Over the last two decades, semantic theory has been marked by a continuing shift from a static view of meaning to a dynamic one. The increasing interest in extending semantic analysis from isolated sentences to larger units of discourse has fostered the intensive study of anaphora and coreference, and this has engendered a shift from viewing meaning as truth conditions to viewing it as the potential to change the "informational context". One of the central problems of discourse analysis is the
treatment of anaphorical expressions, of discourse pronouns and definite NPs.
The traditional solution is to take pronouns as bound variables and to analyze
definite NPs by means of the Russellian iota inversum operator; and this
solution is usually taken over also by those semantic theories which take the
dynamics of language at face value. However, as we try to show, theories
falling in with this approach necessarily stop half way in the dynamic
enterprise and, therefore, cannot achieve a satisfactory semantical analysis of
anaphora. In the present paper we propose to apply the dynamic approach
uniformly to all expressions. We give a dynamic treatment not only to meanings
of sentences and supersentential units of discourse, but also to those of
pronouns and NPs. This is possible by exploring the intuitive idea of salience
and by its formalization by means of choice functions. A definite NP In an utterance such as The present theory is worked out as a modification of
Groenendijk & Stokhof's (1991) dynamic logic and is hence based on the idea
of treating the meaning of a sentence as the relation between its
"input" and "output" referential links; it differs from Groenendijk
and Stokhof's classical version of dynamic logic in formalizing referential
links not by means of assignments of objects to discourse markers, but rather
by means of choice functions. This makes it possible to analyze NPs not via
quantifiers, but instead simply as terms (already Hintikka 1974); and
consequently to dispose of any kind of variables or discourse markers, thus
making the structure of semantic representation straightforwardly correspond to
that of the analyzed discourse The paper is organized in the following way. In section 2
we discuss the motivation for dynamic semantics as rooted in the shift from
sentence semantics to discourse semantics. Connected with this shift we can
perceive at least three kinds of important, and mutually interrelated problems:
the basic problem of the compositional analysis of an anaphora-laden discourse,
the problem of how to account for crossreference - where possible - by
linguistic means, and the problem of a uniform representation of anaphorical
expressions. We show that dynamic logic, although it solves - besides other -
the first of these problems in a very elegant way, does not really tackle the
other two. Anaphorical NPs continue to be analyzed via Russellian definite
descriptions governed by the problematic uniqueness condition, and the right
resolution of anaphorical links is simply presupposed. In section 3 we propose
our modification of dynamic logic that would solve the latter two of the
mentioned problems. We give the definition of the apparatus and we show how to
apply it. Then we hint at various ways of extending the apparatus with the aim
to cover further linguistic data; and, of course, we cannot avoid giving our
own account of donkey sentences. Finally, we discuss further extensions of our
theory to other quantifiers and events. In section 4 we give the conclusion.
The new problems brought about by extending analysis from isolated sentences to larger units of discourse and texts were extensively discussed by Geach (1962). He used classical predicate logic; examining (1) he arrived at the straightforward analysis (1'). A man walks. He
whistles
(1) It is easy to see that (1') is an adequate analysis of (1); and we should expect the two sentences constituting (1) to be analyzable in such a way that the two analyses would add up to (1'). However, although there is no problem with analyzing (2) as (2'), there is no straightforward way to analyze (3): A man
walks
(2) This is the most general problem; we can call it The development of a new family of semantic theories, all
of which are based on a dynamic notion of meaning, was stimulated especially by
a desire to solve the general problem of compositionality. Some of the most
influential among them (esp. Groenendijk & Stokhof, 1991) adhere to the
Geachian tradition in representing anaphorical pronouns as bound variables. An
alternative view stems from Evans and represents discourse pronouns as definite
descriptions (Evans 1977, Slater 1988, Neale 1990, Heim 1990, Chierchia 1992,
van der Does 1993). However, such theories fail, or have difficulties, in
solving the problem of definite descriptions.
Groenendijk & Stokhof (1991) offered an elegant
solution to the problem of compositionality they introduced a "dynamic
version" of the existential quantifier, a version which binds variables
even outside what would traditionally be considered as its scope. (The side
effect of this move is that variables loose something of their essential
character and that they become what Groenendijk & Stokhof call $ The key to the dynamization of $ is that instead of taking the semantic value of a sentence to be a truth value it is taken to be a relation between assignments of values to discourse markers. However, Groenendijk & Stokhof's solution is less
satisfactory on closer inspection. Our success in analyzing (2) and (3) was
conditioned by the fact that we used the same discourse marker in both formulas
- if we used d In fact, this is only a new reincarnation of the
traditional problem of how to get the antecedent of an anaphorical term. The
traditional way to handle this is to interlock corefering expressions by means
of coindexing - but this clearly does not solve the problem, but only moves it
outside of the theory. It is assumed that our formal model starts where coindexing,
and hence anaphora resolution, is already done. Groenendijk & Stokhof
accepted this common policy; they assume that the choice of right discourse
markers is something which is either taken for granted, or at least a job for
someone else.
Discourse representation theories (Kamp 1981, Heim 1982) reduce definiteness to the principle of familiarity according to which a definite expression refers to a familiar or already introduced discourse referent. In this way, definite and indefinite NPs can both be represented as terms (or as variables). However, such theories usually do not distinguish between different familiar discourse referents, i.e. they do not solve the problem of coreference. Dynamic logic, by contrast, usually keeps - in effect - to the classical representation of indefinite NPs as existential quantifiers and definite NPs as Russellian iota terms. Even the dynamic interpretation of indefinite NPs with nondeterministic
programs, as proposed by van Eijck, although in some aspects similar to our
employment of choice functions, still cannot escape this Russellian
predicament. Van Eijck (1993, p. 240) writes: "[...] in a dynamic setup
the interpretation of an indefinite description can be viewed as an act of picking
an arbitrary individual, i.e. as an indeterministic action". Therefore,
instead of employing the (dynamic) existential quantifier, he uses a dynamic
eta term for representing indefinite NPs. The eta operator is not interpreted
as a term-creating operator, but rather as dynamic quantifier: "Note that h and i are program
building operators (in fact, dynamic quantifiers) rather than term building
operators, as in the logic of Hilbert & Bernays" (van Eijck 1993, p.
245). Furthermore, definite noun phrases are treated according to the
Russellian uniqueness condition interpreting definite noun phrases: "It is
not difficult to see that this [i.e. the interpretation conditions for i] results in the
Russell treatment for definite descriptions." (ibid.). This indicates that
even this modification of dynamic logic does not suceed in applying the dynamic
approach in a way such as to dispose of the classical framework which would
seem to be responsible for posing the classical problems.
Here we would like to attack the classical problem via a new approach; we propose a modification of dynamic logic which makes do without discourse makers and hence without quantifiers. The idea, taking its cue from the considerations of von Heusinger (1992), is simply to replace assignment of values to discourse markers by Hilbertian choice functions. Roughly speaking, we can say that if we encounter an indefinite noun phrase, then we do not introduce anything like a discourse marker that would be assigned a referent; instead we let the referent be associated with the noun phrase itself via the extension of its predicative part. Formally, we introduce The basic idea is that an occurrence of an indefinite noun
phrase
Let us assume that we have the non-empty universe U of
individuals. An DEF1. EPS We further define DEF2. UPD = {f | D(f)=EPS´U´Pow(U) and R(f)ÍEPS} The basic epsilon-update, which we shall denote as upd DEF3. upd upd We shall write e'=e Now we can define a fragmentary language to illustrate how epsilon functions can be used to build dynamic semantics. We do it in three steps, describing the lexicon, the syntax and the semantics. DEF4a. (lexicon) There are the following categories of
expressions; the constants of each category (if any) are listed in brackets: Note that there are neither variables, nor discourse markers within our language. This is made possible by the fact that we represent indefinite NPs as terms expressing epsilon updates, and that we do not treat anaphorical relations in any way evocative of the binding of predicate logic. DEF4b. (syntax) DEF4c. (semantics) ||T|| if T is a constant term We extend the function || || to the categories of terms and sentences so that if E is a term or a sentence, then ||E||ÍEPSxEPS: 1a. || Let us add some explanations to bring out the insights
behind the apparatus. An indefinite NP A definite NP The atomic sentence is semantically characterized via its potential to change the current epsilon function e to the updated function e' by way of the subsequent application of the updates expressed by its terms. Thus, e and e' must be connected by a sequence of epsilon functions such that the adjacent pairs of the sequence fall into the respective updates expressed by the terms; and the referents of the terms must fall into the extension of the predicate. Here we differ essentially from usual dynamic logic in that we consider atomic sentences as internally and externally dynamic. The logical operators The semantic interpretation for the quantifier
In this section we discuss some simple examples
illustrating the mechanism of establishing anaphorical links by linguistic
means. As already mentioned, we assume that the semantic contribution to
finding the adequate antecedent can be described in terms of the salience
change potential of expressions. We start our analysis with a simple atomic sentence with an indefinite NP. A man
walks
(4) Sentence (4) is assigned the formula (4'); this formula is
then interpreted according to the definitions given above. As we have noted, a
pair of epsilons <e,e'> falls into the update expressed by an atomic
sentence iff e and e' are connected by a sequence of epsilons such that the
adjacent pairs of the sequence fall into the respective updates expressed by
the terms and the referents of the terms fall into the extension of the
predicate. Since we have only one term in (4), this reduces to the condition
that <e,e'> falls into the update expressed by The man
whistles
(5) Sentence (5) with the definite NP The analysis of the concatenation of (4) and (5) now shows
how the referent of the anaphorical NP A man walks. The man
whistles
(6) <e,e'> falls into the update expressed by (6') if and only if there is an epsilon function e'' such that <e,e''> falls into the update expressed by (4') and <e'',e'> falls into the update expressed by (5'). Using the results of the above analyses and eliminating redundancies, we reach the result that <e,e'> falls into the update expressed by (6') iff e' differs from e at most in the representative of the class of men and this representative is a walker and a whistler. This derivation shows the basic mechanism of relating anaphorical expressions to their antecedents; it illustrates how the semantic principle of familiarity can be explicated in terms of the salience change potentials of the expressions involved. The information that a new object is raised to salience can be thought about as passed on by the updated epsilon functions; and since the subsequent definite term is interpreted according to these updated epsilon functions, it comes to refer to the very same object. Let us now turn our attention to sentences about farmers and donkeys. A farmer owns a
donkey
(7) The farmer beats the
donkey
(8) The atomic sentence (7) contains two indefinite NPs. Since both terms lead to modifications of the actual epsilon function, the eventual output function is modified in respect to its values for both the set of farmer and the set of donkeys. For each of the two sets, a new representative is chosen; the two representatives must be such as to stand in the relation of owning. The sentence (8) does not lead to any updating, since both its terms are definite and act as mere tests. Therefore, its output epsilon function is identical with the input one. Again, the concatenation of the two sentences shows how the epsilon function updated by the first conjunct creates a new context according to which the definite expressions in the second conjunct are interpreted. A farmer owns a donkey. The farmer beats the
donkey. (9) These simple examples show the general principles of
linking anaphorical expressions to their antecedents. The indefinite expression
changes the informational context so that the subsequent definite expression
refers to the object which it has selected. The last example has demonstrated
that the epsilon function can carry information about more than one object.
We have seen that classical examples are adequately analyzable by our apparatus introduced so far; however, there are still simple cases which remain outside its scope. We are now going to show that such cases are analyzable by way of various simple enhancements of the apparatus. We saw that we were able to adequately analyze (6); and in
force of DEF4c (1c) the same holds for (10) - the point is that the semantics
of A man walks. He
whistles
(10) This does not work in case of (11) or (12) - A farmer walks. He
whistles
(11) However, this can be improved by replacing our epsilon
update upd DEF5. Let CAT={M,W,T} be a subset of Pow(U) such that M=||man||, W=||woman|| and T=||thing||. Let Ct be the function from Pow(U) into CAT such that if there is a cÎCAT such that sÍc, then Ct(s)=c and Ct(s) is undefined otherwise. upd If we now modify 1a of DEF4c by substituting upd Walk(a(farmer));
Whistle(he)
(11') Own(a(farmer),a(donkey));
Beat(he,it) (12') We can, of course, go further and let an indefinite noun
phrase DEF6. upd Plugging upd A boxer walks. The sportsman whistles. (12) The right solution seems to be somewhere in between upd Another kind of problem is posed by proper names. We are as
yet unable to account for (14). However, this is easy to repair. Let us add 1f. ||T|| = {<e,e'>| e'=e This allows us to give the analysis of (14) (note that if
e'=e John walks. He
whistles
(14)
Donkey sentences serve as a touchstone for every semantic
theory since they combine several intricate problems. In what follows, we try
to show how we can cope with this kind of sentence within the proposed
framework. The main problem is how an indefinite NP embedded in the restrictive
clause can furnish the antecedent for a subsequent expression. This is the case
of the indefinite NP Every farmer who owns a donkey beats it (15) Before we analyze this sentence, let us inspect a simpler
one to see how the definition of the universal quantifier works. Every man who is a farmer is
boring
(16) The analysis of the sentence yields a test that succeeds if every output epsilon function of the first clause can serve as an input to the second clause. One branch of the test can be imagined as, first, picking up a new representative of the set of men such that he is a farmer (the first clause), and, second, testing whether this representative is boring (the second clause); the whole test is successful if all such possible branches are. The testing is independent of an input epsilon function, i.e. either every epsilon function passes it, or none does. The crucial problem of donkey sentences is solved by
quantifying over epsilon functions instead of over individuals, which is the
result of our disposal of variables and binding. The problem concerns the
anaphorical link between the embedded indefinite NP every(Own(a(farmer),a(donkey)),Beat(he,it))
(15') Again, the whole expression acts as a test that succeeds if
every epsilon function resulting from interpreting the first clause can
successfully act as input for the second clause. Since there are two indefinite
NPs within the first clause, they both modify the input epsilon function, and
so we have to apply the test connected with the second clause to all such
functions differing from the basic output of the whole sentence in the
representatives of the class of farmers and donkeys which pass the test
connected with the first clause. Thus the anaphorical relation is encoded in
the context information and in this way it can cross the scope of the
indefinite NP.
Another way to enhance our apparatus is by the introduction
of events and event-quantification. A way to do this is to introduce the new
category E coinciding with the extension of the new predicate
Then we can analyze (17) as follows: If a farmer owns a donkey, he beats
it.
(17) The relationship between (15') and (17') now depends on how
we treat events, namely which kind of relationship between sets like
{<x,y>| farmer(x)&donkey(y)&own(x,y)} and {<x,y,e>|
farmer(x)&donkey(y)&event(e)&own(e,x,y)} we stipulate. It seems to
be clear that the latter set can be projected on the former and hence that
(17') implies (15'); the other direction of inclusion and implication is,
however, doubtful - but we do not elaborate on this here.
Even after the enhancements outsketched in this chapter, the framework remains in many respects too simple to allow for a satisfactory analysis of some statements and pieces of discourse. However, this was - at least partly - intentional: we wanted first and foremost to bring out the basic principles of the way we think the idea of the exploitation of salience can be formalized and put to work. Let us now mention - briefly - directions in which the possibility of further elaboration of the framework seems to be obvious.
In this paper we have proposed a modification of
Groenendijk and Stokhof's dynamic logic, a modification based on the employment
of choice functions. Dynamic logic was developed to give a compositional
semantic account of discourse, i.e. primarily to cope with the problem of
coreference extending sentence boundaries. However, it did not tackle two
important problems - the problem of resolving linguistically detectable
anaphorical links and the problem of a uniform representation of anaphoric
expressions, i.e. of definite NPs and pronouns. We have tried to show that
modifying it in the proposed way can yield a feasible account of both these
problems. Both definite and indefinite NPs are represented as proper terms;
they are taken to express choice-function-updates and to refer to actually
chosen representatives. The definiteness lies in the fact that a definite NP
refers to the already established representative, whereas an indefinite NP
picks up a new element, which then becomes the new representative. In this way,
the anaphorical linkage is encoded in the contextual information, which is
passed on by epsilon functions. No preceding coindexing is necessary.
Furthermore, the formalism yields a uniform representation of all definite
expressions, i.e. of definite NPs and pronouns.
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1. 2. See Sgall et al. (1986), esp. ¡ì1.35. Choice functions, as we employ
them, can plausibly be seen as a way of explicating "the hierarchy of
activation of the elements in the stock of knowledge that the speaker assumes
to be shared by the hearer" (ibid., p.55). See also Peregrin (1992a). 3. Let us clear away a possible misunderstanding, which
often occurs with respect to dynamic semantics. To facilitate readability, we
sometimes speak about the meaning of a dynamically interpreted statement as if
about a process - but it must be kept in mind that this is nothing more than a
metaphor. The sole aim of logical analysis and logical formalization, which is
what we pursue in this paper, is the articulation of truth conditions, it is
not a description of any processes which might go in speakers' heads or
wherever. The shift from truth conditions to a context change potential must
not be literally seen as a shift to the description of some mental events
underlying discourse utterances - we do not see semantics as a part of
psychology. The specificum of dynamic logic is that it realizes that truth
conditions of certain larger units of discourse ( 4. This accomplishment is what Quine (1986, p.15) considers
"a major step in conceptual analysis". Cf. also Peregrin 1992b. 5. The idea of employing Hilbert's epsilon operator for the
purpose of analyzing indefinite NPs in a way analogous to the employment of
Russell's iota inversum for the purpose of analyzing definite NPs by means of
his iota inversum is of course far from really new. However, this idea is not
easy to exploit, as the nature of the epsilon operator differs from that of the
iota inversum one (the former clearly does not render the indefinite article as
the name of a definite function in a way corresponding to the latter's
rendering the definite article as the name of the function mapping singletons
on their single elements). 6. In contrast to the theories of Kamp and Heim,
Groenendijk and Stokhof tried to retain as much traditional logic as possible;
and they managed to show that the dynamics of discourse does not necessarily
call for a framework which would eschew the basic principles of traditional
logical analysis. See also Peregrin (1992a). 7. Epsilon functions have been extensively studied by
Hilbert and Bernays (1939); however, let us keep in mind that Hilbert's
motivations, and consequently also his theory of epsilon functions, was quite
different from the present one. 8. Some further restrictions might be added; we might for
example require that for every epsilon function e, e(A) is undefined, i.e. that
the empty class is never assigned a representative. Or we might stipulate that
for every epsilon function e and for every element d of the universe, e({d}) =
d (i.e. that the singleton of d is always automatically represented by d); this
would make our analysis work also in cases where the classical Russellian
mechanism for interpreting definite descriptions were required. We do not
explore this here -since this paper aims only to bring out the basic principles,
not to elaborate details. 9. e' = e 10. This semantics provides a unified treatment of definite
and indefinite NPs. Both determine their referents by using an epsilon
function. This similarity of definite and indefinite NPs was pointed out by
Egli (1991). 11. This view meets the Etype analysis of discourse
pronouns; it could be seen as stemming from Quine (1960, 102f.): "Often
the object is so patently intended that even the general term can be omitted.
Then, since 'the' (unlike 'this' and 'that') is never substantival, a 12. Obviously there are several other nonlinguistic
factors that influence and rule the relation between the antecedent and the
corresponding anaphorical expression, but they are not subject to the present
considerations. 13. In the English case - in contrast to languages like
German or Czech - this is not wholly correct, for a NP like 14. It might seem that we need an update changing not only
supersets of the class referred to by the predicative part of the indefinite
NP, but also its subsets. We may follow the utterance 15. Note that as our semantics is such that sentences are assigned
the same type of semantic values as terms (namely sets of pairs of epsilon
functions), it would be only a trivial matter to extend our quantifiers so as
to accept as arguments not only sentences, but also terms. Then (16') could be
alternatively stated as 16. This treatment of donkey sentences uses a kind of
unselective binding and, therefore, it cannot represent asymmetric readings. As
in other theories, additional constraints must be imposed to license asymmetric
readings. |
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