cases in which ψ
hold. For (17), we arrive at:
(19)
(∃x; professor(x); h; famous(x); professor(x); i; ∃y; book(y); write(x, y))
⇒ (makealotofmoney(x))
The appositive says that values for x should be famous professors. The conditional
says that all cases in which a professor x writes a book, are cases in which this x
makes a lot of money. The two dimensions together result in a further restricted
reading which says that famous book-writing professors make a lot of money.
In sum, since strong quantifiers involve the comparison of possible value assignments that result from their scope, any apposition included in this scope domain
will influence the comparison. The fact that outside the quantifier’s scope appositive and quantifier pose conflicting conditions on the variable explains why
appositives are generally not quantified. As we will see next, however, the data are
not as straightforward as this.
5. Discussion of further data
Recently, it has been observed that, in some cases, appositives come with quantificational anchors. Potts 2005 (p. 124), for instance, gives the following example.
(20)
Every climber, all experienced adventurers, made it to the summit.
There is a problem, however, with the intuitions behind an example like this.
With universal quantification it is impossible to distinguish two kinds of anaphoric
options that result from quantifiers: one being exhaustive reference to the intersection of the quantifier’s restriction and scope, the other being reference to the
presupposed domain of quantification. Consider, for instance, (21-a) and (21-b).
(21)
a.
b.
c.
Most students attended the conference. Some of them presented a
paper.
Most students failed to show up at the conference. Some of them,
however, came and presented a paper.
Most semanticists in the Dutch department praised their joint research proposal.
In (21-a) the most natural way to understand the pronoun ‘them’ is for it to refer to
the students that attended the conference. In (21-b), however, another kind of antecedent is picked up by the pronoun. In the second sentence of (21-b), the pronoun
cannot refer to the intersection of restrictor and scope of the antecedent sentence,
for this would result in the contradictory claim that some of the students that failed
to show up at the conference, did show up and presented a paper. The pronoun
instead refers to the domain of quantification of the antecedent sentence, namely
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.13
14
RICK NOUWEN
the set of (salient) students. This type of reference is sometimes called reference to
the maximal set, or, in short, maxset reference. A similar reading is available for
(21-c). As we can observe from (21-b) and (21-c), this type of anaphora can occur
both within and outside the scope of quantification. Presumably, this is so because
strong quantifiers presuppose their domain.
For universal quantifiers like that in (20), however, there is no difference between referring to the maxset (the set of climbers) or the set of climbers that made
it to the summit, since the sentence itself claims these two sets are identical. And
so, we could claim that (20) is a case of maxset reference, which given the fact that
this type of reference is felicitous within the scope of the quantifier, renders (20)
completely unproblematic.
It is quite easy, however, to come up with examples involving non-universal
quantifiers. For instance, the appositive in (22) is quantified without this leading to
the example being infelicitous.
(22)
Less than half the climbers, all French nationals, made it to the summit.
On at least one reading, this example says that less than half the climbers made it
to the summit and that all these successful climbers were French nationals. This
reading is comparable to a discourse anaphoric reading where a pronoun refers
to the intersection of restriction and scope, exactly the kind of anaphora I have
claimed did not exist within the scope of a quantifier. The problem is worse still,
since given the semantics for apposition we gave above, we would expect that
if felicitous at all, an example like (22) would result in an interpretation where
the appositive actively participates in the comparison between sets denoted by the
quantifier. In other words, we would expect, (22) to say that less than half the
climbers were French national who made it to the summit. Of course, this is not
what the example conveys.
A clue to the solution of this problem is the fact that the appositive in this case
is plural. This means it could never be bound by the quantifier, since the quantifier
only takes singular values into account. (Recall that strong quantifiers are distributive. Compare also to footnote 10.) This suggests that, initially, the appositive is
not co-indexed with the quantifier, and so ends up not being bound at all.
(23)
< 21 x[climber(x)][h; ∃y; french-pl(y); i; reachsummit(x)]
Reference resolution may then at a later stage connect the plurality formed by the
quantifier with the plurality described by the appositive. Only when this happens
after the quantificational sentence has been processed is this possible. So, (24-a)
results in an empty relation, because there is no way the plural value for y can be
identical to the singular value for x, but (24-b) gives us exactly the reading we are
after. (Here, ’french-pl’ is an abbreviation of a test on y that is successful only if y
is a plurality of French nationals).
(24)
a.
< 12 x[climber(x)][h; ∃y; french-pl(y); x = y; i; reachsummit(x)]
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.14
APPOSITIVES AND DYNAMIC BINDING
b.
15
< 12 x[climber(x)][h; ∃y; french-pl(y); i; reachsummit(x)]; x = y
Note that a similar strategy is not possible for singular appositives. The strategy
corresponding to (24-b) would in that case be out because of the clash between
singular y and plural x. The strategy corresponding to (24-a) would be equally
infelicitous because the appositive content contains the condition that the values of
x and y are identical. This identity cannot be maintained outside the scope of the
quantifier, however.
An approach like the one in (23) only works for nominal appositives, however.
As soon as we take a close look at appositive relatives, it becomes clear that not all
appositions are alike.
(25) ??Most students, who (all) arrived very late, brought a bottle of wine.
Del Gobbo, however, observes that the position of an appositive relative matters.
(26)
a. #Many students, who arrived very late, brought a bottle of wine.
b. They invited many students, who arrived very late.
(del Gobbo 2003a, p. 26)
According to (26-b), many students were invited and all of the invited students
arrived very late. An obvious problem with the contrast in (26), however, is that it is
not clear at all that many is a strong quantifier. It is well-known that one of the uses
of many is weak, as in There are many students here. This means that one could
argue that (26-b) happens to trigger the weak reading, whereas the subject position
of (26-a) results in a strong reading for many. But if we use an unambiguously
strong quantifier in many’s place, then it becomes apparent that the contrast in (26)
is a more general phenomenon.
(27)
a.
b.
Less than half the climbers, who (by the way) were (all) French
nationals, made it to the summit.
They interviewed less than half the climbers, who (by the way) were
(all) French nationals.
It seems that for (27-a), only a maxset reading is available (that is, saying that all
climbers are French), whereas for (27-b) a reading exists where the interviewed
climbers are all French. If this data is trust-worthy, then this may show that the
referential link between an appositive relative and its anchor is stronger than the
link between a nominal appositive and its anchor. What I mean is that relatives
seem to have to be co-indexed with their anchor.9 The 2D semantics developed
above then suggests a straightforward account of the contrast in (27), based on
the assumption that sentence-final appositives can be interpreted outside the quantificational structure. So, (27-a) and (27-b) correspond respectively to (28-a) and
(28-c).
(28)
a.
< 12 x[climber(x)][h; french-pl(x); i; reachsummit(x)]
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.15
16
RICK NOUWEN
b.
c.
< 12 x[climber(x)][reachsummit(x); h; french-pl(x); i]
< 12 x[climber(x)][reachsummit(x)]; h; french-pl(x); i
The form in (28-b) is a possible logical form for (27-b), but it is rejected because it
results in denoting the empty relation, just like (28-a). The form that corresponds to
(27-a) in its felicitous reading is (28-a) but with french-pl(x) replaced by predication
over whatever variable corresponds to the presupposed set of climbers.
6. Conclusion
The formalism presented in this paper facilitates a comparison between discourse
anaphora and quantification into appositives. The result is that if we stick to a
semantics that yields the proper predictions for anaphoric possibilities in discourse,
the most unshakeable observations on quantified apposition can be explained in
terms of multidimensionality and the sharing of referential resources. However,
as the last section shows, the landscape of referential possibilities is never easily
captured. I showed that the proposal allows for amendments that yield desirable
results. Undoubtedly, however, there will be extra-semantic aspects of apposition
that will not be accommodated so easily. Syntactic, discoursal and pragmatic constraints, and their interaction with semantics, have been largely overlooked in this
paper and have to be left to further research.10
Acknowledgements
Earlier versions of this paper have been presented at the DGfS workshop on Parentheticals in Bielefeld in February 2006 and the LENS workshop in Tokyo in
June 2006. I am indebted to the audience of these workshops for valuable feedback. I would like to thank Carla Umbach, Alastair Butler, Eric McCready, Linton
Wang, Christopher Potts and an anonymous reviewer for helpful comments and
thought-provoking critical remarks, some of which I was unfortunately not able to
address in this article. Any mistakes are mine. This work was funded by a grant
from the Netherlands Organisation for Scientific Research (NWO), which I hereby
gratefully acknowledge.
Notes
1
The use of separate dimensions in semantics is not new. The locus classicus is Karttunen and
Peters 1979; other examples include the focus-semantics literature based on Rooth 1985 and the
many-dimensional representation framework proposed by Geurts and Maier (2003).
2
For the sake of exposition, I focus on the propositional level here only. In a compositional
derivation, however, multidimensionality applies to any model-theoretic type.
3
One possible characterisation of the added dimension is to see it as a place where conventional
implicatures are stored. The link between appositives and conventional implicatures (Grice, 1975) is
discussed in detail by Potts (2005) (see also Jayez and Rossari 2005).
appositives_dynamic_rolc_versionbeta.tex; 3/07/2006; 14:32; p.16
APPOSITIVES AND DYNAMIC BINDING
17
4
Note that this definition predicts that any existential quantifier that may occur in the restrictor
ϕ or scope ψ yields maximal anaphora in discourse. This prediction is borne out by examples like:
“Every student in the Dutch class wrote a paper. {#It wasn’t/They weren’t} well-written, though.”
5
This can be demonstrated on the basis of the following examples from Kamp and Reyle 1993.
(a) Two lawyers (each) hired a secretary they interviewed.
(b) Most lawyers hired a secretary they interviewed.
The example in (a) is ambiguous. In the first reading, both lawyers hired a secretary he or she
interviewed. In the second, they each hire a secretary that was interviewed by both of them. This
ambiguity involves the possibility of the plural pronoun ‘they’ to be construed as a collective subject
for ‘interview’. In other words, in the first reading the secretary was interviewed by the lawyer that
hired him, while in the second he was interview by the two lawyers. In contrast to (a), (b) lacks
this latter reading. One cannot use (b) to convey that a majority of lawyers each hired a (different)
secretary they collectively interviewed.
6
This final condition might seem a superfluous at first sight, but it is not. Without it the dynamics
of the non-actual dimension will be completely lost. The test whether the quantificational relation
holds is fully based on the summation of successful assignment functions. But, if we happen to
evaluate this relation with respect to a possibility in which the appositive content within the quantifier’s scope is false, this could potentially verify the quantificational statement. For instance, an
example like “No friend of Sue, who by the way is Cody’s sister, is rich” should not turn out true in
a possibility in which all of Sue’s friends are rich, but Sue is not Cody’s sister. The final condition in
the semantics for quantified sentences eliminates such possibilities.
7
For ease of exposition, I assume that the ‘one’-anaphor can be interpreted as a repetition of the
noun phrase ‘Dutch boxer’.
8
In (29), I represent the appositive as part of the nuclear scope of the quantifier. It is not obvious at
all that this is correct. However, for the analysis that follows this choice is immaterial. What matters is
that both the appositive and the host sentence give conditions on the same variable, which eventually
leads to a clash.
9
Assuming that the anchor carries an index for both the maximal set (the domain of quantification)
and the running variable of quantification, this predicts that relatives are either bound or domainreferring.
10
One of the most pressing issues appears to be the interaction of apposition with presupposition.
Eric McCready and Linton Wang (p.c.) provided me with the following example: “John, a good
tennis player, is a good golfer too”. Admittedly, it is not immediately clear how such examples are to
be treated in the dynamic formalism provided here. Much will depend, however, on the more general
question of what exactly the pragmatic status of appositive content is, which is an issue I have not
commented on at all in this paper.
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