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\begin{document}

\title{Ambiguity and Coherence}
\author{Alex Lascarides,\\
{\small Centre for Cognitive Science,}\\
{\small University of Edinburgh,}\\
{\small 2, Buccleuch Place,}\\
{\small Edinburgh, EH8 9LW,}\\
{\small Scotland, UK}\\
{\small {\tt alex@cogsci.ed.ac.uk}}
\and
Ann Copestake\\
{\small CSLI}\\
{\small Stanford University,}\\
{\small Cordura Hall,}\\
{\small Stanford, CA 94305,}\\
{\small USA}\\
{\small {\tt aac@csli.stanford.edu}}
\and
Ted Briscoe\\
{\small IRCS, University of Pennsylvania,}\\
{\small Suite 400C, 3401 Walnut Street,}\\
{\small Philadelphia,}\\
{\small PA 19104--6228,}\\
{\small USA}\\
{\small {\tt ejb@linc.cis.upenn.edu}}}

\maketitle

\begin{abstract}
Several recent theories of linguistic representation
treat the lexicon as a highly structured
object, incorporating fairly detailed semantic information, utilising
various types of relationships between entries, especially inheritance,
and allowing multiple aspects of meaning to be represented in a
single entry (e.g. Pustejovsky, 1991; Copestake, 1992; Copestake
and Briscoe, 1995).  One consequence of
these approaches is that word senses cannot be thought of as discrete
units which are in one-to-one correspondence with lexical entries.  
This has many advantages in allowing an account of systematic polysemy
and of graded distinctions between senses (Cruse, 1986, 1995), but leaves the
problem of accounting for effects such as zeugma and
the absence of crossed readings, which have traditionally been
explained in terms of multiple lexical entries, but which can also
arise in examples where other criteria demand that a single lexical
entry be involved.
Copestake and Briscoe (1995) claimed that these cases could be
explained in terms of discourse coherence, but did not
describe how this might work.  We remedy this here, by developing a general
account of zeugma and the non-availability of crossed readings.
\end{abstract}

\section{Introduction}

This paper is part of a general attempt to
explicitly and formally investigate
interactions between the lexicon and pragmatics.   
We assume that lexical and linguistic processing 
is informationally encapsulated and 
utilises relatively simple `taxonomic' lexical semantic knowledge,
in contrast with pragmatic reasoning which
is open-ended and involves arbitrary
knowledge and interactions.
Previous papers have discussed how pragmatics affects
lexical processes such as
metonymy (Lascarides, 1995; Lascarides and Copestake, 1995), disambiguation 
(Asher and Lascarides 1995a, Asher and Sablayrolles 1995),
metaphor (Asher and Lascarides, 1995b)
and blocking (Briscoe {\em et al}, 1995).  
Here we extend that work by formalising a general principle of
discourse interpretation
which captures the
intuition that hearers prefer not to infer
things that have an adverse effect on the quality of coherence in the
discourse.  In particular, this principle formulates how discourse
coherence can constrain how words in the discourse are interpreted.
One effect of this is to give an account of 
some rhetorical effects, which have frequently been used as tests
for lexical ambiguity (e.g. Zwicky and Sadock, 1975).

The traditional picture of lexical organisation
is that distinct word senses should be represented as discrete units
(e.g.~Kempson, 1977:81f).
This is taken to account for the markedness of (\ref{stockings})
which demonstrates the figure of speech 
traditionally known as zeugma, 
and also for the impossibility of crossed readings: (\ref{file1}) and
(\ref{file2}),
for example, cannot have readings where John has a tool designed 
for shaping and Bill has a folder of documents.\footnote{We use {\em sense} 
pretheoretically, to correspond to something like the
distinctions which might be made by lexicographers. We use zeugma to
refer to a construction in which a single word is applied to two (or
more) others in such a way that 
its sense(s) are appropriate to each in different ways.} 
\begin{examples}
\item   \label{stockings}
He was wearing a pair of women's stockings 
and a look of considerable 
embarrassment. \\
(adapted from Cruse, 1986)
\item   \label{file1}   John and Bill each have a file.
\item   \label{file2}   John has a file and Bill does too.
\end{examples}

Copestake and Briscoe (1995) argue that an account of polysemy based on
distinct lexical entries alone
is inadequate because it
does not adequately capture the systematic nature of much polysemy
and does not allow for gradations in distinctions (see e.g. Cruse, 1986, 1995).
Furthermore, in many cases it seems that multiple aspects of a word's
meaning should be encoded in a single lexical entry --- for example,
{\it book} can be treated as having a dual nature as both a physical
object and an abstract representation, which are both encountered in
examples such as (\ref{book}a,b).
\begin{examples}
\item \label{book}
The books on the top shelf are about syntax.
\item   \label{thesis0}
That thesis has thousands of pages and is quite unreadable.
\end{examples}
Such examples (which are are classified as cases of {\em constructional
polysemy} by Copestake and Briscoe)
cannot be easily accounted for if we assume multiple lexical
entries for {\it book} and {\it thesis}. 
Some recent theories of the lexicon 
attempt to account for constructional polysemy
and other cases of systematic polysemy 
by treating the lexicon as a highly structured
object, incorporating fairly detailed semantic information, utilising
various types of relationships between entries, especially inheritance,
and allowing multiple aspects of meaning to be represented in a
single entry (e.g. Pustejovsky, 1991, 1995; Copestake, 1992; Copestake
and Briscoe, 1995).
However, if the lexical entries are not distinct, we are left with no
formal explanation of the oddness of sentences such as (\ref{zbook})
and the absence of crossed readings in (\ref{cbook}), which 
cannot refer to a situation where
Kim sold two works to a publisher and Sandy sold
two spare paperbacks to a book dealer. 
\begin{examples}
\item \label{zbook} Kim's thesis is orange and unreadable.
\item \label{cbook} 
Kim and Sandy both sold two books.
\end{examples}

Copestake and Briscoe claimed that such cases could be
explained in terms of discourse coherence, but did not
describe how this might work.  
In this paper, we will provide an account of zeugma and the 
absence of crossed readings which will cover examples such as those shown.
We will claim that although the existence of a multiple lexical entry
({\sc mle}) is part of the explanation for some cases of
zeugma and non-crossed readings, inferences about pragmatic
interpretation and discourse coherence can cause this effect, even
when there is a {\em single} lexical
entry.  Our account will thus complement and expand the traditional one.

We will first discuss zeugma and crossed readings and their 
use as a test for ambiguity, briefly discuss the traditional
{\sc mle} approach 
and illustrate its shortcomings as an account of these phenomena. 
We will then advocate a complementary
approach, which involves recourse to pragmatic reasoning,
constraints on discourse coherence and psychological models of
incremental interpretation.  This revised account will extend the {\sc
  mle} approach by explaining how zeugma and non-crossed readings can
occur in the absence of distinct lexical entries.  
We describe this informally in \S\ref{inf} by means of some examples.
By studying how information
in the discourse leads to zeugma and non-crossed readings, we
will come to some general conclusions about the way in which lexical
organisation should be linked to other components of the linguistic
interpretation model.
The theories of discourse structure and pragmatics are
introduced in the next section, followed by a formal account of these
examples in \S\ref{formal}.


\section{Zeugma and crossed readings}
\label{mle}

For the purposes of this paper, we will assume that zeugma is a word
pun or a figure of speech, in which two terms are inappropriately
linked together.
Sentences (\ref{bank1}) to (\ref{eg6})
are all examples of zeugma, and they indicate
that it can occur with 
a wide variety of syntactic constructions.
\begin{examples}
\item   \label{bank1}   Some dam busters blew up banks and so did some
bank robbers.
\item   \label{plane}   I tried to take the plane to 
Chicago, but it was too heavy.\\
{\em Groucho Marx}
\item   \label{eg4}     Heseltine left with a smile, a wave 
and his wife.\\
{\em BBC news}
\item   \label{eg5}     Mr. Pickwick took his hat and his leave.\\
{\em Dickens, Pickwick Papers}
\item   \label{eg6}
That whole committees of people, whole corridors and floors of
professionals thought it worth making defies the licence fee and
belief.\\
{\em A. A. Gill, The Sunday Times, January 29th, 1995}
\end{examples}
In all of these sentences, a single occurrence of a word is used in
incompatible ways.  In the attributed examples, this is being
done deliberately as a pun.   In the {\sc mle} account,
the zeugma in all of these cases would have arisen from the use of an
ambiguous lexical item: {\it bank} in (\ref{bank1}),
{\it take} in (\ref{plane}) and (\ref{eg5}), {\it with} in 
(\ref{eg4}) and {\it defy} in (\ref{eg6}).  In (\ref{eg5}) and 
(\ref{eg6}) one of the usages is part of an idiom ({\it
take one's leave}, {\it defy belief}).

It is important to be careful about which contexts are used to test
for zeugma.  It is clearly inappropriate to use contexts where
contradictory statements are made of the same entity.  Thus sentence
(\ref{teacher1}) is inconsistent, but not zeugmatic:
\begin{examples}
\item \label{teacher1}The teacher was pregnant and had a beard.
\end{examples}
Rather, 
zeugma must arise from linking two mutually
satisfiable assertions.  For example, it's possible to 
bank money and then bank a
plane, but these assertions are inappropriately linked in
(\ref{bank20}), resulting in zeugma.
\begin{examples}
\item \label{bank20} John banked the money and then the plane.
\end{examples}

Zeugma and the absence of
crossed readings are often taken to be
the most reliable tests for lexical ambiguity 
as opposed to vagueness or generality of meaning (e.g., Cruse 1986,
Zwicky and Sadock 1975).
So, for example, the fact that we don't interpret (\ref{file1}) and
(\ref{file2}) as John having a dossier and Bill a tool
(even though they might possess different physical objects)
indicates that {\em file} is ambiguous
between the sense meaning {\em dossier} and that meaning {\em tool}.
\begin{examples}
\item   [\ref{file1}]   John and Bill each have a file.
\item   [\ref{file2}]   John has a file and Bill does too.
\end{examples}
So
on the assumption that lexical ambiguity implies
{\sc mle}s, (\ref{file1}) and (\ref{file2}) indicate {\sc mle}s
for {\em file}.
Similarly, (\ref{bank2}) is zeugmatic and (\ref{bank30}) doesn't
permit a non-crossed reading.  So
the word {\em bank} has {\sc mle}s; one for its mound of earth
sense, and one for its financial institution sense.
\begin{examples}
\item   \label{bank2}   Some dam busters and bank robbers blew up 
banks.
\item   \label{bank30}  Texas and Alabama have preservation orders on their
most beautiful banks.
\end{examples}
On the other hand, {\em teacher} is neutral or general between male and female
{\em teachers} in (\ref{teacher2}b) can refer to a mixture of male and
female teachers.
\begin{examples}
\item \label{teacher2}
\begin{subexamples}
\item   Teachers may take maternity or paternity leave.
\item   John and Bill both like their teachers.
\end{subexamples}
\end{examples}
And because {\em brush} in (\ref{generalbrush}) can
refer to a mixture of paint brushes, cleaning brushes, tooth brushes
and so on, there must be a lexical entry for {\em brush}, which is
neutral with respect to the type of brush being referred
to.
\begin{examples}
\item \label{generalbrush} Nylon bristles and plastic
handles make cheap durable brushes.
\end{examples}

Zeugma is often taken to be symptomatic of lexical ambiguity as
opposed to lexical vagueness, but the concept of vagueness is
heterogeneous. In a multiple inheritance hierarchy, multiple aspects
of meaning can be combined in a single entry and since lexical entries
themselves are not simply listed but are hierarchically arranged, a
general entry can be given for a word usage which subsumes more
specific usages. The neutral or general senses discussed above are
treated in this way. Their underspecified senses can be specialised by
linguistic context, and therefore they are termed constructionally
polysemous in Copestake and Briscoe (1995).  Below we argue that this
(sub)type of lexical vagueness can lead to zeugma.  This contrasts
with the {\sc mle} view of
zeugma, as being caused by distinct senses of a word that
lack 
a more general subsuming sense (e.g., {\em bank}, {\em file}). Below we give
some detailed arguments for this, but first we review the {\sc mle}
explanation.

\subsection{The MLE account of zeugma and the absence of crossed
readings}

The {\sc mle} explanation of zeugma and absence of crossed
readings goes roughly like this.
First, there are at least two entries for {\em bank} in the lexicon:
call these {\em bank}$_{\mbox{\tiny ground}}$ 
(the mound sense) and {\em bank}$_{\mbox{\tiny org}}$
(the financial organisation sense).  Moreover, there is no entry for
{\em bank} which is general between these two senses.  So there are
two 
possible logical forms, for both (\ref{bank2}) and (\ref{bank30}),
which in each case are identical except for the sense of {\it bank}
involved.
  
We
give these below in Discourse Representation Theory ({\sc drt})
(Kamp and Reyle, 1993), since our account of zeugma will be formally
explicated using {\sc drt}.  
Throughout this paper we'll assume that logical forms are
built via a constraint-based grammar such as Head Driven Phrase
Structure Grammar ({\sc hpsg}) (Pollard and Sag, 1994), with a
semantic component encoded in {\sc drt} (cf. Frank and Reyle,
1995), and that
the process by which the {\sc drs} is built is monotonic.\footnote{It
should be stressed at this stage that the
way we represent multi-sentence discourse will go beyond {\sc drt}, in
that we will use rhetorical relations such as {\em Parallel} and {\em
Contrast} to connect {\sc drs}s together.  We do this so that
coherence constraints on these relations can predict zeugma.
Furthermore, we assume that the rhetorical
relations that connect {\sc drs}s are computed via
the semantic content of those {\sc drs}s.  This means that 
the semantic representation of a
clause must be evaluated against a model before we reason about how
it connects to the semantic representation of the text so far.  This
is in further contrast to traditional {\sc drt}, where evaluation
against a model doesn't occur until the semantic representation of the
whole discourse is built.}
\begin{examples}
\item   [\ref{bank2}$'$]
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, bank-robber(x$_1$),
  dam-buster(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t}{bank$_{\mbox{\tiny ground}}$(y)\\ blow-up(e,x,y)\\
  hold(e,t)\\ t$\prec$now}}
\item   [\ref{bank2}$''$]
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, bank-robber(x$_1$),
  dam-buster(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t}{bank$_{\mbox{\tiny org}}$(y)\\ blow-up(e,x,y)\\
  hold(e,t)\\ t$\prec$ now}}
\item [\ref{bank30}$'$]
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, Texas(x$_1$),
  Alabama(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t,z}{bank$_{\mbox{\tiny ground}}$(y)\\ pres-order(z) \\
  have(e,x,y,z)\\ 
  hold(e,t)\\ t$\subseteq$ now}}
\item [\ref{bank30}$''$]
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, Texas(x$_1$),
  Alabama(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t,z}{bank$_{\mbox{\tiny org}}$(y)\\ pres-order(z) \\
  have(e,x,y,z)\\ 
  hold(e,t)\\ t$\subseteq$ now}}
\end{examples}
We now immediately see why {\em banks}
in (\ref{bank30}) doesn't
refer to a mixture of earth mounds and financial institutions:
neither logical form would be satisfied in this case.  
There is no possibility of a single occurrence of {\em bank} in a sentence
having both readings simultaneously, because there is no lexical entry
for {\em bank} which is general between the senses.
Similarly the logical forms of (\ref{bank2}) lead to zeugma.
Intuitively, the explanation runs along the following lines.
As with lexical ambiguity in general, the interpreter uses
pragmatic information in an attempt to decide whether
(\ref{bank2}$'$) or (\ref{bank2}$''$) corresponds to the natural
interpretation of
(\ref{bank2}).  Knowledge
about bank robbers
favours interpreting {\em bank} as {\em bank}$_{\mbox{\tiny org}}$, 
so this favours
the logical form
(\ref{bank2}$''$) over (\ref{bank2}$'$).  But, on the other hand, 
knowledge about dam busters favours interpreting {\em bank} as
{\em bank}$_{\mbox{\tiny ground}}$, and so this favours (\ref{bank2}$'$) over
(\ref{bank2}$''$).  The interpreter cannot decide which of these two pieces
of knowledge should take precedence and this irresolvable
conflict in pragmatics produces a word pun effect: the conflict means
the semantic ambiguity can't be resolved.
{\sc mle}s alone do not provide a formal framework in which
this pragmatic line of reasoning can be modeled.  
To do this it is necessary to link
the compositional semantics to a logic for pragmatic
reasoning, which predicts irresolvable conflict in appropriate cases.
We will describe such a logic in \S\ref{logic}.

In contrast to (\ref{bank2}), 
the predicate for {\em teacher} in logical form will be
non-committal as to whether the teacher referred to is male or female.
So there will be single
logical forms for (\ref{teacher2}a) and (\ref{teacher2}b) respectively.
Therefore (\ref{teacher2}a) won't be interpreted as
zeugmatic and (\ref{teacher2}b) will permit the domain of teachers
that is quantified over
to include a mixture of both male and female teachers.

If the connection with
pragmatics can be formalised,
the {\sc mle} account of zeugma and non-crossed readings appears to
be adequate for homonymous words such as {\em bank}, where the
financial institution and mound (of earth) senses are unrelated, and it is
a coincidence that they share the same word form. 
The results of
the ambiguity tests seem 
quite clear cut in these cases.  But although examples like this
are often taken as standard cases of ambiguity, they are actually
quite infrequent compared to the cases where senses are related.  For
example, {\em bank} also has senses which correspond
to the building used by a
financial institution, the financial institution itself, the act of
depositing money and the stock of tokens or money used in some games.
At least the first two of these senses are regular and
predictable, from
the financial organisation sense,
in that similar polysemy is found for 
other organisation denoting nouns or noun compounds
such as {\em company}, {\em building society}, {\em credit union} and
so on (Copestake and Briscoe 1995).

Unfortunately, zeugma and non-crossed readings often do not provide reliable
ambiguity tests when applied to related senses; nor do {\sc mle}s
provide satisfactory explanations of
zeugma and non-crossed readings for
polysemous words.  We will demonstrate this by discussing some problems
with the {\sc mle} approach.

\subsection{Problems with tests}
\label{aspects}

As is well known, ambiguity tests are difficult to apply consistently. 
Some of the problems point to inadequacies in the {\sc mle} account of zeugma.
The zeugma and crossed readings tests can conflict with
other tests for {\sc mle}s.  Consider the
sentences in (\ref{brush1}):
\begin{examples}
\item   \label{brush1}
\begin{subexamples}
\item   Nylon bristles and plastic handles are used to make cheap 
brushes.
\item   Rembrandt used a brush and so did our janitor.
\item   Rembrandt and our janitor used a brush.
\item   The grandparents gave the two children brushes for 
Christmas.
\end{subexamples}
\end{examples}
The word {\em brush} is being used quite generally in (\ref{brush1}a),
indicating that even if there are separate entries in the lexicon for
{\em paint brush}, {\em hair brush}, and so on, there must also be a
lexical entry that is general between these items.  But in contrast,
according to the zeugma and crossed readings
tests, examples (\ref{brush1}b,c,d) indicate no such entry for {\em
  brush} which is general in this way.
For example, (\ref{brush1}b) and (\ref{brush1}c)
are both zeugmatic.  But if there were a
single lexical entry for {\it brush} they would each have a single logical
form.  So the {\sc mle}
explanation of zeugma we outlined above would not apply in this case.
Similarly, (\ref{brush1}d) does not have a mixed reading, but there is
nothing to prevent this under the {\sc mle} account.
So the results of the zeugma and crossed reading tests for
ambiguity conflict with the evidence of (\ref{brush1}a).

As another example, consider {\em thigh}, which by default might be
taken to refer to part of the human body.  But it is possible to use
it in a more general way, as illustrated by (\ref{thigh}a).  In
contrast, while discussing a barbecue, {\em thigh} might be understood
to refer to a piece of chicken meat, and in this context
(\ref{thigh}b) involves a pun.
\begin{examples}
\item   \label{thigh}
\begin{subexamples}
\item   Thighs are the most muscular part of the vertebrate 
hind limb.
\item   Bill likes mustard on his thighs but John prefers sun 
tan lotion.
\end{subexamples}
\end{examples}
Again, the evidence for ambiguity provided by the zeugma test
conflicts with the other evidence.  

It appears that the  zeugma in (\ref{brush1}b,c) and
(\ref{thigh}b) must be explained with more complex machinery than the
{\sc mle}s alone.  (\ref{brush1}a) and (\ref{thigh}a)
are evidence
that there are general entries for {\em brush} and for {\em thigh}, which
if used to represent the logical forms of these sentences 
would predict that there was no
zeugma.  So, there must be some reason why {\em brush} and {\em thigh}
are not simply left underspecified in the 
representation of the sentences (\ref{brush1}b,c) and (\ref{thigh}b).
This idea can be fleshed out
when one considers how real world knowledge and constraints on discourse
coherence affect
interpretation.  If Rembrandt is using a
brush, then by default, he is using a paint brush, and if a janitor is
using a brush, then by default it is a cleaning brush.  If one models
this specialisation of {\em brush} to {\em paint brush}
in sentences like {\em Rembrandt used a brush},
then one can explain
why (\ref{brush1}b) is zeugmatic: there are
conflicting messages about specialisation.  Note that this account of
zeugma does not appeal to distinct lexical entries for {\em
  paint brush} and {\em cleaning brush}; it appeals to {\em pragmatic}
knowledge resources, and their
effect on the interpretation of words like {\em brush} in a discourse
context.   The general idea is that even though there is a
single lexical entry for {\em brush}, and so sentences like
(\ref{brush1}c) are not semantically ambiguous, one still gets irresolvable
conflict in the pragmatic component---namely, conflict about how the
sense of {\em brush} is specialised---making (\ref{brush1}c)
{\em pragmatically} ambiguous.
Conflicting clues about specialisation will also explain the pun
effect in (\ref{thigh}b).

Another slightly different set of examples
involve contexts where two related senses
of a polysemous word can either yield zeugma or not, depending on the
particular assertions being made, and how those assertions are
presented.  For example, 
words such as {\em thesis} and {\em book} systematically
have usages which refer to the 
physical object, but can also be used to refer to the abstract contents
(as illustrated in (\ref{book})).
This gives rise to zeugma in some sentences
and not in others: (\ref{thesis1}a) and (\ref{thesis1}b) involve
predication over both the physical object ({\em has thousands of
  pages} and {\em is orange}) and the content ({\em is unreadable}).
And yet (\ref{thesis1}a) and (\ref{thesis1}c) are better
than (\ref{thesis1}b):
\begin{examples}
\item   \label{thesis1}
\begin{subexamples}
\item   That thesis has thousands of pages and is unreadable.
\item   That thesis is orange and unreadable.
\item   That orange thesis is unreadable.
\end{subexamples}
\end{examples}
So, according to the zeugma test, (\ref{thesis1}a) indicates that there
is a lexical entry for {\em thesis} that is general between physical
object and content, whereas (\ref{thesis1}b) indicates that there
isn't any such lexical entry.
Under standard assumptions about coordination and constraint based
processing, it is impossible to account for (\ref{thesis1}a) if there
are two different lexical entries for the different usages of {\it thesis}.
Pustejovsky (e.g. 1991) and Copestake and Briscoe (1995) have argued
that a single lexical entry can be used to represent both aspects of
the word.
Briefly, these accounts claim that 
lexical entries for nouns like
{\it thesis}
contain distinct attributes for form and constituency, which 
in the case of {\it thesis} will reflect the 
physical object / abstract content duality.  Different predicates 
select for different attributes --- for example, colour-denoting predicates
such as {\it orange} select for the value of the form attribute, while
{\it unreadable} selects for the contents.

Given this, one 
possible explanation of the contrast between (\ref{thesis1}a,b) 
is that even though both involve copredication---i.e., conjunction of
predicates---on different aspects
of {\em thesis}, the size of books is clearly relevant to their
readability, and the colour of their covers is not.  In other words,
whether coordinating constituents is zeugmatic or not will depend not only on
whether the lexical structure permits such co-predication, but also on
how the assertions rhetorically link together, to
contribute to the coherence of discourse.  In this case, a close
rhetorical connection can be found in (\ref{thesis1}a) but not in
(\ref{thesis1}b), because in (\ref{thesis1}a), the causal connection
between a book's size and its readability is inferrable.  In contrast,
(\ref{thesis1}c) doesn't involve co-predication, so
there is no need to find a rhetorical relation
and coherence constraints on rhetorical relations don't
influence interpretation.  We will discuss this approach in more detail
in \S\ref{dicoex} and formalise it in \S\ref{formal}.

\subsection{Word Repetition}
\label{wordrep}

The {\sc mle} approach also fails to
explain why zeugma and the absence of
crossed readings tend to remain when the word form
is repeated, as in (\ref{bank3}) and (\ref{file3}) (see e.g. van
Deemter 1990):
\begin{examples}
\item   \label{bank3}   John banked the money and then
banked the plane.
\item   \label{file3}   John had a file and Bill had a file.
\end{examples}
The word pun effect in (\ref{bank3}) is not as pronounced as in
(\ref{bank2}), but there is still a pun effect which the {\sc mle} account
fails to predict.
According to the {\sc mle} approach, (\ref{bank3}) has
four possible logical forms since each of 
the two occurrences of {\it bank} could correspond to either
{\em bank}$_{\mbox{\tiny turn}}$ or
{\em bank}$_{\mbox{\tiny deposit}}$.
Similarly, (\ref{file3}) also has four logical forms.  In
the case of
(\ref{bank3}) the knowledge that you 
{\em bank}$_{\mbox{\tiny deposit}}$  money
rather than {\em bank}$_{\mbox{\tiny turn}}$ 
it and that you {\em
  bank}$_{\mbox{\tiny turn}}$ 
planes rather than {\em bank}$_{\mbox{\tiny deposit}}$ 
them favours the logical
form where the predicate which takes the money as an argument is {\em
  bank}$_{\mbox{\tiny deposit}}$ 
and the predicate which takes planes as an argument is
{\em bank}$_{\mbox{\tiny turn}}$.  
So, unlike (\ref{bank2}), there is no irresolvable
conflict in pragmatics induced by the constraints imposed by logical
form, and indeed, pragmatics 
successfully disambiguates the sentence.  So,
contrary to intuitions, the {\sc mle} account predicts that
(\ref{bank3}) isn't zeugmatic.
Furthermore, two of the candidate logical forms for (\ref{file3})
invoke crossed readings, but intuitively these seem to be excluded, or,
at least, strongly dispreferred.  

The fact that (\ref{bank3}) is 
zeugmatic and the interpretation of (\ref{file3}) isn't crossed must
be explained by something other than the distinct
lexical entries for the various senses of {\em bank} and {\em file}.
We will claim that the natural interpretations of (\ref{bank3}) and
(\ref{file3}) essentially involve constraints on
{\em discourse coherence} imposed by the way the phrases are rhetorically
linked together.  
And it is these discourse coherence constraints that ultimately
contribute
to the zeugma in (\ref{bank3}) and the absence of crossed readings in
(\ref{file3}).  For example in (\ref{file3}), the rhetorical
link between the two clauses is closer
under the non-crossed interpretation than the
crossed one.  One can see this intuitively by considering the topic of
the discourse.  If both John and Bill have a dossier sense of {\em
  file}, then the topic is {\em humans have a dossier}.  If John has a
dossier and Bill a tool, the topic is much more general: {\em humans
  have a physical artefact}, for instance.
One prefers interpretations where discourse constituents
are as closely connected as possible---this is a manifestation of
Grice's Maxim of Relevance (Grice 1975, Sperber and Wilson, 1986),
which can roughly be paraphrased as: the current 
constituent is as relevant as possible to the discourse context,
subject to the constraints imposed by the cost of processing and inference.
And so this Relevance heuristic of discourse interpretation explains 
why the non-crossed reading of (\ref{file3}) is preferred.  

In
(\ref{bank3}), the two bits of real world knowledge concerning banking
money and banking planes don't work independently, because 
there are constraints on how the two constituents ({\em
John banked the money} and {\em John banked the plane})
are rhetorically linked together.
There is a closer
rhetorical connection between the two constituents if the senses of {\em
  bank} are the same for the two word forms.  Again, this can be seen
by considering the specificity of the topic of the discourse: it is
{\em John banking things} if the senses of {\em bank} are the same,
compared with {\em John doing things} if they're different.   The
Relevance
heuristic of aiming for as close connections as possible between the
clauses creates a feeling of irresolvable
conflict among the various knowledge resources recruited, 
as in (\ref{bank2}).  To obtain an adequate
rhetorical connection one must drop the pragmatic implicature
that the money is
bank$_{\mbox{\tiny deposit}}$ed, 
or the plane is bank$_{\mbox{\tiny turn}}$ed, so as to ensure that both
senses of {\em bank} are the same.  But one doesn't know which
implicature to drop.  So even though
there is word repetition, thereby yielding a
logical form for (\ref{bank3}) which permits 
crossed readings, this is blocked by pragmatic considerations
concerning discourse coherence.  So, whereas (\ref{bank2}) doesn't permit a
mixed interpretation of {\em bank} on semantic grounds,
(\ref{bank3}) rules it out on pragmatic grounds.  And both
(\ref{bank2}) and (\ref{bank3}) yield
irresolvable ambiguity.
We will give a detailed, formal account of how
this discourse machinery can complement the {\sc mle} approach, so as
to predict the correct interpretations of (\ref{bank3}) and
(\ref{file3}) in \S\ref{formal}.

\section{Introducing a Pragmatic Approach}
\label{inf}

We have noted several problems with the {\sc mle} account of zeugma and
non-crossed readings.
We have claimed that it is necessary to
complement the {\sc mle} account with a model of how
pragmatic knowledge resources
and discourse structure contribute to the overall interpretation.  
We have also shown that zeugma can occur even in the
absence of {\sc mle}s.  In this section we will illustrate three
examples of zeugma in the absence of {\sc mle}s and illustrate
informally how they arise in slightly different ways
from the interaction of lexical semantics and
pragmatics. 


\subsection{Weak coherence}
\label{dicoex}

In (\ref{thesis1}a), {\em has thousands of pages} is true of the
physical thesis, whereas {\em unreadable} predicates of the content
of the thesis.
\begin{examples}
\item   [\ref{thesis1}]
\begin{subexamples}
\item   [a]     That thesis has thousands of pages and is unreadable.
\item   [b]     That thesis is orange and unreadable.
\item   [c]     That orange thesis is unreadable.
\end{subexamples}
\end{examples}
This can be achieved with a single lexical entry that has 
conventionalised aspects of an appropriate type, as outlined in
\S\ref{aspects}. 
But we suggested above that
this is not by itself a sufficient condition for acceptability.  
We must also find a
rhetorical connection between the coordinated
constituents in the {\sc vp}.  Intuitively, this is a way of checking
that the Gricean Maxim of Relevance is being adhered to.

In (\ref{thesis1}a), real world knowledge can be used to infer a causal
link between the number of pages of the  thesis and its
readability.  This causal link permits us to link the constituents
{\em the thesis has thousands of pages} and {\em the thesis is unreadable}
with the rhetorical relation {\em Result}.  They are also
linked with {\em Parallel}, because of the structural and semantic
relationship between the conjuncts (Scha and Polanyi 1988, Thompson
and Mann
1987, Asher 1993a).  

Inferring {\em Parallel} for
(\ref{thesis1}b) is also possible, since the structural and semantic
relationship is similar.  But one cannot infer {\em Result} anymore,
because our real world knowledge does not support a causal relationship
between the constituents.

There are constraints that must be met for
a {\em Parallel} relationship to be coherent (Asher, 1993a).  
One is that a common
theme between the constituents
must be found.  This theme must be a proposition which: (a) is
entailed by those that are connected together with {\em Parallel}, and
(b) is simple in the sense that it is described by a single predicate
argument structure
(e.g., {\em x has physical properties} is simple whereas {\em x has an
orange cover and x has thousands of pages} is not).  One can compute
the theme by generalising the predicates and arguments of the
propositions being connected by {\em Parallel} (cf. Grover {\em et
  al}, 1994).  In this case, the common theme is very general relative
to the semantic content of the constituents being connected.  This
is because
the only things that the predicates have in common are {\em properties
  of the thesis}.  Compare this with a
more specific theme such as {\em properties of the physical aspects
  of the thesis}, that would accompany a sentence like {\em That
  thesis is orange and has thousands of pages}.  
This general common theme makes the rhetorical
connection in (\ref{thesis1}b)
weak.  The connection with (\ref{thesis1}a) was stronger,
because although the {\em Parallel} relation had the same general
common theme, there was also the {\em Result} relation which linked
the constituents.  So (\ref{thesis1}a) is better than
(\ref{thesis1}b) because the rhetorical relation is stronger.

In contrast, we don't need a rhetorical connection
for (\ref{thesis1}c) at all, since the {\sc vp} does not contain
coordinated constituents.  The adjective {\em orange} in the {\sc np}
can simply be used to ensure that the {\sc np} successfully refers,
rather than requiring this property of being
orange to have some relevant connection to the
property in the {\sc vp}.
And therefore coherence constraints on
rhetorical relations play no part in predicting whether
(\ref{thesis1}c) is acceptable or not.  Rather, having a single
lexical entry for {\em thesis} that has conventionalised aspects of an
appropriate type will suffice.   

It is important to stress that this informal explanation of the
difference in acceptability between the sentences in (\ref{thesis1})
uses a single lexical entry for {\em thesis} combined with
pragmatic reasoning techniques.  We will formalise this line of
reasoning in \S\ref{formal}.

\subsection{Irresolvable conflict}
\label{ircoex}

(\ref{brush1}a) is evidence that there is an
entry for {\em brush} which refers to brushes in
general, and covers paint brushes, cleaning brushes, hair brushes and
so on.  We must explain how even so, zeugma occurs in (\ref{brush1}c).
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [a]     Nylon bristles make cheap durable brushes.
\item   [c]     Rembrandt and our janitor used a brush.
\end{subexamples}
\end{examples}
Assume that the general entry for {\em brush} is used to build the
logical form for (\ref{brush1}c).  So in contrast
to (\ref{bank2}), (\ref{brush1}c) has a single logical form
where reference to different kinds of brushes used by the agents is
permitted.  As in
(\ref{thesis1}a,b), we must calculate the rhetorical connection
between the propositions associated with the items linked by the
coordination, in order to ensure that Grice's Maxim of Relevance is
satisfied.  In the case of (\ref{brush1}c), this entails finding a
rhetorical connection between {\em Rembrandt used a brush} and {\em
  our janitor used a brush} (since the reading being considered here
is distributive).  The only candidate relation is {\em Parallel}
because of the semantic content and structural properties of these
constituents, and this rhetorical relation comes with a coherence
constraint: there must be a common theme.

Now, these two propositions verify two default laws from real world
knowledge: the first is that Rembrandt, if he's using a brush, is
normally using a paint brush; and the second is that our janitor, if
he's using a brush, is normally using a cleaning brush.  We will
assume that the common theme is determined by the {\em pragmatic}
interpretation of the constituents as opposed to only their truth
conditional content, since themes are essentially a pragmatic property
of the discourse.  So
suppose these
defaults from real world
knowledge were to fire.  Then we would be attempting to compute a
common theme between the propositions {\em Rembrandt used a
  paint brush} and {\em Our janitor used a cleaning brush}.  

Using generalisation on the propositions, the common theme
is {\em humans use a painting or cleaning brush} (to the exclusion of
all other kinds of brushes).  This is a bad theme, because it violates
Grice's maxim of Quantity.  Quantity stipulates that one must use
referring expressions that are neither too general nor too specific
for the purposes of the discourse.  But in this discourse context, we cannot
calculate why the discourse is about painting or cleaning brushes, but
not any other kind of brushes (i.e., not brushes in general), and not
about painting brushes alone, or cleaning brushes alone.  
So it seems
that a referring expression has been used that is either not specific
enough, or not general enough.
In other words, this theme involves an interpretation of {\em brush}
that violates Quantity.

The discourse is coherent, because we could find a common theme and so
satisfy the coherence constraints on the rhetorical relation.  But if
the above world knowledge contributes to the interpretation of the
sentence, then the 
coherence is weak, because the resulting theme is a bad one due to Grice's
Maxim of Quantity.  In such cases, one
attempts to maintain a stronger coherence, by computing an
interpretation of the constituents which avoids
the weakness.  In this case, 
changing the pragmatic interpretation of the
constituents will change the common theme.
Coherence would
be improved by dropping at least one of the default
implicatures about the kinds of brushes that are
being used.  If one did this, then the range of denotations for {\em
  brush} in the 
theme would change, and so the theme wouldn't violate Quantity anymore.  But
one can't decide which pragmatic implicature to drop: should one drop
the implicature that Rembrandt is using a paint brush, or that the
janitor is using a cleaning brush, or both?  This {\em pragmatic
  ambiguity} is irresolvable, leading to the observed word pun in
(\ref{brush1}c).

Thus, the difference between (\ref{bank2}) and (\ref{brush1}c) is that
the former constitutes irresolvable semantic ambiguity, whereas the
latter, while semantically unambiguous, constitutes irresolvable
pragmatic ambiguity, in that one cannot decide which implicatures
apply.


\subsection{Reinterpretation}

The previous examples exploited both coherence constraints on
discourse and
lexical structure to predict zeugmatic effects.  Here, we
consider an example where psychological mechanisms of sentence processing also
play a part.  Consider again sentence (\ref{brush1}c).
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [b]     Rembrandt used a brush and so did our janitor.
\end{subexamples}
\end{examples}
We assume that the interpreter calculates meaning
incrementally and dynamically (e.g. Marslen-Wilson and Welsh 1978,
Crain and Steedman 1985, Altmann and Steedman 1988).  In other words, in
(\ref{brush1}b), the interpreter will calculate the pragmatic
interpretation of {\em Rembrandt used a brush}, before going on
to process the rest of the sentence.  So, using real world knowledge,
one specialises {\em brush} to {\em paint brush} before
interpreting the elided clause, and attempts to attach this to the
preceding one with a rhetorical relation.

Upon attempting to attach this elided clause, one must use
the rhetorical relation {\em Parallel} (because of the structural
properties of the sentence) and must use this relation to resolve the
ellipsis (Asher 1993a, Kehler 1994).  The only possibility is to
interpret the elided clause as {\em our janitor used a brush}.  Now
one must calculate the pragmatic implicature of this whole sentence.
This in part involves checking the pragmatic implicatures that were
inferred for the first phrase (i.e., the pragmatic implicature that
Rembrandt uses a paint brush).  If one did not check this, then it
would never be cancelled by subsequent context, and so the pragmatic
inference could never be nonmonotonic.

By carrying out this check, one finds that there are clues from
real world knowledge that the janitor uses a cleaning brush.  But if
Rembrandt uses a paint brush, then as in (\ref{brush1}c), the common
theme for the {\em Parallel} relation is bad, because Quantity is violated.
Consequently, to maintain coherence one aims for a different pragmatic
interpretation.  And as in (\ref{brush1}c), it is not possible to
decide which implicature to drop: Rembrandt using a paint brush, or
the janitor using a cleaning brush, or both. We assume that the need
for reinterpretation or reanalysis can create a `garden path' effect
(e.g. Frazier 1979), and that such effects can result from incremental
pragmatic as well as syntactic or semantic processing decisions (e.g.
Crain and Steedman, 1985). So the zeugmatic effects in (\ref{brush1}b)
are a consequence of both reinterpretation and indecision.
First, reinterpretation takes place (the interpreter cannot rely on
Rembrandt's brush being a paint brush anymore).  And second, there is
now indecision about how to interpret {\em brush}.  In other words, as
in (\ref{brush1}c), although there is no semantic ambiguity, there is
irresolvable pragmatic ambiguity. 

Reinterpretation can also account for the word pun effects in
(\ref{thigh}b) and (\ref{plane}).
\begin{examples}
\item [\ref{thigh}]
\begin{subexamples}
\item [b] Bill likes mustard on his thighs, but John prefers suntan
  lotion.
\end{subexamples}
\item   [\ref{plane}]   I tried to take the plane to 
Chicago but it was too heavy.
\end{examples}
In these cases, the most plausible senses available at the end of the
initial clause for {\em thighs} and {\em take} must be revised when
the second clause has been processed in order to infer the {\em Parallel}
relation and resolve the ellipsis and anaphor, respectively. However,
the zeugmatic effect derives from the retraction of the initial
plausible lexical senses rather than from the impossibility of
arriving at a consistent (though bizarre) reading.

\subsection{The Three Components of Zeugma}

In the three informal examples we gave above, we described three
distinct ways in which zeugmatic effects can be created:
\begin{enumerate}
\item   The rhetorical connection between constituents in the
discourse is weak, and this weak coherence makes the discourse sound odd.
\item   Constraints on discourse coherence can produce
  irresolvable conflict in the pragmatic interpretation of
  constituents
in the discourse, and this irresolvable pragmatic ambiguity gives a
word pun effect.
\item   Incremental processing in pragmatics and 
discourse coherence force a reinterpretation.
\end{enumerate}
Furthermore, these can be intermixed, so for instance some examples involve
both an effect of reinterpretation and irresolvable conflict.
This is a heterogeneous account of
zeugma, but there are common threads.
Pragmatic reasoning is central to zeugma, regardless of the 
presence of {\sc
mle}s. And in all cases, the account of zeugma rests on the
interaction between three components: lexical structure, (incremental)
pragmatic 
reasoning and the rhetorical structure of discourse. 

Formalising the above informal accounts therefore rests on providing
general axioms that guide the information flow between lexical
structure, pragmatic reasoning and constraints on discourse
coherence.  To model this information flow, we
must link a theory of lexical structure to a theory of the rhetorical
structure of discourse and the coherence constraints it
imposes, and a theory of how pragmatics affects the
interpretation of discourse.  The rest of this paper is concerned with
integrating these three components, in order to demonstrate how this
integration leads to an account of zeugma and non-crossed readings for
all
the above examples.  In the next section, we'll
review the rhetorical structure and pragmatic frameworks that
we'll use and then give the formal account of the examples we described in
this section in
\S\ref{formal}.

\section{Discourse Structure and Pragmatic Reasoning}

This section first 
describes the background theory of discourse structure assumed
here and the logic which is used to represent real world knowledge
and also to compute rhetorical relations.  We then go on to introduce a
new discourse constraint on word meaning, which generalises the one
assumed by Asher and Lascarides (1995a) in order to account for sense
disambiguation in a discourse context.  
We discuss this in detail here because the new form of the rule will
contribute to the explanation of zeugma involving weak coherence.

\subsection{Discourse Relations}

We need a theory of discourse structure which specifies discourse
coherence constraints, and we use these to explain zeugmatic effects
and the preference for non-crossed
readings as suggested in \S\ref{inf}.  We will use
Asher's (1993a) Segmented Discourse Representation
Theory ({\sc sdrt}) for this purpose.  {\sc sdrt} is an extension of 
{\sc drt} (Kamp and Reyle, 1993) that assumes
richer discourse representations.
In {\sc sdrt}, an {\sc nl} text is represented by a
segmented {\sc drs} ({\sc sdrs}), which is a pair of sets containing
the {\sc drs}s or {\sc sdrs}s (representing respectively sentences and
text segments) and discourse relations between them.  
Discourse
relations, which link constituents together,
are modeled after those proposed by Hobbs (1985), Polanyi
(1985) and Thompson and Mann (1987). 
We will mention four: {\em Narration}, 
{\em Result}, {\em Contrast}
and {\em Parallel}.  Examples of these are given below: 
\begin{examples}
\item \label{ex}
\begin{subexamples}
\item Max stood up.  John greeted him.\\
{\em Narration} (because the textual order of events matches temporal order)
\item John pushed Max.  Max fell.\\
{\em Result} (because the pushing caused the falling).
\item Max has black hair, but Mary has brown hair.\\
{\em Contrast}
\item Max has black hair, and Mary has black hair too.\\
{\em Parallel}
\end{subexamples}
\end{examples}
These discourse relations place constraints on 
discourse coherence, which are defined in terms of the structures and
semantic contents of
the constituents being attached together.  Asher (1993a) and
Lascarides and Asher (1993) discuss and
motivate the coherence constraints for each of the above relations in detail.
For example, {\em  Narration} entails that 
the textual order of events must match temporal
order.  There must also be a distinct common topic, that summarises
what the narrative describes.  In contrast,
{\em Result} doesn't require a distinct common topic, but it
does require a causal relation between the events
described in the constituents.  These constraints are formally
specified below: in what follows, $e_\alpha$ stands
for the main 
eventuality described in the constituent
$\alpha$ and $\gamma\Downarrow\alpha$ means
$\gamma$ is a topic or summary of $\alpha$:
\begin{itemize}
\item {\tt Axiom on Narration}\\
$\Box(\mbox{{\em Narration}}(\alpha,\beta) \rightarrow e_\alpha\prec
e_\beta)$
\item {\tt Distinct Common Topic}\\
$\Box(\mbox{{\em Narration}}(\alpha,\beta) \rightarrow
\exists 
\gamma(\gamma \Downarrow\alpha \wedge \gamma \Downarrow\beta 
\wedge \neg(\alpha\Downarrow\beta) \wedge \neg(\beta\Downarrow\alpha)))$
\item {\tt Axiom on Result:}\\
$\Box(\mbox{{\em Result}}(\alpha,\beta) \rightarrow  \mbox{{\em
    cause}}(e_\alpha,e_\beta))$ 
\end{itemize}

{\em Contrast} and {\em Parallel} both require a partial 
structural isomorphism between the constituents.  In addition, {\em
  Contrast} requires a contrasting theme, and {\em Parallel} a common
theme.  Asher (1993a) defines precisely how to calculate the structure
of each constituent and also specifies which mappings between the
structures
are
sufficiently close to bijections that the structural constraint
on {\em Parallel} and {\em Contrast} is met.  We will explicate these
definitions in \S\ref{formal}, when we use them in our
formal analysis of zeugma.  We
also need a theory for computing themes, so that we can check that the
constraints on themes are satisfied too --- 
an approach similar to that of Grover {\em et al.} (1994).  They 
use constraint generalisation on the constituents, as defined by the
semantic type hierarchy that accompanies the lexicon.
We refine this slightly: the common theme will be the generalisation
of the propositions corresponding to the {\em pragmatic}
interpretations (with respect to the discourse context) of the
constituents being connected together.  So for example, the common
theme of (\ref{rubens2}) will be the generalisation of {\em Rembrandt
used a paint brush} and {\em Rubens used a paint brush}, as opposed
to the generalisation of {\em Rembrandt used a brush} and {\em Rubens
used a brush}.
\begin{examples}
\item   \label{rubens2} Rembrandt and Rubens used a brush.
\end{examples}
This is because the implicature that the brushes are paint brushes
survives in this context.  And so the common theme for (\ref{rubens2})
is {\em artists used a paint brush}.

Asher shows how one can resolve {\sc vp} ellipsis as a byproduct of
checking the coherence constraints on {\em Contrast} and {\em
Parallel}.  For example in (\ref{rubens1}), the elided clause is
resolved to {\em and Rubens used a brush} because of the structural
coherence constraints on {\em Parallel}.  
\begin{examples}
\item   \label{rubens1} Rembrandt used a brush and so did Rubens.
\end{examples}
The details of how the coherence
constraints contribute to the interpretation of {\sc vp} ellipsis will be
given in \S\ref{formal}, with respect to the zeugmatic sentence
(\ref{brush1}b):
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [b]     Rembrandt used a brush and so did our janitor.
\end{subexamples}
\end{examples}

\subsection{Pragmatic Reasoning}
\label{logic}

We need to represent default real world knowledge, such as {\em If
Rembrandt uses a brush, then normally it's a paint brush}.
We also need a theory which computes the rhetorical connections
between meanings of segments of text via the interpreter's background
knowledge.  For example, we want to 
be able to predict that the constituents in
(\ref{thesis1}a) are connected together with {\em Result} and {\em
Parallel}, whereas in (\ref{thesis1}b) the relation is {\em
Parallel} alone.
\begin{examples}
\item   [\ref{thesis1}]
\begin{subexamples}
\item   That thesis has thousands of pages and is unreadable.
\item   That thesis is orange and unreadable.
\end{subexamples}
\end{examples}

The informal lines of reasoning about zeugma given in
\S\ref{inf} cast light on the
kind of intuitively compelling
reasoning patterns we need to validate in the logic that 
underlies pragmatic reasoning.  We must use a logic
which 
automatically calculates the interaction between various knowledge
resources such as lexical knowledge, real world knowledge and Gricean
style pragmatic maxims which takes place during interpretation.  
In particular, it must detect
conflict, and predict when it's resolvable and when it's not.  This is
because, on occasion, the various knowledge resources yield conflicting
messages about how to interpret a word, and the logic must predict
`automatically' if and how this conflict is resolved (cf. our
discussion of (\ref{bank2})).


We will exploit state-of-the-art techniques in commonsense reasoning
from {\sc ai} to reason about pragmatic
interpretation.  The particular
theory we'll use is {\sc dice} (Discourse In
Commonsense Entailment)
(Lascarides and Asher 1991, 1993).
{\sc dice} represents as default rules
the interpreter's background knowledge about how syntax,
compositional semantics, Gricean style pragmatic maxims
and real world knowledge affect overall interpretation.  
These are used to compute both the
pragmatic interpretation of constituents and the rhetorical relations
between them.  It is the theory of discourse attachment by which {\sc
  nl} text is translated into discourse representations in {\sc sdrt}.

The logic that underlies {\sc dice} is Asher and 
Morreau's (1991) commonsense entailment ({\sc ce}).  This is a
conditional logic in which a formula of the form $A>B$ is supplied
with a model theoretic and proof theoretic interpretation, which
roughly means ``If $A$, then normally $B$''.  Default knowledge
in {\sc dice} is represented as conditionals of this form.

{\sc ce} exploits two
notions of validity, a monotonic one ($\models$) and a nonmonotonic one
($\csmodel$).  There are at least six attractive properties of
$\csmodel$.  
\begin{enumerate}
\item All the consequents of $\models$ are computed
before one starts to calculate the consequents of $\csmodel$. 
\item $\csmodel$ validates Defeasible Modus Ponens ({\sc dmp}):
when the default laws whose antecedents are verified all have 
consequents that are consistent with the knowledge base ({\sc kb}) and
with
each other, then all the consequents are nonmonotonically
inferred.
\item  $\csmodel$ validates Specificity:
in cases of conflicts among defaults where one has a more
specific antecedent than the others, $\csmodel$ will verify the
conclusions of the more specific default.  
\item $\csmodel$ validates Skepticality: in cases of conflicts among
defaults where there is no default that has a most specific antecedent
relative to the others, $\csmodel$ won't verify any conclusions of the
defaults.  
Because of Specificity and Skepticality, {\sc ce} 
automatically predicts when
knowledge conflict can be resolved and when it can't.
\item $\csmodel$ is robust in that if $\Gamma\csmodel \phi$ then
$\phi$ will survive as a consequence of the premises $\Gamma$
augmented with logically independent information.  
\item For each
deduction $A\csmodel B$ there is a corresponding embedded default in the object
language (that is, a formula in which one occurrence of the
connective $>$ occurs within the scope of another) which links
boolean combinations of the formulae $A$ and $B$, and which is
verified to be true.  We gloss this embedded default formula 
as $\emb(A,B)$.  So $\emb(A,B)$ means $A\csmodel B$.
This amounts to a weak deduction theorem.  The object language
formula $\emb(A,B)$ means that $A$ nonmonotonically yields $B$ in the
metalanguage. 
\end{enumerate}

We will also on occasion use the notation $\emb_{KB}(A,B)$ to mean
$\emb(KB\wedge A,B) \wedge \neg\emb(KB,B)$.  In other words,
$\emb_{KB}(A,B)$ means that $B$ nonmonotonically follows from the {\sc kb}
augmented with A, but not from the {\sc kb} alone.
The details of the logic can be found in
(Asher and Morreau 1991, Asher 1993b).

We can use {\sc ce} to represent 
the knowledge recruited to compute the
pragmatic interpretation of discourse.  This knowledge is of many
different types: knowledge of the real world; knowledge of Gricean-style
pragmatic maxims; knowledge of the communicative intentions of the
author, and so on.  We have examined in detail elsewhere
how to represent these
different pieces of knowledge as formulas of {\sc ce} in {\sc dice}
(Lascarides and
Asher 1991, 1993; Lascarides and Oberlander, 1993; Asher and
Lascarides, 1994, 1995a).  

Real world knowledge such as {\em When Rembrandt uses a
  brush, it's normally a paint brush} can be represented via the
schemata below:
\begin{itemize}
\item   {\tt Brush Purpose}:\\
$\Box(\mbox{\em brush}(y) \rightarrow \mbox{\em purpose}(y, \mbox{\em
  brush-purpose}(e)))$ 
\item   {\tt Rembrandt:}\\
$agent(e,x) \wedge \mbox{{\em rembrandt}}(x) \wedge
\mbox{{\em instrument}}(e,y) \wedge \mbox{{\em 
purpose}}(y,
\sqsupseteq \mbox{{\em paint}}(e')) \\
\hspace*{0.2in}> \mbox{{\em 
purpose}}(y,\mbox{{\em
paint}}(e'))$\\
where $\mbox{{\em purpose}}(y, \sqsupseteq \mbox{{\em paint}}(e'))$ is
an abbreviation for: 
$\mbox{{\em purpose}}(y, P(e')) \wedge (\mbox{{\em paint}}(e') 
\rightarrow P(e'))$
\end{itemize}
That is, when Rembrandt is the agent of an event where the instrument
is something whose purpose is more general than
painting, then normally, the purpose of that
instrument in that particular event is painting. 
Information about purposes of brushes---as given in Brush Purpose---has to be part of common sense
knowledge for independent reasons.  For our current concerns it is
irrelevant whether or not this information is also accessible to the
lexicon. 

There is also a
similar rule, which links the purpose of agents to the purposes of
their instruments: Specialisation will predict that when a janitor
(whose `purpose'/occupation is to clean) 
uses a brush, then normally it is a
cleaning brush:
\begin{itemize}
\item   {\tt Specialisation}:\\
$agent(e,x) \wedge \mbox{{\em purpose}}(x, P(e')) \wedge
\mbox{{\em instrument}}(e,y) \wedge \mbox{{\em purpose}}(y,
\sqsupseteq P(e')) \\
\hspace*{0.2in}> \mbox{{\em purpose}}(y,P(e'))$ 
\end{itemize}
These laws enable {\sc ce} to predict that {\em Rembrandt used a
  brush} is pragmatically interpreted as {\em Rembrandt used a paint
  brush}, and similarly for {\em janitor} and cleaning brush.  
The pattern of inference is {\sc
  dmp} on Rembrandt and Specialisation respectively.\footnote{
We don't rely on Specialisation to hanlde the inference about
Rembrandt, because we wish to remain agnostic as to wether Rembrandt's
`purpose' is to paint.}
We would need to
extend these representations of real world knowledge to handle quantified
sentences such as {\em No artist used a brush}, but for the sake of
simplicity and brevity, we ignore this issue here.\footnote{It would
be relatively
straightforward to extend the rules to handle these cases; one need
only use {\sc drs} duplex conditions in the antecedents of these
rules, where the quantifier is left underspecified.}

In addition, we can use {\sc ce} to infer rhetorical relations---such
as {\em Contrast}, {\em Parallel}, {\em Result} and {\em Elaboration},
among others---that connect the meanings of segments of text together.
{\sc dice} uses
the interpreter's background knowledge to build {\sc sdrs}s
from {\sc nl} text (Lascarides
and Asher 1991, 1993).  The way this is done is as follows.  
First, a constraint-based grammar with a {\sc drt} semantic component
(cf. Frank and Reyle, 1995) is used to build up the {\sc drs} for the
clause that is currently being processed.  Thus {\sc drs}s for clauses
are built in a monotonic fashion via information from syntax, much as
in traditional {\sc drt}.  However, in contrast to traditional {\sc
drt}, 
one then reasons in {\sc
dice} about how this {\sc drs} attaches to the preceding discourse
structure with a rhetorical relation.  
And in contrast to the monotonic
construction of the {\sc drs} representing the clause, the inferences
in {\sc dice}
underlying the choice of rhetorical relation for attachment is
nonmonotonic, because rhetorical relations aren't always
linguistically marked.  Rather, they are inferred on the basis of
clues from real world knowledge, the discourse context, communicative
intentions and Gricean style pragmatic maxims.  A further contrast to
traditional {\sc drt} is that once the semantic representation of the
clause currently being processed is built by the grammar, it is
evaluated against the model.  This is because {\sc dice} uses the 
semantic content of the clause as a clue about which rhetorical
relation to use to attach it to the representation of the discourse
built so far.  In traditional {\sc drt}, semantic interpretation
doesn't occur until the representation of the whole discourse is built.

As well as real world knowledge such as Rembrandt and Specialisation,
there are default rules like Result in {\sc dice} for computing
rhetorical relations:
if $\beta$ is to be attached to $\alpha$ with a
rhetorical relation where $\alpha$ is part of
the discourse structure $\tau$ already, and the event described in
$\alpha$ caused that described in $\beta$, then normally, the
rhetorical relation is {\em Result}.
\begin{itemize}
\item   {\tt Result:}\\
$\langle \tau,\alpha,\beta\rangle \wedge \mbox{{\em
    cause}}(e_\alpha,e_\beta) > \mbox{{\em Result}}(\alpha,\beta)$
\end{itemize}
Result uses
information about the constituents $\alpha$ and
$\beta$ to come to nonmonotonic conclusions about how they're
connected together.  In general, axioms in {\sc dice}
that are about computing rhetorical relations are of this general form:
$\langle\tau,\alpha,\beta\rangle \wedge
\phi > R(\alpha,\beta)$, where $\phi$ is
information about $\alpha$, $\beta$ and the
discourse
context $\tau$, and $R$ is a particular
discourse relation.  

\subsection{A Discourse Constraint on Word Interpretation}

{\sc dice} is sufficiently rich to encode how the rhetorical structure
of the discourse affects the way words are interpreted.  Indeed, we'll show
that the rule in {\sc dice} which encodes how rhetorical structure
affects lexical sense disambiguation can also create zeugma.  To
explicate how rhetorical structure affects word meaning---and in
particular, disambiguation---we give a detailed analysis of some
cases which involve the rhetorical relation {\em Elaboration}.
As described in
Asher and Lascarides (1995a), $\mbox{{\em Elaboration}}(\alpha,\beta)$
can be inferred by reasoning that the event condition in $\beta$ is a
subtype of that in $\alpha$.  
\begin{itemize}
\item   {\tt Elaboration}\\
$\langle\tau,\alpha,\beta\rangle \wedge \mbox{{\em
    Subtype}}(\alpha,\beta) > \mbox{{\em Elaboration}}(\alpha,\beta)$
\end{itemize}
Elaboration states that if $\beta$ is to be attached to $\alpha$ with
a discourse relation, where $\alpha$ is part of the {\sc sdrs} $\tau$
built already, and $\beta$ is a subtype of $\alpha$, then normally,
the discourse relation is {\em Elaboration}.
For the sake of brevity, we don't repeat the rules by which one infers
$\mbox{{\em Subtype}}(\alpha,\beta)$, since the details of this aren't
central to our main concerns here.  These rules used lexical
information, and are powerful enough to infer that $\mbox{{\em
Subtype}}(\mbox{\ref{elab}a},\mbox{\ref{elab}b})$ 
holds for the text (\ref{elab}), because
learning to use a paint brush is a subtype of learning to paint, and
the janitor is plausibly a member of school staff.
\begin{examples}
\item   \label{elab}
\begin{subexamples}
\item   All the school staff learned how to paint.
\item   Even the janitor learned to use a paint brush.
\end{subexamples}
\end{examples}
So Specificity in {\sc dice}
between Narration given below (from Lascarides and Asher, 1991)
and Elaboration predicts that
(\ref{elab}a) and (\ref{elab}b) are linked by {\em Elaboration}.
\begin{itemize}
\item   {\tt Narration}: $\langle\tau,\alpha,\beta\rangle > \mbox{{\em
      Narration}}(\alpha,\beta)$ 
\end{itemize}

Inferring that (\ref{elab}) was an elaboration crucially uses the
information that the brush in (\ref{elab}b) is explicitly stipulated
to be a paint brush.  Whereas learning to use a paint brush is a
subtype of learning to paint according to the rules in Asher and
Lascarides (1995a), learning to use a (general) brush
isn't---the brush in question may be a hair brush and 
learning to use a hair brush is not a subtype of learning to paint.  
And so without the
information that the brush is a piant brush, we can't infer $\mbox{{\em
Subtype}}(\alpha,\beta)$ so that the rule Elaboration applies.
But what about the discourse (\ref{brushelab}), where there is no such
explicit stipulation about the brush, and indeed, where real world
knowledge 
about janitors favours cleaning as opposed to paint
brushes?  
\begin{examples}
\item   \label{brushelab}
\begin{subexamples}
\item   All the school staff learned how to paint.
\item   The janitor learned to use a brush.
\end{subexamples}
\end{examples}
Intuitively, this text is still an elaboration, and one {\em infers}
that the brush used is a paint brush, because of the discourse context.
We explain here how the discourse context wins over the real world
knowledge about the brushes janitors use, because the rules in {\sc
dice} that encode these discourse effects on word meaning
will also create zeugma in the cases we've discussed.

Asher and Lascarides (1995a) provided rules in {\sc
  dice} which encoded interactions between words and discourse, in
service to sense
disambiguation in a discourse context.  They paid special attention
to cases where information in the discourse context
conflicts with information within the sentence as to how a word
should be interpreted.  Asher and Lascarides argued in favour
of a rule they called Lexical Impotence.  This rule ensured that, if
disambiguating a word according to the information from within a
sentence led ultimately to discourse incoherence, this disambiguation
was blocked and, instead, the word was disambiguated via information in
the preceding discourse context.

This was used to model the lexical disambiguation in (\ref{bar}).
\begin{examples}
\item   \label{bar}
\begin{subexamples}
\item   The judge asked where the defendant was.
\item   His lawyer said he was drinking across the street.
\item   The court bailiff found him slumped underneath the bar.
\end{subexamples}
\end{examples}
Suppose one disambiguates {\em bar} in (\ref{bar}c) according to the
clues from within (\ref{bar}c) itself (so the bar is the courtroom
bar).  Then one cannot connect this to (\ref{bar}b) with a rhetorical
relation.  This is because the only candidate relation is {\em
  Narration}, but if the defendant is at the courtroom bar in
(\ref{bar}c), then a spatial constraint on {\em Narration} is violated
because he's across the street in (\ref{bar}b).  Lexical Impotence
ensured that the discourse information and need for coherence wins out
in this case, and {\em bar} is disambiguated to the pub sense, so that
{\em Narration}'s spatial constraints aren't violated and coherence is
established.

But Lexical Impotence cannot explain the interaction between words and
discourse in (\ref{brushelab}), because if the brush is a cleaning
brush (as the real world knowledge in (\ref{brushelab}b) favours), then a
narrative, list or parallel relation can be inferred.  So a rhetorical
connection can be found when the sentential information drives the
interpretation of the word {\em brush}; this sentential information
doesn't lead to incoherence.  So Lexical Impotence wouldn't apply when
interpreting (\ref{bar}), and we must model why the preference
triggered by the real world knowledge about janitors is blocked in some
other way.  
We'll generalise Lexical Impotence here, to increase the empirical
coverage and further explicate how rhetorical structure influences how
words are interpreted.  The rule for interaction given here will  
handle both (\ref{bar}) and (\ref{brushelab}).
And it will capture the following intuition: we prefer not to infer
things that have an adverse effect on the quality of coherence in the
discourse. 

Consider (\ref{brushelab}) once again.  Suppose that Specialisation
fired, so that the brush in (\ref{brushelab}b) is interpreted as a
cleaning brush.  Then the event (\ref{brushelab}b) isn't a
subtype of that in (\ref{brushelab}a), because the janitor is learning
to use a cleaning brush (i.e., learning to clean) in
(\ref{brushelab}b), rather than learning to paint.  So the only rule
for computing the rhetorical
connection that would be verified is Narration.  Narration
has coherence constraints: one must find a distinct common topic.
This topic must summarise both $\alpha$ and $\beta$.  The only
candidate here is {\em school staff learning to paint or
clean}.  But one
could argue that, as in
(\ref{brush1}b), although this is coherent, it is only weakly so,
because the topic is a disjunction of the individual sentential
topics. It is not clear what the discourse is `about'.

Assume that $\mbox{{\em weak}}(\tau)$ means that the
{\sc sdrs} $\tau$ is at best only weakly coherent.
Assume
furthermore that a weak theme or topic yields only weak discourse
coherence.  
Then,
intuitively, one wants to avoid discourse interpretations that yield
an {\sc sdrs} $\tau$ where $\mbox{{\em weak}}(\tau)$ is true.\footnote{Providing
  formal truth conditions for $\mbox{{\em weak}}(\tau)$ would involve
  complex reasoning about the semantic content of $\tau$ and the
  cognitive states of the participants in the discourse.  This goes
  beyond the purposes of this paper, and so we rely in this paper on
  intuitive predictions of when $\mbox{{\em weak}}(\tau)$ is true,
  such as the intuition that the discourse theme violates a Gricean maxim.} 
In principle, the interpretations of constituents that yield weak
discourse coherence can be
blocked, if (a) the weak coherence hinges essentially on a
particular pragmatic inference on
one of the constituents, and (b) we block those pragmatic inferences
in those cases, so that we can attempt to improve overall discourse
coherence.  In other words, a pragmatic implicature of a particular
constituent may be blocked, if that pragmatic implicature has an
adverse effect on overall discourse coherence. 

We can capture this intuitively compelling
interpretation strategy as a rule in {\sc dice}.  We will gloss all
monotonic information about the constituents $\alpha$ and $\beta$ as
$\mbox{{\em Info}}(\alpha)$ and $\mbox{{\em Info}}(\beta)$ (sometimes
abbreviated to $\mbox{{\em Info}}(\alpha,\beta)$).\footnote{
We assume that in cases where $\alpha$ is, for example, {\em Rembrandt
  used a brush}, $\mbox{{\em Info}}(\alpha)$ will contain the
information $\mbox{{\em brush}}(y) \wedge \mbox{{\em
    purpose}}(y,\mbox{{\em brush-purpose}}(e'))$, because this is
monotonically inferrable from the truth conditional content of
$\alpha$.  However, $\mbox{{\em Info}}(\alpha)$ will {\em not} contain
the information $\mbox{{\em purpose}}(y,\mbox{{\em paint}}(e'))$ since
this is only nonmonotonically inferrable.}
Let
$\mbox{{\em KB}}$ be the interpreter's knowledge base: this includes the update
function $\langle\tau,\alpha,\beta\rangle$, the semantic content of
$\tau$, $\alpha$ and $\beta$, and the rules of {\sc dice}.  
Then the Interpretation Constraint below captures the intuition that
people try not to infer propositions that lead to weakly coherent or
incoherent discourse. It states: if (a) $\beta$ is to
be connected to $\alpha$ with a discourse relation, and $\beta$ and
$\alpha$ are both true,
and (b) if the KB that includes not only the update
task of $\beta$ to $\alpha$, but also the information $\beta'$,
nonmonotonically leads to a discourse of only weak coherence or no
coherence at all, then normally (c) $\beta'$ doesn't hold.
\begin{itemize}
\item   {\tt Interpretation Constraint}\\
\begin{tabular}[t]{l l}
(a) & $\langle\tau,\alpha,\beta\rangle \wedge \mbox{{\em
    Info}}(\alpha,\beta)\wedge$ \\
(b) & $\beth_{KB}(\beta', \mbox{{\em
weak}}(\tau\cup\beta))) $\\
(c) & \hspace*{0.1in}$> \neg\beta'$
\end{tabular}
\end{itemize}
This rule captures a dependence between pragmatic inferences from
information within the sentence and information from the
discourse context.  Consider the case
when $\beta'$ is a pragmatic implicature of $\beta$ in isolation of
the discourse context (e.g., $\mbox{{\em Info}}'(\beta)>\beta'$
is a rule in the {\sc kb}, where $\mbox{{\em Info}}'(\beta)$ is a gloss
for some relevant information pertaining to the syntagmatic
representation of $\beta$).
If assuming $\beta'$ yields 
only weak discourse coherence when one attempts
to attach the constituents together, then the Interpretation
Constraint blocks this pragmatic inference, because its consequent
will conflict with that of
the default rule $\mbox{{\em Info}}'(\beta) >\beta'$,
and it is also more specific than this
rule (because the antecedent of
the Interpretation Constraint entails $\mbox{{\em Info}}'(\beta)$
since it entails {\em all} monotonic information about $\beta$).  So
$\beta'$ would follow via {\sc dmp} from a {\sc kb} that
included just $\beta$ in isolation of
discourse $\tau$.  But in the context of $\tau$, $\beta'$ does not
follow because the Interpretation Constraint blocks it.

The Interpretation Constraint can now replace Lexical Impotence
because it predicts that sentential clues about disambiguation are
blocked when they lead to discourse incoherence.  To see this,
consider (\ref{bar}) again.  (\ref{bar}c) on its
own is pragmatically interpreted as involving the courtroom bar.  
For the sake of simplicity, assume this is encoded in {\sc dice} as
$\mbox{{\em Info}}'(\gamma) >\gamma'$, where $\gamma$ is the semantic
representation of 
(\ref{bar}c), and $\gamma'$ asserts that the {\em bar} is the
courtroom bar.\footnote{In Asher and Lascarides (1995a), this rule was
represented
as $\mbox{{\em FS}}_{bar} > \mbox{{\em FS}}_{bar_1}$, where
$\mbox{{\em FS}}_{bar}$ is all the information in the feature
structure that represents sentence (\ref{bar}c), and $\mbox{{\em
    FS}}_{bar_1}$ is that feature structure, with {\em bar} replaced
by its courtroom sense.  We have changed it here 
for the sake of simplicity, to the more general
notation above; nothing hinges on this difference.}
Now, we must attach this $\gamma$ to the semantic
representation $\beta$ of (\ref{bar}b).  The only candidate is {\em
  Narration} (via the axiom Narration).  But {\em Narration} has spatial
constraints.  These were formally spelled out in Asher and Lascarides
(1995a); informally, they capture the intuition that objects stay where
they are in a narrative, unless one can infer otherwise.  If the bar
is the courtroom bar---i.e., the {\sc kb} contains $\gamma'$---then 
{\em Narration}'s spatial
constraint is violated, so that no discourse relation can be
inferred.  Using this same knowledge here, this means that
$\emb_{KB}(\gamma',\mbox{{\em weak}}(\tau\cup\gamma))$
holds---$\tau\cup\gamma$ is incoherent when $\gamma'$ is part of the
{\sc kb}.  So
clauses (a), (b) and (c) of the Interpretation
Constraint
are verified, with $\beta$, $\gamma$ and $\gamma'$ substituted for
$\alpha$, $\beta$ and $\beta'$ respectively in the schema.  So
normally, $\neg \gamma'$ holds.  This conflicts with the rule
$\mbox{{\em Info}}'(\gamma)>\gamma'$ 
which also forms part of the {\sc kb}.  But the
Interpretation Constraint is more specific, and so $\neg\gamma'$ is
inferred.  That is, the bar is not the courtroom bar.  This was exactly
the inference that arose from Lexical Impotence, and so the
Interpretation Constraint can replace this rule.

The Interpretation Constraint can also explain how the rhetorical
structure of (\ref{brushelab}a,b) affects the interpretation of {\em
  brush}.  It leads to an
inference that the brush in question is a {\em paint brush}, thereby
overriding the real world knowledge about the kinds of brushes that
janitors normally use, given in Specialisation. Let
the {\sc drs}s representing the compositional semantic content of
(\ref{brushelab}a,b) be $\alpha$ and $\beta$ respectively.  
Then let $\beta'$ be the condition that the brush
is a cleaning brush.  
Specialisation can be glossed as $\mbox{{\em Info}}'(\beta)>\beta'$,
where $\mbox{{\em Info}}'(\beta)$ is the relevant information about
$\beta$ that ensures the antecedent of Specialisation is verified.  So
$\mbox{{\em Info}}'(\beta)>\beta'$ applies.

But
the condition (b)
of the Interpretation Constraint is verified with this
$\beta'$, because as we've stated, interpreting the
brush as a cleaning brush 
yields a nonmonotonic inference that the discourse relation is
{\em Narration}, with a weak common topic and so $\mbox{{\em
    weak}}(\tau\cup\beta)$ is inferred with respect to this
information.  That is, $\beth_{KB}(\beta',\mbox{{\em
    weak}}(\tau\cup\beta))$
holds.
So, the Interpretation Constraint applies,
and its consequent is $\neg\beta'$;
that is, the brush is not a cleaning brush.  This
conflicts with Specialisation.  But 
the Interpretation Constraint is more specific, because its
antecedent contains as a conjunct all the 
information about $\beta$ that's in the 
antecedent of Specialisation.
So by Specificity,
$\neg\beta'$ is inferred.  That is, the brush is not a cleaning
brush.  Informally, because the pragmatic interpretation of
(\ref{brushelab}b) in isolation of (\ref{brushelab}a) has an adverse
effect on discourse coherence (although not making it totally
incoherent) when one attaches it to (\ref{brushelab}a), this pragmatic
interpretation is blocked by the Interpretation
Constraint, so as to improve overall coherence.

Indeed, the antecedent of the Interpretation Constraint applies for each
$\beta'$ where the brush is a
tooth brush, a hair brush and so on (hence we {\em don't} remain
neutral about what kind of brush the janitor used).  So for these
$\beta'$, {\sc dmp} on the Interpretation Constraint yields the
conclusion that the brush is not a tooth brush, a
hair brush and so on.  But the Interpretation Constraint does {\em
  not} apply for the $\beta'$ where the brush is a paint brush.  When
this $\beta'$ is added to the {\sc kb}, then one can infer
$\mbox{{\em Subtype}}(\alpha,\beta)$ holds, and thus Elaboration
applies and by Specificity on this and Narration,
$\mbox{{\em Elaboration}}(\alpha,\beta)$ is inferred.   Thus adding
this $\beta'$ to the {\sc kb} does not yield a weak discourse, and so
the antecedent of the Interpretation Constraint doesn't apply for this
$\beta'$, and so
one doesn't conclude $\neg\beta'$ in this case.  So the Interpretation
Constraint ultimately restricts the brush to being a paint brush,
because all other possible interpretations of {\em brush} are ruled
out by the Interpretation
Constraint, on the basis that they yield a discourse that is only weakly
coherent.
So {\sc dice}, with the aid of the Interpretation
Constraint, correctly predicts that (\ref{brushelab}) is an
elaboration, and the brush in question is a paint brush.
We'll see in \S\ref{formal}
that the Interpretation Constraint not only disambiguates
words and causes sense shifts via rhetorical structure, but it also
plays a crucial part in creating zeugma in the absence of
multiple lexical entries in the lexicon.



\section{Integrating the Components}
\label{formal}

In the previous section we saw how rules 
at the discourse level can
have effects on interpretations; 
for example the Interpretation Constraint causes {\em
  brush} to be interpreted in a particular way in (\ref{brushelab}),
which conflicts with the clues about how it should be interpreted
given our knowledge about the kinds of brushes janitors use.
We'll now explain further how lexical information, compositional 
semantics, pragmatics and discourse coherence interact, in 
order to explain how zeugma and the absence of crossed readings occur.

\subsection{Homonyms}

We start with an example that's zeugmatic due to homonymous ambiguity.
We assume that unrelated homonyms are encoded by 
completely distinct lexical entries.  
For example {\it bank} has two completely unrelated lexical entries
corresponding to the financial institution and raised ground senses.
Because the structures are distinct, and there is no lexical entry
which corresponds to their generalisation,
there is no possibility of a
single use of {\it bank} in a sentence having both readings 
simultaneously.
So, consider sentence (\ref{bank2}) and its logical forms
once more.\footnote{For the sake of brevity, we ignore the
collective readings and concentrate on the distributive
ones. The two logical forms thus arise because of the two
lexical entries for {\it bank}.
The notation $\iplus$ denotes an operator which forms plural individuals
(Link 1983).}
\begin{examples}
\item   [\ref{bank2}]   Some dam busters and bank robbers blew up 
banks.
\end{examples}
\begin{examples}
\item   [\ref{bank2}$'$]
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, bank-robber(x$_1$),
  dam-buster(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t}{bank$_{\mbox{\tiny ground}}$(y)\\ blow-up(e,x,y)\\
  hold(e,t)\\ t$\prec$now}}
\item   [\ref{bank2}$''$]       
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, bank-robber(x$_1$),
  dam-buster(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t}{bank$_{\mbox{\tiny org}}$(y)\\ blow-up(e,x,y)\\
  hold(e,t)\\ t$\prec$ now}}
\end{examples}
To check the coherence of this sentence, we must do two
things: (a) resolve the semantic ambiguity; and (b) check the
coherence of linking the coordinated constituents together.  We
concentrate here on the former task, because the latter depends on it
being accomplished.

We must consider the pragmatic inferences that accompany (\ref{bank2}).
Two pieces of knowledge are
relevant.  First, if a bank robber is blowing up an object,
which is either a financial institution or a mound of earth but we
do not know which, then in the absence of further information
to the contrary, we jump to the conclusion that the object in
question is a financial institution.  Conversely, for dam busters, the
default is that the object is a mound of earth.
These two rules are represented
in {\sc ce} as follows:
\begin{itemize}
\item   {\tt Bank Robber}:\\
$\mbox{{\em bank-robber}}(x) \wedge \mbox{{\em 
blowup}}(e,x,y) \wedge
(\mbox{\em bank}_{\mbox{\tiny ground}}(y) 
\vee \mbox{{\em bank}}_{\mbox{\tiny org}}(y)) >  
\mbox{{\em bank}}_{\mbox{\tiny org}}(y)$
\item   {\tt Dam Buster}:\\
$\mbox{{\em dam-buster}}(x) \wedge \mbox{{\em blowup}}(e,x,y) 
\wedge
(\mbox{{\em bank}}_{\mbox{\tiny ground}}(y) \vee 
\mbox{{\em bank}}_{\mbox{\tiny org}}(y)) > 
\mbox{{\em bank}}_{\mbox{\tiny ground}}(y)$
\end{itemize}
Bank Robber and Dam Buster
are intended for expository purposes in this paper.  They
will do for the sentences involving {\em bank} that we consider, but
would have to extended to deal with quantified sentences, such as
{\em No modern bank robber blows up banks}.
The rules above
could be derived from more general rules about the kinds of
things that dam busters and bank robbers blow up (see Asher and
Lascarides (1995a) for an account of how this is done).  But we gloss
over this for now since it's not relevant to the account of zeugma.
For our purposes the only crucial point concerns the logical
relationship between the rules; we must ensure that the interaction
between this and the compositional semantics of (\ref{bank2}) has the
desired effect.
These rules ensure that although {\em A
  bank robber blew up a bank} and {\em A dam buster blew up a bank} are
ambiguous sentences at the semantic level, they are disambiguated at
the pragmatic level (under the assumption that the interpreter's {\sc kb}
contains the knowledge that one or other of the logical forms is true).

How do Bank Robber and Dam Buster affect the pragmatic interpretation
of (\ref{bank2})?  We assume the KB contains the information that
(\ref{bank2}$'$) or (\ref{bank2}$''$) is true, but not both.  So the
antecedents of Bank Robber and Dam Buster are verified.  Now,
we are assuming that (\ref{bank2}) is distributive, 
so the
embedding functions that verify the {\sc drs}s (\ref{bank2}$'$) and
(\ref{bank2}$''$) permit bank robbers and dam busters to be agents to
different events, involving different banks.  But even so,
neither logical forms
permit a {\em mixture} of banks to be blown up; $\ref{bank2}'\vee
\ref{bank2}''\wedge \neg(\ref{bank2}'\wedge \ref{bank2}'')$ thus
entails that either all the banks involved are financial, or they're
all earth banks.  So even though the
antecedents of Bank Robber and
Dam Buster could be verified by different agents, different banks and 
different events, the rules are nevertheless 
dependent.  Because the logical forms don't permit bank robbers to
blow up financial institutions while the dam busters blow up earth
banks, the above real world knowledge conflicts.  Moreover, it is
irresolvable conflict, because the antecedents of Bank Robber and Dam
Buster are logically unrelated.  Consequently by Skepticality in {\sc
  ce}, the interpreter comes to no conclusions about whether {\em bank} should
be interpreted as {\em bank}$_{\mbox{\tiny org}}$ or {\em bank}$_{\mbox{\tiny ground}}$.
So even after having
brought pragmatic reasoning to bear, it is not possible to 
decide how to resolve
the lexical ambiguity.
In other words, irresolvable
conflict in pragmatics produces a word pun effect in (\ref{bank2}).
The semantic ambiguity of (\ref{bank2}) is irresolvable, because of
Skepticality in Pragmatics.  The interpreter does not even come to the point
of checking how the coordinated constituents rhetorically link
together, because there was no resolution of the truth conditional
content
of the
constituents. 

\subsection{Syntax and Pragmatics:
A Single Lexical Entry}
\label{weakf}

Now consider an example where there is a single lexical entry, but
nevertheless there is a feeling that the sentence involves a word pun:
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [c]     Rembrandt and our janitor used a brush.
\end{subexamples}
\end{examples}
We assume that there is a single lexical entry for {\em
  brush}, which refers to paint brushes, cleaning brushes, tooth
brushes and so on.  These subclasses of brush can also be represented
in the lexicon, but they would be specialisations of the more general entry.
Therefore there is only one logical form for (\ref{brush1}c):
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [c$'$]  
\topdrs{X,x$_1$,x$_2$}{X=x$_1\iplus$x$_2$, rembrandt(x$_1$), janitor(x$_2$)\\
\dup{x}{x$\in$X}{x}{every}{e,y,t}{use(e,x,y)\\ brush(y)\\hold(e,t) \\
  t$\prec$ now}}
\end{subexamples}
\end{examples}
However, pragmatics still yields a word
pun effect.  
This is how it occurs.  

We must check the coherence of coordinating the constituents in
(\ref{brush1}c) together.  {\sc sdrt} assumes that rhetorical
relations relate propositions rather than properties.  Extending {\sc
  sdrt} to incorporate rhetorical connections between abstracted {\sc
  drs}s is straightforward, but it is a technical exercise that
doesn't cast light on the points we want to make about zeugma.  So for
the sake of clarity, we make the simplifying assumption that in cases
like (\ref{brush1}c), the task is to compute the rhetorical connection
between the propositions {\em Rembrandt used a brush} and {\em
  our janitor used a brush}.
Let these propositions be $\alpha$ and $\beta$ respectively.  Using
the mechanisms available in {\sc sdrt}, one infers in the monotonic
component of {\sc ce} that the relation is {\em Parallel}.  We must
check that the coherence constraints on {\em Parallel} are verified:
that is, we must find a common theme.  

Now, suppose that we stipulate that $\beta'$ is the
{\sc drs} conditions that the brush that Rembrandt is using is a paint
brush, and the brush that the janitor is using is a cleaning brush.
Then, using the method for computing the common theme that we
discussed earlier, we infer that the
common theme for the {\em Parallel} relation is {\em humans used a
paint or cleaning} brush.  As we argued in \S\ref{ircoex},
this violates Grice's Maxim of
Quantity, at least in this discourse context, making it a weak
theme.
Grice's Maxim
of Quantity stipulates that one should not use a word which has
an overspecific or underspecific denotation for whatever it is the
speaker is trying to communicate.  For example, in most 
contexts, {\em Give me my thing} cannot be used to mean 
{\em Give me my pen}.  A detailed formal account of when a theme
produces good quality coherence and when it doesn't would involve
complex reasoning involving speech acts and the communicative
intentions of the participants, because for a theme to be of good
quality it must be possible to infer why the speaker is talking about
this theme.  This intentional reasoning is beyond the scope of this
paper, and so we rely here on intuitive predictions about when a theme
is bad, and when it violates Quantity.

What's overspecific or underspecific obviously depends on context.
Here, the general brush purpose is replaced with the purpose of
painting or cleaning.  Consequently, hair brushing and all the other
possible purposes that are more specific than brush-purpose have been
ruled out in pragmatics.  Contexts where the speaker intends to convey
{\it paint brush or cleaning brush} and nothing more are rare.  
We cannot compute why painting or cleaning is the topic of the
discourse, to the exclusion of all else in this context.  It should be
stressed, however, that this
violation of Quantity is context sensitive: referring to
{\it paint brush or cleaning brush} in the context below is fine,
because we can work out why the theme involves painting or cleaning
brushes from the discourse context.
\begin{examples}
\item   \label{whackybrush}
Several artists, two hairdressers and the janitor have been 
visited by
a representative from the ACME wonder brush company.  One 
artist and
our janitor now use ACME brushes.
\end{examples}

Since adding $\beta'$ to the {\sc kb} produces a weak discourse,
clause (b) of the Interpretation Constraint is verified with this
$\beta'$.  Indeed, the whole antecedent of the Interpretation
Constraint is verified.\footnote{
If $\alpha$ involves a paint brush but $\beta$
involves a general brush, or if $\alpha$ involves a general brush and
$\beta$ a paint brush, then the common theme would be {\em humans
used a (brush-purpose) brush}, and this wouldn't violate Quantity
in the way that {\it paint-or-clean brush} does (because the latter is
neither specific enough, nor general enough, whereas the former is
general enough).  So the $\beta'$ as stipulated is the only
combination of candidate interpretations of $\alpha$ and $\beta$ that
yield weak coherence.
}
This conflicts with Rembrandt and
Specialisation taken together, since together these latter rules
nonmonotonically yield $\beta'$---that 
the brush Rembrandt uses is a paint brush
and the one the janitor uses is a cleaning brush, but the consequent
of the
Interpretation Constraint is that this doesn't hold.  The result
is irresolvable conflict between Rembrandt and Specialisation, via the
conditions imposed by the Interpretation Constraint.  One cannot
decide which implicature to drop: whether it should be the one that
Rembrandt is using a paint brush, or the janitor using a cleaning
brush, or both.  So although (\ref{brush1}c) is not {\em semantically}
ambiguous in that there is only one logical form, it
is {\em pragmatically} ambiguous.  This pragmatic ambiguity cannot be
resolved and this produces a word pun effect.

Consider (\ref{whackybrush}) again.  As long as discourse attachment
is stipulated as intuitions would dictate, the same $\beta'$ 
(that the artist uses a paint brush and the
janitor a cleaning brush) {\em doesn't} produce a weak discourse
in (\ref{whackybrush})
because the common theme doesn't violate Quantity in this particular
context.  Therefore, unlike (\ref{brush1}c), the Interpretation
Constraint {\em isn't} verified for this same $\beta'$ with respect to the
{\sc kb} for interpreting (\ref{whackybrush}), and so the
Specialisation rules fire via {\sc dmp}, rather than being in
irresolvable conflict with each other via the Interpretation
Constraint.  So (\ref{whackybrush}) is correctly predicted by this
theory to be a discourse context where the zeugma is ameliorated.

As in 
(\ref{brush1}c), there are real world knowledge clues in (\ref{teacher})
that the entities involved probably have different properties 
(one teacher is male and the other
female).
\begin{examples}
\item   \label{teacher} In 1950 the typist and the managing director both
  married teachers.
\end{examples}
However, our theory predicts that, in spite of this common property of
(\ref{brush1}c) and (\ref{teacher}), 
(\ref{teacher}) isn't zeugmatic.  As in
(\ref{brush1}c), the relevant constituents must be attached with {\em
  Parallel} in (\ref{teacher}).  Suppose that the managing director
marries a female teacher and the typist a male one.  Then
generalisation yields the common theme {\em humans married a
(general---i.e., male or female) teacher}.  This isn't a weak theme,
because Quantity isn't violated.  So in contrast to (\ref{brush1}c),
interpreting $\alpha$ and $\beta$ in the way real world knowledge clues would
predict doesn't  produce weak coherence.  So the Interpretation
Constraint doesn't block the pragmatic
implicatures.  Consequently, {\sc dmp} on the relevant 
laws yields the inferences that the managing director
married a female teacher, and the typist a male one.  

Sometimes, there is no scope for re-interpreting a constituent in
order to improve discourse coherence, because the truth
conditional content of the constituents yields weak coherence,
rather than the weakness hinging on pragmatic interpretations alone.
Consider (\ref{thesis1}b).
\begin{examples}
\item   [\ref{thesis1}]
\begin{subexamples}
\item   [b]     That thesis is orange and unreadable.
\end{subexamples}
\end{examples}
As with coordinated {\sc np}s,
we assume that a {\sc vp} which contains some coordinated
constituent (in this case, a coordinated
{\sc ap}), is coherent only if one can
find a rhetorical connection that links the relevant constituents
together; in this case {\em that thesis is orange} and {\em that (same)
  thesis
is unreadable}.

The interpreter's knowledge about syntax, compositional semantics and
pragmatics entails that the only candidate is the relation {\em
  Parallel}.  So one must check that its
coherence constraints are satisfied.  In other words, the constituents
must be structurally isomorphic, and there must be a common theme.
We will give more details of the structural isomorphism condition in
the next section --- it is met here straightforwardly.
The common theme must now be computed, by
generalising on the two constituents {\em That thesis is orange} and
{\em That thesis is unreadable}.  The first constituent predicates the
physical aspect of {\em thesis}, and the second one
predicates its content.  Assuming that generalising over the semantics
for an individuated object and an
abstract representation gives something which is basically empty of content,
the generalisation of the two constituents is {\em Properties of the 
thesis}.  So, the common theme constraint is also met.  But this
common theme is very general relative to the semantic content of the
constituents being connected, and makes the rhetorical connection
weak.  Compare this theme with one that's more specific, such as {\em 
Physical properties of the thesis}.  This would have been the common
theme for (\ref{thesis5}), for example:
\begin{examples}
\item   \label{thesis5}
That thesis is orange and has thousands of pages.
\end{examples}
So although (\ref{thesis1}b) was found to be coherent, the rhetorical
connection is weak.  Moreover, this connection is weak when only the
{\em monotonic} or compositional semantics of the constituents are
assumed.  So the Interpretation Constraint fails to allow us a
revision of interpretation to improve coherence.  This is because
clause (b) in the antecedent of the Interpretation Constraint doesn't
hold.  For suppose $\beta'$ is instantiated to the truth conditional
content of $\alpha$ and $\beta$.  Then it is indeed the case that
$\beta'$ yields weak discourse coherence, but even so
$\beth_{KB}(\beta',\mbox{{\em weak}}(\tau\cup\beta))$ doesn't hold.
This is because $\beth_{KB}(\beta',\mbox{{\em weak}}(\tau\cup\beta))$
means $\mbox{{\em weak}}(\tau\cup\beta)$ is nonmonotonically inferred
from the KB augmented with $\beta'$, but {\em not} from the KB alone.
Here, since the KB entails $\beta'$, $\mbox{{\em
weak}}(\tau\cup\beta)$ is inferrable from the KB alone.
Since the Interpretation Constraint doesn't apply, and since
there is no scope for shifting the interpretation of the constituents
so that coherence is improved, the sentence sounds odd.

Now compare (\ref{thesis1}b) with (\ref{thesis1}a).
\begin{examples}
\item   [\ref{thesis1}]
\begin{subexamples}
\item   [a]     That thesis has thousands of pages
and is unreadable.
\end{subexamples}
\end{examples}
As before, (\ref{thesis1}a) is coherent only if one can compute a
rhetorical connection between the {\sc vp} constituents {\em has
  thousands of pages} and {\em is unreadable}.  Just as in
(\ref{thesis1}b), the same linguistic clues allow the interpreter to
compute a {\em Parallel} relation between them, with the same mapping
% Alex - as above
% between the {\sc eet}s 
and the same common theme.  But, unlike
(\ref{thesis1}b), that's not all.  Real world knowledge also allows the
interpreter to infer a causal link between the constituents.  This can
encoded formally in {\sc dice}.  Let $\mbox{{\em
    Info}}(\alpha,\beta)$ be a gloss for $\alpha$ is {\em x has
  thousands of pages} and $\beta$ is {\em x is unreadable}.  Then
Unreadable states that if one tries to connect such constituents
together in some way, then normally there is a causal relation
between them.
\begin{itemize}
\item   {\tt Unreadable:}\\
$\langle\tau,\alpha,\beta\rangle \wedge \mbox{{\em
    Info}}(\alpha,\beta) > \mbox{{\em cause}}(e_\alpha,e_\beta)$
\end{itemize}
This, together with Result, allow one to infer
$\mbox{{\em Result}}(\alpha,\beta)$ in {\sc ce}.
So there is a stronger rhetorical connection between the constituents
in (\ref{thesis1}a) than there was in (\ref{thesis1}b): as well as the
weak {\em Parallel} connection, the interpreter's background knowledge has
permitted the computation of a {\em Result} connection.  So our theory
correctly predicts that (\ref{thesis1}a) is better than (\ref{thesis1}b).

\subsection{Discourse Coherence and Incremental Processing}

It is well established that interpretation of language
proceeds incrementally (e.g., Marslen-Wilson and Welsh 1978,
Frazier 1979, Crain and Steedman 1985). Here, we investigate how this
affects lexical processing in a discourse context.  Consider
(\ref{brush1}b).
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [b]     Rembrandt used a brush and so did our janitor.
\end{subexamples}
\end{examples}
Now, suppose we assume that we calculate the semantic content {\em
and} pragmatic implicature of the text as soon as enough information
is available to allow an initial decision to be made.  In
(\ref{brush1}b), this means that the interpreter calculates the pragmatic
implicatures of {\em Rembrandt used a brush}, before parsing
{\em and so did our janitor}.\footnote{It is not our intention to
advance fine-grained hypotheses concerning the temporal organisation
of human language processing, since these can only be distinguished by
psycholinguistic experimentation.  Furthermore, the gross (and
uncontroversial) assumption that pragmatic implicatures are calculated
at least on a clause by clause basis is sufficient for our purposes.}

So, using a monotonic compositional semantic construction procedure
such as that in Frank and Reyle (1995), the {\sc drs} $\alpha$ for the first
clause {\em Rembrandt used a brush}, and $\alpha$ is
not only evaluated against the model at this stage, but also the pragmatic
implicatures of this semantic content are calculated in {\sc dice} as well.
\begin{examples}
\item   [$\alpha$]
\topdrs{$x,y,e,t$}{$\mbox{{\em rembrandt}}(x)$\\
$\mbox{{\em brush}}(y)$\\
$\mbox{{\em use}}(e,x,y,t)$\\
$\mbox{{\em holds}}(e,t)$\\
$t\prec\mbox{{\em now}}$}
\end{examples}
So, using the default rule Rembrandt, {\em brush} is specialised
to {\em paint brush}.  Only then does the interpreter begin to parse
{\em and so 
  did our janitor}.  
Again, one uses the constraint-based
grammar to construct the {\sc drs} $\beta$ for
this clause:
\begin{examples}
\item   [$\beta$]
\topdrs{$z,e',t'$}{$\mbox{{\em janitor}}(z)$\\ $\mbox{{\em
      do}}(e',z,c_1)$\\$\mbox{{\em
      holds}}(e',t')$\\ $t' \prec\mbox{{\em now}}$\\$c_1\approx ?$}
\end{examples}
The condition $c_1\approx ?$ marks the fact that (\ref{brush1}b) contains an
elided clause, which must be resolved via the discourse context.
That is, the anaphor $c_1$ must be resolved to a property that $z$ is doing.
We must attach $\beta$ to $\alpha$ with a discourse relation.  This
will then help resolve the `property' anaphor and hence the elided clause.
In this case, coordination constrains the applicable relations to
{\em Parallel} or {\em Contrast} and the use of {\em and} selects {\em
Parallel}.

We must
check that {\em Parallel} is being used coherently: there must be a partial
structural isomorphism and a common theme.
Asher (1993a) defines structural isomorphism
precisely, using the notion of an {\em extended embedding tree} ({\sc eet}) .
An {\sc eet} amounts to the usual {\sc drs} structure, expanded
out so that the {\sc vp} information, which is represented as an
abstracted {\sc drs} (the subject {\sc np} and event and time
variables being abstracted out) is
subordinate to the {\sc drs} as a whole.  So, for example, {\em
  Rembrandt used a brush} is assigned the following
two-level {\sc eet}, where
$\alpha$ is the {\sc drs} above, 
and $P_1$ is the abstracted {\sc drs} given below, which intuitively
corresponds to the property of using a brush (in the past):

{\modsmalltree{1}{
{\node{k1}{$\alpha$}}\\
{\node{P1}{$P_1$}}}
\nodeconnect{k1}{P1}}

\begin{examples}
\item   [$P_1$]
$\lambda e\lambda t\lambda x$\topdrs{$e,x,y,t$}{
$\mbox{{\em brush}}(y)$\\
$\mbox{{\em use}}(e,x,y)$\\
$\mbox{{\em holds}}(e,t)$\\
$t\prec\mbox{{\em now}}$}
\end{examples}

Similarly, the {\sc eet} for the second clause is given below, where
$\beta$ is the {\sc drs} corresponding to {\em so did our janitor}, and $c_1$
is the predicative {\sc drs} anaphor in $\beta$:

{\modsmalltree{1}{
{\node{k2}{$\beta$}}\\
{\node{c}{$c_1$}}}
\nodeconnect{k2}{c}}

Now, we determine the degree of structural isomorphism, by finding
the mapping between the nodes of these {\sc eet}s that is as close to a
bijection as possible. The formal details of how to do this when an
actual bijection isn't possible are to be found in Asher (1993a;286--291).
These details can be skipped over here, however, because in this case
there is a bijection.  Here,
$\beta$ is mapped to $\alpha$ and $c_1$ to $P_1$.  Because $c_1$ is mapped
to $P_1$, the propositional anaphor is resolved to the property
described by $P_1$; i.e., the property of use a brush.  So, the
event $e'$ that the janitor did is use a brush (in the past).  
That is, the {\sc
  drs} in question with the propositional anaphor resolved is $\beta_1$:
\begin{examples}
\item   [$\beta_1$]
\topdrs{$z,e',t'$}{$\mbox{{\em janitor}}(z)$\\ $\mbox{{\em
      do}}(e',z,c_1)$\\$\mbox{{\em 
      holds}}(e',t')$\\ $t' \prec\mbox{{\em now}}$\\$c_1\approx 
\lambda x''\lambda e''\lambda t''\drs{\mbox{$e'',x'',y',t''$}}{
\mbox{$\mbox{{\em
      use}}(e'',x,y')$}\\ \mbox{$\mbox{{\em brush}}(y')$} \\ \mbox{$\mbox{{\em
      hold}}(e'',t'')$} \\ \mbox{$t''\prec${\em now}}}$}
\end{examples}
The model theoretic interpretation of $\beta_1$ as described in Asher
(1993a;251) is equivalent to $\beta_2$---that is, one can reduce the
abstract property the janitor does to the `concrete' event that he
uses a brush:
\begin{examples}
\item [$\beta_2$]
\topdrs{$z,y',e'',t''$}{$\mbox{{\em janitor}}(z)$\\
$\mbox{{\em use}}(e'',z,y')$\\
$\mbox{{\em brush}}(y')$\\
$\mbox{{\em holds}}(e'',t'')$\\
$t'' \prec\mbox{{\em now}}$}
\end{examples}
In this way,
the {\sc vp} ellipsis is resolved as a byproduct of checking the
coherence constraints on {\em Parallel}.  This is compatible with the
approach taken in Hobbs (1985), but the mechanisms described in Asher
(1993a) go beyond those in
Hobbs (1985), in that they handle examples where the clauses have
complex structures (e.g., as those to be found in propositional
attitude reports).

Now, according to our assumption about the process of incremental
interpretation, the interpreter now calculates the pragmatic implicatures
of what has been processed so far.  Because pragmatic implicature is
nonmonotonic in nature, it is important to retract old pragmatic
inferences before calculating the new ones in the light of the new
information.  In this case, the specialisation of {\em brush} to {\em
paint brush} that occurred earlier isn't considered to be fixed.  
So new information has the ability to force a reinterpretation of this word.
This happens here because, just as in (\ref{brush1}c),
the default rules Rembrandt and Specialisation apply, and if they fire
then the common theme that is demanded by the coherence constraints on
{\em Parallel} violates Grice's Maxim of Quantity. So the
Interpretation Constraint applies as well, where $\beta'$ in that
constraint is the information that Rembrandt uses a paint brush and the
janitor a cleaning brush.   
So just as in (\ref{brush1}c), there is irresolvable conflict between
Rembrandt, Specialisation and the Interpretation Constraint.  We
cannot decide whether to assume
Rembrandt doesn't use a paint brush, or the janitor doesn't use a
cleaning brush, or both.

The interpreter has therefore been forced to reinterpret the word {\em
brush}, in the light of processing the subsequent phrase {\em and so
did our janitor}.  Now, there is indecision about how to interpret it,
where there wasn't indecision before the elided clause was parsed.

Calculating pragmatic implicature as one processes constituents,
rather than doing it all at the end, is a risky but necessary strategy,
because of limitations of memory, the utility of rapid comprehension,
and so forth.  Here, the risk has not paid off, and reinterpretation
has been forced.  Hence there is a word pun effect.  In addition,
because there are conflicting messages about how to specialise {\em
brush} once all the relevant information has been processed in
(\ref{brush1}b), this contributes further to the word pun effect.  The
effect is more pronounced in (\ref{brush1}b) than it was in
(\ref{brush1}c), exactly because reinterpretation as well as
irresolvable conflict occurs.

Now consider an example involving word repetition.
\begin{examples}
\item   [\ref{bank3}]   John banked the money and then he 
banked the plane.
\end{examples}
As before, we calculate the pragmatic interpretation of {\em John
  banked the money}, before parsing the second constituent.
Background knowledge allows one to infer in {\sc ce} that {\em bank}
in this phrase should be interpreted as {\em bank}$_{\mbox{\tiny deposit}}$ (that
is, the verb with the financial sense).  Now we parse the second
constituent.  We must attach it to the first.  The similarity in
syntactic structure and word forms used prompt one to infer a {\em
  Parallel} relation (as well as {\em Narration}).  Both require
coherence constraints to be met.  We'll concentrate here on those for
{\em Parallel}.  We must find a common theme.
The common theme for John bank$_{\mbox{\tiny deposit}}$ed
the money and bank$_{\mbox{\tiny turn}}$ed the
plane is {\em John doing things}, which is very general relative to
the semantic content of the constituents being connected, and so yields
weak discourse coherence.  Checking the coherence constraints on {\em
Narration} is similar, and so yields no improvement.  
Therefore, the Interpretation Constraint applies
where $\beta'$ stipulates that John bank$_{\mbox{\tiny deposit}}$ed the money and
bank$_{\mbox{\tiny turn}}$ed the plane.  
But there is real world knowledge that yields these
interpretations of the two occurrences of {\em banked} in
(\ref{bank3}).  The Interpretation Constraint thus blocks both pieces
of real world knowledge from firing simultaneously, much like it did in
(\ref{brush1}c).  But the relevant rules have
unrelated antecedents.  So there is irresolvable conflict between the
real world knowledge that money is bank$_{\mbox{\tiny deposit}}$ed  
rather than bank$_{\mbox{\tiny turn}}$ed and
planes are bank$_{\mbox{\tiny turn}}$ed rather than 
bank$_{\mbox{\tiny deposit}}$ed, thanks to the
Interpretation Constraint.  The listener therefore fails to conclude what
the interpretations of the word forms {\em
  bank} should be; she cannot work out which implicature from real world
knowledge
to drop.  
This
produces a word pun effect.  Again, two things  contribute to it:
reinterpretation of the word {\em bank} is forced, and there is ultimately
irresolvable conflict about how to interpret it.  As we mentioned
in \S\ref{wordrep}, {\sc 
  mle}s alone don't predict the word pun in (\ref{bank3}) because of
the word repetition.
We have formally encoded
discourse interpretation principles and shown how they contribute to
the word pun.

Exactly the same line of reasoning explains the word pun in
(\ref{brush2}):
\begin{examples}
\item   \label{brush2}       
Rembrandt used a brush and our janitor used a brush.
\end{examples}
So (\ref{brush2}) and (\ref{brush1}c), which are truth
conditionally equivalent, are both zeugmatic but for slightly 
different reasons.  (\ref{brush2}) involved reinterpretation where
(\ref{brush1}c) did not, but both ultimately yield irresolvable
pragmatic ambiguity via conflicts involving Rembrandt and Specialisation.

\subsection{The Absence of Crossed Readings}

In cases where there are {\sc mle}s, such as for {\em file}
and {\em bank}, the
account of non-crossed readings when there is a single occurrence of
the word form in the sentence is the same as the traditional {\sc mle} approach:
\begin{examples}
\item   [\ref{file1}]   John and Bill each have a file.
\item   [\ref{bank30}]  Texas and Alabama have preservation orders on
  their most beautiful banks.
\end{examples}

However, by adding information flow between lexical organisation and
discourse inference, we can also provide a complementary account of
non-crossed readings in certain cases where there aren't {\sc mle}s, or
where there is word repetition.
Consider (\ref{brush1}d).
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [d]
The grandparents gave the children brushes for Christmas.
\end{subexamples}
\end{examples}
Our theory predicts that 
this sentence doesn't have a reading where one child got a paint
brush, and the other a hair brush.  This is because we have assumed that
there is a general lexical entry for {\em brush},
which refers to paint brushes, hair brushes, tooth brushes and so on.
Because of this, the word {\em brush} fails to have a specialised
meaning unless something in the context gives it one.  In
(\ref{brush1}d), there is nothing in the context which causes
specialisation.  Since {\em brushes} is general we
don't interpret the sentence specifically as one child getting a paint
brush, and the other a floor brush, for example.  Rather, the sentence
is interpreted as vague or general about what kind of brushes the
children received.  Here, the absence of a crossed reading is
epiphenomenal.  And with further context, a crossed reading can be
obtained:
\begin{examples}
\item   [\ref{brush1}]
\begin{subexamples}
\item   [d]     The grandparents gave the two children brushes for
  Christmas.  \\One got a hair brush and the other a paint brush.
\end{subexamples}
\end{examples}
Now consider an example where there is word repetition.
\begin{examples}
\item   [\ref{file3}] John had a file and Bill had a file.
\end{examples}
There are four possible logical forms of this sentence, corresponding
to the four permutations of $\mbox{{\em file}}_{\mbox{{\tiny tool}}}$
and $\mbox{{\em file}}_{\mbox{{\tiny dossier}}}$ 
for the two occurrences of {\em file} in the sentence.
The Interpretation Constraint rules out the crossed readings, however.
We must attach the constituents {\em John had a file} and {\em Bill
  had a file} together with a rhetorical relation, to check the
coherence of the coordination.  The only candidate relation is {\em
  Parallel}.  If the reading is crossed, then the common theme which
the {\em Parallel} relation demands is weak, because it's too general
({\em humans have physical artefacts}).  So, let $\beta'$ be the
information that the files Bill and John have are `crossed'.  Then the
Interpretation Constraint applies with respect to the {\sc kb} which
contains the {\em disjunction} of attachment tasks corresponding to
the four alternative logical forms.  Thus {\sc dmp} on the
Interpretation Constraint yields $\neg \beta'$; i.e., the files
aren't `crossed'.  Consequently, the interpreter non-monotonically infers
that two of the four alternative logical forms is ruled out by the
Interpretation Constraint.  The remaining two logical forms amount to:
either John and Bill both have files$_{\mbox{{\tiny tool}}}$, or they
both have files$_{\mbox{{\tiny dossier}}}$. 
The interpreter cannot choose between these two alternatives, but he does at
least infer that (\ref{file3}) doesn't have a crossed reading.

In contrast, (\ref{teacher2}b) can have a crossed reading, because
there is only one logical form for (\ref{teacher2}b) since the
predicate {\em teacher} is neutral about gender, and the
Interpretation Constraint doesn't apply when $\beta'$ is {\em The
  teachers John likes are male and those that Bill likes are female},
because the common theme is acceptable ({\em humans like their
  (general)
teachers}).
\begin{examples}
\item   [\ref{teacher2}]
\begin{subexamples}
\item   [b]     John and Bill both like their teachers.
\end{subexamples}
\end{examples}

\section{Conclusion}

We have argued that viewing lexical ambiguity as 
caused by the existence of discrete lexical
units is too simplistic and that this view of the lexicon can tell only
part of the story concerning word puns and non-crossed readings.  Our
complementary account formalises the pragmatic
reasoning underlying the {\sc mle} approach to zeugma and non-crossed
readings involving homonymous ambiguity, but also
explains how zeugma and non-crossed readings occur in cases involving
word repetition and constructional
polysemy.  In these cases, zeugma is caused by
information flow between
lexical organisation, grammatical
structure, pragmatic reasoning, coherence constraints on
discourse structure and incremental interpretation.  
Similarly, the absence of crossed readings was not due to lexical
organisation alone, but the 
information flow between lexical organisation and
discourse coherence constraints.
This is in accord with the assumptions in Copestake and Briscoe (1995) that
zeugma can arise in the absence of multiple lexical entries.  

We studied all these interactions in 
a formal setting, by providing links
between a lexicon with hierarchical organisation, {\sc sdrt} and
{\sc dice} and showed how the logic of {\sc ce} which underlies
{\sc dice} validates the inferences which predict
zeugma and non-crossed readings in the right
cases.  This led to some general conclusions about linking the
various components.  First, discourse coherence constraints don't work
independently from lexical ambiguity.  The Interpretation Constraint
records this dependency because it ensures that pragmatic inferences
about what a word means can be blocked if they have an adverse effect on
the overall coherence of the discourse.  Word pun
effects are caused by (a) irresolvable conflict in pragmatic
interpretation, created by axioms that link lexical interpretation with the
interpreter's background knowledge; (b) weak rhetorical connections between
the constituents; and (c) reinterpreting incremental interpretation,
which is forced by subsequent information in the discourse.  In future
work, we will show that this account can provide an adequate analysis
of the effects attributed to
degrees of discreteness in readings of words such as {\em brush}
(compared with {\em teacher}) discussed in Cruse (1995), without
making structural distinctions in the lexicon.

{\small
\section*{References}
% first argument - i.e. #1 is always author
% #2 is title
% last is date
% agreed author format is, for example
% Alshawi, H., D. Carter, M. Rayner and B. Gamb\"ack
% 
% the only formatting commands that should appear in fields
% are those to mark idiosyncratic information like accents
\newcommand{\book}[4]{\item #1 (#4) {\it #2,} #3.}
% #3 is publisher
\newcommand{\unpub}[4]{\item #1 (#4) `#2', #3.}
% for tech reports etc
% #3 is information about institution etc
\newcommand{\bookart}[7]{\item #1 (#7) `#2' in #5 (ed.), {\it #4,} #6, pp.~#3.}
% published book article (old \barticle)
% #3 is page numbers 
% #4 book title
% #5 editors
% #6 publishers
\newcommand{\bookartnoed}[6]{\item #1 (#6) `#2' in {\it #4,} #5, pp.~#3.}
% book article where editors are unknown
% #3 is page numbers 
% #4 book title
% #5 publishers
\newcommand{\bookartnopp}[6]{\item #1 (#6) `#2' in #4 (ed.), {\it
#3,} #5.}
% book article where page numbers are unknown
% #3 book title
% #4 editors
% #5 publishers
\newcommand{\forthart}[6]{\item #1 (#6, forthcoming) `#2' in #4 (ed.), {\it #3,} #5.}
% article in forthcoming book (old \farticle}
% #3 book title
% #4 editors
% #5 publishers
\newcommand{\journart}[6]{\item #1 (#6) `#2', {\it #3,} {\bf #4},
#5.}
% article in journal (old \jarticle)
% #3 name of journal
% #4 volume
% #5 pages
\newcommand{\forthjournart}[4]{\item #1 (in press) `#2', {\it #3}}
% article in journal no page numbers
% #3 name of journal
% #4 volume
\newcommand{\journartnopp}[5]{\item #1 (#5) `#2', {\it #3,} 
{\bf #4}.}
% article in journal no page numbers
% #3 name of journal
% #4 volume
\newcommand{\procart}[6]{\item #1 (#6) `#2', {\it Proceedings of the
#3,} #4, pp.~#5.}
% article in proceedings (old \particle)
% #3 name of conference
% #4 venue
% #5 pages
\newcommand{\pubprocart}[7]{\item #1 (#7) `#2', {\it Proceedings of the
#3,} #4, ,#5, pp.~#6.}
% #3 name of conference
% #4 venue
% #5 publishers
% #6 pages
\newcommand{\procartnopp}[5]{\item #1 (#5) `#2', {\it Proceedings of the
#3,} #4.}
% article in proceedings without known page numbers
% #3 name of conference
% #4 venue
\newcommand{\pubprocartnopp}[6]{\item #1 (#6) `#2', {\it Proceedings of the
#3,} #4, #5.}
% article in proceedings without known page numbers
% #3 name of conference
% #4 venue
% #5 publishers
\newcommand{\procartnoppnoplace}[4]{\item #1 (#4) `#2', {\it Proceedings of the
#3}.}
% article in proceedings without known page numbers without place
% where meeting happened
% #3 name of conference
\newcommand{\procartnoplace}[5]{\item #1 (#5) `#2', {\it Proceedings of the
#3,} pp.~#4.}
% article in proceedings where the venue is not known
% #3 name of conference
% #4 pages
%\newcommand{\acqwp}{ESPRIT BRA-3030 ACQUILEX WP NO.}
%\newcommand{\aunpub}[5]{\item #1 (#4) {\it #2.}\\ \acqwp #3}


\begin{list}{}
   {\leftmargin 1.6em
    \itemindent -\leftmargin
    \itemsep 0pt plus 1pt
    \parsep 0pt plus 1pt}

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\journart{Asher, N. and P. Sablayrolles}
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%van Deemter, 1990)
% OTS-WP-CL-91-001
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    {de Gruyter, Berlin}
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recognition in continuous speech}
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Meeting of the Chicago Linguistics Society}
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        {}
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%
\journart{Pustejovsky, J.}
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%Pustejovsky (1994)
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\bookart{Zwicky, A. M. and J. M. Sadock}
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\end{list}
}

\end{document}









