Jun 23, 1993 - Barwise, Jon and Robin Cooper (in press) Extended Kamp Notation: a Graphical Notation for Situation Theory, in Situation Theory and its ...
Towards a general semantic framework Robin Cooper Centre for Cognitive Science and Human Communication Research Centre University of Edinburgh 23rd June 1993
0 Introduction I believe that if we are to develop general semantic formalisms for use in computational semantics, we should only do this on the basis of a deep understanding of the relationships between the various semantic theories on o er. In fact, I think it is more important that we develop a general semantic framework with associated algorithms and computational techniques which o ers the opportunity to de ne reusable theory independent semantic modules than to de ne a formalism as such. There may indeed be a single canonical formalism but it may also be the case that there is a family of formalisms all based on the same framework which are suitable for di erent applications. In this paper I shall o er a few introductory remarks on what such a semantic framework might be and then focus on a contribution that might be made by one mathematical tool recently developed by Peter Aczel and Rachel Lunnon.
1 What is a semantic framework? A semantic framework should be a formal system which abstracts core notions which are shared by the various approaches to natural language semantics which are on o er. At the same time the framework should be rich enough to specify the particular proposals of the An earlier version of this paper was presented at the Dagstuhl Seminar \Semantic Formalisms in Natural Lanugage Processing", 22nd-26th February, 1993. This document is part of a deliverable presented in the ESPRIT BR Project 6852, DYANA-2
1
individual approaches so that it becomes clear in formal terms exactly where the approaches di er. The computational advantages of such a framework are that it enables the implementation of modules relating to the core notions, perhaps parameterized in order to account for di erences in the di erent approaches. This means that there will be some general computational techniques available which can be used no matter which theory you choose to base your implementation on. Also, it means that implementations need not be restricted to one particular theory. It should give us the opportunity to mix and match analyses cast within various approaches. The fact that everything is embedded in a general framework should enable us determine which analyses from the di erent approaches are compatible. There are three strategies for creating such a framework which might be pursued and I feel that they should ultimately complement each other:
semantic operators This is the approach introduced by Johnson and Kay (1990). It invol-
ves abstracting out various operators such as APPLY, NEW INDEX, CONJOIN and interpreting these operators di erently for di erent semantic theories such as Montague semantics, discourse representation theory and situation semantics. formal speci cation This would involve the application of speci cation techniques developed in computer science as described, for example, in work on algebraic speci cation (Sannella and Tarlecki, 1992) and in research on the Logical Framework (Harper, Honsell and Plotkin, 1987). semantic metatheory This involves the development of a general theory in which the various individual semantic theories can be cast. In this paper I want to develop an example relating to the last of these options, since I feel that making progress on this front would contribute to the aims of the other two strategies.
2 Aczel-Lunnon abstraction in situation theory I am going to concentrate on the notion of abstraction and try to argue that the particular kind of abstraction proposed by Aczel and Lunnon (1991) and Lunnon (1991) in connection with situation theory gives us a view of how Montague's semantics and discourse representation theory could be comprehended within a single metatheory which makes the two approaches interact and blend in an interesting way. For the sake of concreteness, I will use situation theory as the metatheory. While this gives us the additional advantage of drawing some parallels with situation semantics, I think it should become clear that Aczel-Lunnon abstraction is independent of situation theory1 and that ultimately one may wish to consider a general semantic metatheory which is more general than situation theory but includes Aczel-Lunnon abstraction. Certainly current presentations of Aczel-Lunnon abstraction are not in terms of situation theory but generalized set theory, set theory with the addition of abstracts as non-sets. See Lunnon(1992). 1
2
Aczel and Lunnon developed their notion of abstraction originally in order to provide a model of abstraction as it had been described in informal accounts of situation theory. The idea is that abstracts are a particular kind of object in a structured universe. This contrasts with the standard2 view of abstracts received from Montague's semantics as functions or rather -expressions which are interpreted as functions. While this will be important for what we are going to do there are two other features of this kind of abstraction which I would like to emphasize:
simultaneous abstraction Any number of parameters in a parametric object may be ab-
stracted over simultaneously. While in standard -notations one may have expressions such as x; y; z[ (x; y; z)] this is to be construed as an abbreviation for x[y[z [ (x; y; z)]]] In Aczel-Lunnon abstraction, however, it is the set which is abstracted over. Thus arguments to the abstract can be supplied simultaneously and there is no required order. indexing This feature is closely related to the previous one. Since abstraction over parameters results in an object in which those parameters do not occur3 , we have to have some way of determining how arguments are to be assigned to the abstract in the case where more than one parameter has been abstracted over. Aczel and Lunnon achieve this by de ning the abstraction operation in terms of indexed sets of parameters, i.e. one-one mappings from some domain (\the indices") to the parameters being abstracted over. An important aspect of this for us is that we can use any objects in the universe as the indices.
In the remaining sections of this paper I shall sketch the following results:
Montague's combinatory techniques using abstraction can be recaptured using Aczel-
Lunnon abstraction in a straightforward and unsurprising way since Montague only requires unary rather than simultaneous abstraction. Embedding this is situation theory gives us an interesting perspective on partial Montague grammar and the problem of identity of logically equivalent propositions. Essentially it shows that it is hard to recreate this classical problem in a structured universe even if you think of propositions as abstracts over possible worlds rather than situations. We can use Aczel-Lunnon abstraction to model Kamp's discourse representation structures by reconstructing them as predicates and exploiting the fact that we have simultaneous abstraction with indexing. This gives us an interesting perspective on discourse 2 3
at least for formal semanticists This is important in order to achieve -equivalence, i.e. x[ (x)] = y[ (y)]
3
representation theory and allows its integration with Montague's combinatory techniques. The feature of non-selective binding so important to classical discourse representation theory is captured by introducing quanti cation over simultaneous abstracts, in a manner similar to the introduction of quanti cation in the standard -calculus. Discourse anaphora can be achieved by exploiting the fact that arbitrary indices are used in the abstracts. While the parameters are bound within an abstract the indices are freely available and can be used to encode discourse anaphoric relations. A predicate (DRS) corresponding to the discourse so far can be combined with a predicate corresponding to the current sentence in a way that merges the roles in the two abstracts which have the same index. In the remainder of the paper I will give an outline of the techniques used to achieve this.
3 Some technical preliminaries I will now brie y present these notions using the terminology and notation from Barwise and Cooper (1991, in press). This is a very informal sketch and the reader is referred to this work and the Aczel and Lunnon references for more precise details. First let us deal with some preliminaries. We represent conjunction of infons and propositions by listing the representations of the objects in a box. (1)
a b
For convenience we will also list representation of a single object in a box even though this is not strictly a conjunction. (2)
a
We represent the disjunction of two infons or propositions a and b in the standard logical way. (3)
a_b 4
An important part of the situation theory we have developed is the theory of restricted objects. Any object can be restricted by a proposition. The object a restricted by the proposition p is represented (4)
a
p
This is the object a if p is true and is unde ned otherwise. Let a(X; Y ) be a parametric object with the parameters X and Y . (We use upper case letters to represent parameters.) Then we represent the result of abstracting over X and Y and indexing the two roles of the resulting abstract with the role indices r1 and r2 as
r1 ! X , r2 ! Y (5)
a(X; Y )
Notice that (5) is identical with (6). Once the parameters are abstracted over they are not present in the object and thus it does not matter which parameters you started from in the parametric object. (This is known as -equivalence.)
r1 ! Z , r2 ! W (6)
a(Z; W )
The order in which the parameters and their indices are written in the tab at the top of the box is not signi cant. It is only the association of indices with the parameters which matters. Thus (7) again represents the same object.
r2 ! Y , r1 ! X (7)
a(X; Y )
In the more standard linear notation of Aczel and Lunnon an abstract is represented as in (8) where is a parametric object and F is an indexed set (of the parameters in ). (8) F 5
Thus our example might be represented as (9) f< r1; X >; < r2; Y >g a(X; Y ) Abstracts can be applied to assignments. An assignment for an abstract is a function whose domain includes the role indices of the abstract. In EKN we represent assignments as in (10). (10)
"
#
r1 ! b r2 ! c
The result of applying (5) to (10) is represented in (11).
r1 ! X , r 2 ! Y (11)
a(X; Y )
"
#
r1 ! b r2 ! c
This is identical with the result of substituting the objects b and c for the parameters in the parametric object a(X; Y ) that was abstracted over, i.e. (12). (12)
a(b; c)
Within situation theory there are two particular kinds of abstracts that play an important role: infon abstracts which are called relations (or properties if they are unary) and proposition abstracts which are called types. (13) is an example of an infon absract where the parametric infon abstracted over is a conjunction.
r1 ! X , r2 ! Y (13)
see(X; Y ) see(Y; X )
Relations are a kind of predicate and to represent their special status as abstracts which are predicates we have a special additional notation for them in EKN where the tab goes all the way across the top of the box: 6
r1 ! X , r2 ! Y see(X; Y ) see(Y; X )
(14)
Relations can be applied to assignments in the way that we indicated for abstracts in general. However, they are called predicates because in addition they can be predicated of assignments. This means that they can be used as the relation in an infon as in (15).
r1 ! X , r2 ! Y see(X; Y ) see(Y; X )
(15)
( r1 ! a, r2 ! b)
Note that the infon represented in (15) is distinct from the result of applying the relation to the assignment which is represented in (16). see(a; b) see(b; a)
(16)
Note that the infon in (15) preserves structure (i.e. the identity of the original relation) which is lost in (16). The other kind of predicate in situation theory is proposition abstracts. Here is an example of a proposition.
s (17)
see(a; b) see(b; a)
This is the proposition that is true just in case the situation s is of the type represented by the infon (18), i.e. if s supports (18). (18)
see(a; b) see(b; a) 7
Here is an example of a proposition abstract.
r1 ! X , r2 ! Y
s (19)
see(X; Y ) see(Y; X )
We call proposition abstracts types and again have a special additional notation for them to represent their status as predicates. Thus (20) is the same type as (19).
s (20)
r1 ! X , r2 ! Y see(X; Y ) see(Y; X )
Just as with relations we can both apply types to assignments and predicate them of assignments. In the case of types predication means that we use them to form a proposition. (21) contains an example.
r1 ! a, r2 ! b s
(21)
r1 ! X , r2 ! Y see(X; Y ) see(Y; X )
Again (21) is distinct from the result of applying the type to the same assignment (given in (22)) in that it preserves the identity of the type.
s (22)
see(a; b) see(b; a)
8
4 Montague's semantics The basic idea behind the reconstruction of Montague's semantics is that we treat Montague's propositions as abstracts. Here I shall think of them as simultaneous abstracts over situations or possible worlds and times as represented in (23a) though it would actually be more faithful to Montague's original to use only unary abstraction as in (23b).
S
(23) a.
S; T
smile(a,T )
T S
b.
S
smile(a,T )
In (23) we are making use of a notational convention introduced in Barwise and Cooper (1991) which suppresses the role indices in an abstract when they are the natural numbers beginning with 1. Thus (23a) is really an abbreviation for (24). (24)
S
1 ! S, 2 ! T
smile(a,T )
Note that in this abbreviatory notation the order of the S and T in the tab in (23a) is signi cant. We shall introduce a similar abbreviatory notation for assignments. Thus (25a) is an abbreviation for (25b). (25) a. [S; T ] b.
"
1!S 2!T
#
This reconstruction is not without interest. If we take S to range over situations and assume a situation theory in which there are possible but non-actual situations I believe that we would obtain a system similar to Muskens' (1989) partial Montague grammar. If we take S to range over complete and coherent situations or possible worlds, we have something 9
corresponding to Montague's non-partial propositions except that we can have two distinct types which hold of the same set of world-time pairs. Using Montague's terminology this means that we can have two distinct logically equivalent propositions (reconstructed as types which hold of the same world-time pairs). In order to recreate the precise logical equivalence problem of classical possible world semantics we would need to model Montague's propositions as equivalence classes of types. It is important to notice that on this view the solution to the logical equivalence problem is driven by the use of Aczel-Lunnon abstracts and seems independent of whether we use situations or possible worlds. In order to indicate how the Montague style combinatorial machinery would work I give a very small sample grammar in (26).
10
(26) NP { fAnna, Claire, Mariag Vi { fsmilesg Vt { fhugsg S ! NP VP VP ! Vi VP ! Vt NP [ Anna ] =
S; T
P
P [S; T ] [a]
And similarly for the other proper names.
X [ smiles ] = S
S; T
smile(X; T )
X; Y [ hugs ] = S
S; T
hugs(X; Y; T )
S; T [ [NP VP]S ] =
[ NP ] [S; T ] [[[ VP ] ]
[ [V NP]VP ] =
X
S; T
S ;T 0
[ NP ] [S; T ] [
Y
0
[ VP ] [S ; T ] [X; Y ] 0
0
]
Using this grammar and several applications of -conversion we can show
11
[ Anna smiles ] =
S
S; T
smile(a,T )
as desired. We provide a de nition of Montague's PTQ fragment in Appendix A.
5 Discourse representation theory In our reconstruction of Montague style semantics we used simultaneous abstraction as provided by Aczel and Lunnon as a convenience for grouping together S and T , but it was not necessary. However, in our treatment of discourse representation theory the simultaneous nature of the abstraction coupled with indexing using arbitrary indices will be crucial. The leading idea is that we model discourse representation structures as abstracts which from the situation theoretic perspective are predicates. If we are working in situation theory, modelling DRSs as predicates gives us two options. They can be either relations or types. The choice is illustrated in (27) with respect to DRS corresponding to a man owns a donkey (ignoring matters of tense). (27) a. Relation
i ! X, j ! Y
man(X ) donkey(Y ) own(X; Y ) b. Type
S
i ! X, j ! Y , k ! S
man(X ) donkey(Y ) own(X; Y ) The di erence is that in the relation there is no role for a situation whereas this is the case in the type. 12
I think that ultimately DRSs need to be modelled as types in order to be deal with the DRT analyses of tense and attitudes. It is natural within situation theory to think of DRS conditions where DRS's are used to classify events and states as corresponding to propositions concerning those events and states (i.e. situations) supporting infons. Consider, for example, the discourse in (28) discussed by Kamp (1990). (28) Last month a whale was beached near San Diego. Three days later it was dead. His DRS for this makes crucial use of a discourse referent for the event of the whale being beached and another discourse referent for the state of the whale being dead and there is a condition relating the temporal occurrence of the two. In his notation in the paper the relevant conditions look as in (29). (29) a. e : : : b. s : : :
beached(x) dead(y )
Often in DRS notations `:' is used instead of `: : : '. In (30) I give a rough reconstruction of Kamp's DRS for the discourse as a situation theoretic type. I have made many arbitrary decisions here, for example, concerning which information is backgrounded as restrictions and the exact representation of temporal relations. My only aim here is to illustrate the relationship between DRT's `:' or `: : : ' and the situation theoretic notion of a situation supporting an infon.
13
(30)
0
00
X; E; P; Z; N; T ; T ; Y; S; T ; R E
P
event
place
E < N T
T
time
time
0
T
E
0
0
T ;
calendar-month beached(X )
S
T
calendar-month[ ] N
succ
T0
R
whale( ) in( ) near( ) named( , \San Diego") at( ) X
E; P
P; Z
dead(Y)
Z
E; T
S
state T S < N
00
time
day[ ] = day[ ]+3 T
00
T
S
T 00
X
=
Y
However, even though I think that in a situation theoretic approach DRSs should ultimately be treated as types, it is nevertheless very attractive to treat simple DRSs that do not involve conditions with `:' or `: : : ' as relations, precisely because this emphasizes that it is not the situations which are central to the modelling of DRSs but rather the Aczel-Lunnon abstraction. This then o ers us the possibility of building on the Aczel-Lunnon abstraction but moving in a direction other than situations to obtain a complete treatment of DRSs, if such a move should be considered desirable. For the remainder of this paper I will talk of DRSs as relations. This means that we will take (27a) as the relation corresponding to the DRS for a man owns a donkey. How now do we get the e ect of non-selective existential quanti cation that is obtained when traditional DRSs are interpreted in a model? We cannot use interpretation 14
in a model since our DRSs are not syntactic objects. However, abstracts are the kind of thing that can be quanti ed over using a variant of the same technique that is used for the introduction of quanti cation into the -calculus as represented in (31). (31) 9(x[ (x)]) We introduce a distinguished property of relations \instantiated" or \realized" represented by 9. This holds of a relation just in case there is some assignment to the roles of the relation which yields something that holds true when the relation is applied to the assignment. We can make this precise in situation theoretic terms as in (32), though, of course, one could choose to do it a di erent way if one wished to avoid the committment to situations.
s (32)
9(r)
is true implies
there is some assignment f appropriate to r such that
s
rf
is true
If desired (32) could be strengthened to a biconditional, though I do not believe that this is necessary. This shows us that, given the relation corresponding to the DRS for a man owns a donkey, we can construct an infon where this relation is existentially quanti ed.
i ! X, j ! Y (33)
X) 9( man( donkey(Y ) own(X; Y )
)
Notice the important e ect of unselective binding here. Since we are using simultaneous abstraction we have simultaneous quanti cation. This is one important ingredient which enables us to capture the classical DRT analysis of donkey anaphora, though we will not have space in this paper to present the details. Now that we have a quanti ed infon it is straightforward to use this to construct a proposition which might correspond to the interpretation of a DRS in classical DRT. If one does this in 15
situation theoretic terms one such proposition is (34), though, of course, one might wish to use a di erent notion of proposition.
s i ! X, j ! Y (34)
X) 9( man( donkey(Y ) own(X; Y )
)
But how, I hear you ask, are you going to achieve the e ect of discourse anaphora if the DRS that you construct for a single sentence is modelled as an abstract where everything is already bound? It is here that the second feature of Aczel-Lunnon abstraction that I highlighted comes into play. The use of arbitrary role indices allows us to bind parameters but at the same time uniquely identify the roles in the abstract and identify roles across di erent abstracts. In designing grammars the strategy that I have been using for the incrementation of discourse representation is to assign a predicate corresponding to a DRS to each new sentence of the discourse and then integrate that predicate with the one obtained for the discourse so far. Basically the integration is predicate conjunction where roles that have the same index are merged. We de ne an operation of predicate conjunction, , which will be the central tool used in the incrementation of one DRS with another DRS (corresponding to the next sentence in the discourse). The idea is best illustrated rst by an example.
16
i ! X, j ! Y (35)
r(X; Y )
i ! W, k ! Y r (W; Y ) 0
i ! X, j ! Y , k ! Z i ! X, j ! Y r(X; Y ) =
i ! W, k ! Y r (W; Y ) 0
2
3
2
3
i!X 6j ! Y 7 5 4 k!Z i ! X7 6 4j ! Y 5 k!Z
i ! X, j ! Y , k ! Z =
r(X; Y ) r (X; Z ) 0
In (35) we have two binary predicates which are conjoined by to form a ternary predicate. The roles indexed by i in the two original predicates are merged in the result. If there had been no overlap in the indices the result would have been a quaternary predicate and if the roles of both binary predicates had been indexed by i and j then the result would have been a binary predicate. Thus even though the parameters are bound, the indices are freely available and can be used to encode anaphoric relations. Note that it is important here that we are allowing arbitrary indices rather than, say, always using an intial segment of the natural numbers to do our indexing. It is the fact that we are allowed to use arbitrary indices which will give us the freedom to use them to encode discourse anaphoric relations. Given the machinery for Aczel-Lunnon abstraction we have sketched here it is quite straightforward to give a general de nition of .
17
(36) De nition of If is a predicate with role indices r 1; : : :; rn, is a predicate with role indices r1; : : :; rm and f is an assignment whose domain fr1; : : :; rk g = fr 1; : : :; rng [ fr1; : : :; rmg which assigns a unique parameter Xi (which is distinct from any free parameter in or ) to each ri in its domain then r1 ! X1 ,: : : , rk ! Xk
=
f f
In Appendix B I have characterized a basic DRT fragment based on part of the fragment de ned in Kamp and Reyle(forthcoming) including donkey anaphora, quanti ed sentences and relative clauses. In characterizing the fragment I exploit the fact that the use of Aczel-Lunnon abstraction allows us to use abstracts not only to recreate DRSs in the way I have presented here but also to combine that with the use of abstraction for compositional interpretation as in Montague's semantics. By way of example I give a sketch of the derivation of a man loves a woman according to the grammar I have.
18
(37) A man loves a woman a man =)
Q
P
2 6 6 6 4
Q[X ] P [X ]
3
X
7 7 7 5
man (X ) 0
P man (X ) P [X ]
=
0
loves a woman =)
X
P
2 6 6 6 4
woman (X ) P [X ] 0
3
Z love (X; Z ) 0
7 7 7 5
X woman (Y ) love (X; Y )
=
0
0
a man loves a woman =) 2
P
6 6 6 6 6 4
man (X ) P [X ] 0
3
X
woman (Y ) love (X; Y ) 0
0
7 7 7 7 7 5
man (X ) woman (Y ) love (X; Y ) 0
=
0
0
Note that what we have obtained in the derivation as represented in (37) is just a parametric infon and not the abstract that we will actually use to model the DRS. In order to obtain 19
the abstract we need to use a rule of discourse interpretation which will abstract over the free parameters. In more complex sentences such as conditionals parameters corresponding to inde nites are abstracted over before the root sentence is reached. Parameters introduced by proper names are always abstracted over at the root sentence level. This fragment is accompanied by a prolog implementation being carried out by Julian Day and Phil Kime which is described in the paper by Julian Day in this deliverable. I believe that there are a number of parallels between this work and that for Manfred Pinkal and his group (Pinkal, 1991) in the use of abstraction to obtain recapture Montague's combinatorial techniques in DRT and also in the way in which DRSs are combined to produce interpretations of discourses. Alan Black takes a di erent approach to the kind of issues I address here, avoiding the use of abstraction. His work, accompanied by a lisp implementation of the computational language ASTL which he has developed (also being presented as a deliverable), is presented in Black(1992, 1993).
6 Conclusion I have presented some general considerations about the nature of a computational semantic framework and given an example of a direction in which a semantic metatheoretic approach might take. While the details of what I have sketched use situation theory for the sake of concreteness, I have tried to emphasize the way in which one might exploit a theory of abstraction with the properties of Aczel-Lunnon abstraction in order to draw together to leading semantic theories which have been developed over the past twenty years or so. I hope I have managed to show that this mathematical tool is potentially useful for our purposes independent of the exact nature of the ultimate framework in which it is embedded.
References Aczel, Peter and Rachel Lunnon (1991) Universes and Parameters, in Situation Theory and its Applications, Vol. 2 ed. by Jon Barwise, Jean Mark Gawron, Gordon Plotkin and Syun Tutiya, CSLI Barwise, Jon and Robin Cooper (1991) Simple Situation Theory and its Graphical Representation, DYANA deliverable R2.1C, Partial and Dynamic Semantics III, ed. by Jerry Seligman. Barwise, Jon and Robin Cooper (in press) Extended Kamp Notation: a Graphical Notation for Situation Theory, in Situation Theory and its Applications, Vol. 3, ed. by Peter Aczel, David Israel, Yasuhiro Katagiri and Stanley Peters, Stanford: CSLI Black, Alan (1992) Embedding DRT in a Situation Theoretic Framework, Proceedings of the fteenth International Conference on Computational Linguistics, coling-92, pp. 1116{1120 20
Black, Alan (1993) A Situation Theoretic Approach to Computational Semantics, PhD thesis, Department of AI, University of Edinburgh. Harper, Robert, Furio Honsell and Gordon Plotkin (1987) A Framework for De ning Logics, in Symposium on Logic in Computer Science, IEEE. Johnson, Mark and Martin Kay (1990): Semantic Abstraction and Anaphora, Proceedings of COLING 90, Helsinki. Kamp, Hans (1990) Prolegomena to a Structural Theory of Belief and Other Attitudes, in Propositional Attitudes: The Role of Content in Logic, Language and Mind, ed. by C. Anthony An derson and Joseph Owens, CSLI. Kamp, Hans and Uwe Reyle (forthcoming) From Discourse to Logic: Introduction to Model Theoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory, Dordrecht: Kluwer Lunnon, Rachel (1991) Generalized Universes, Ph.D. thesis, University of Manchester Lunnon, Rachel (1992) Course notes for LLI Summer School. Muskens, Reinhard (1989) Meaning and Partiality, PhD thesis, University of Amsterdam Pinkal, Manfred (1991): On the Syntactic-Semantic Analysis of Bound Anaphora , Proceedings of the 5th EACL, Berlin Sannella, Donald and Andrzej Tarlecki (1992) Toward Formal Development of Programs from Algebraic Speci cations: Model-theoretic Foundations, Proceedings of the Nineteenth International Colloquium on Automata, Languages and programming, Lecture Notes in Computer Science, Springer Verlag.
A The PTQ Fragment To make the previous discussion more concrete we include a de nition of Montague's PTQ fragment. This de nition assumes familiarity with the notational conventions of Barwise and Cooper(1991).
21
Notation Parameter sorts Parameters S T X; Y; Xi P M P; Q
Sort situation time individual [S; T ] ! (([S; T ] ! ([X ] ! proposition)) ! proposition) i.e. [S; T ] ! type of types of individuals, a noun-phrase \intension" [S; T ] ! proposition i.e. a type of situations and times, a \Montague proposition" [S; T ] ! ([X ] ! proposition) i.e. [S; T ] ! types of individuals
Combination (\Linguistic application") f g = [S; T ]( [S; T ][ ])
Note We give the interpretation of expressions de ned in the left hand column in EKN notation (in the middle column) and linear notation (in the third column). [ 1 ; : : :; n] represents the assignment [ 1 ! 1 ,: : : , n ! n ]
Lexicon 1. proper names [ ] where is a proper name
P
S; T [S; T ][P ](P [S; T ][ ]) 0
P [S; T ][ ] 0
22
2. pronouns [ hei ]
S; T
P
P [S; T ][Xi]
3. common nouns [ ] where is a common noun
S; T
X S
[S; T ][X ](S j= hh ; X; T ; 1ii) 0
(X; T ) 0
4. determiners [ ] where is a determiner
[S; T ][P ](P [S; T ][Xi])
S; T Q
P S
(Q[S; T ]; P [S; T ]) 0
[S; T ][Q][P ](S j= hh ; Q[S; T ]; P [S; T ]; 1ii) 0
5. intransitive verbs [ ] where is an intransitive verb
S; T
X S
0
(X; T ) 0
6. transitive verbs [ ] where is an transitive verb
[S; T ][X ](S j=hh ; X; T ; 1ii)
S; T P
X S
[S; T ][P ][X ](S j=hh ; X; P ; T ; 1ii) 0
(X; P ; T ) 0
23
S; T P
X be
[S; T ][P ][X ](P [S; T ][[Y ](X = Y )])
Y
0
P [S; T ][ X = Y
]
Where: (X = Y ) abbreviates (X; Y : =) 7. verbs taking sentential complements [ ] where is a VS
S; T M
X S
[S; T ][M ][X ](S j= hh ; X; M; T ; 1ii) 0
(X; M; T ) 0
8. verbs taking in nitive complements [ ] where is a VVP
S; T P
X S
[S; T ][P ][X ](S j= hh ; X; P; T ; 1ii) 0
(X; P; T ) 0
9. sentence adverbs [ ] where is a sentence adverb
necessarily
0
M
S; T
[S; T ][M ] M
[S; T ][P ][X ](S j= hh ; X; P ; T ; 1ii) 0
0
S; T [S; T ][M ](2M )
2M
24
10. VP adverbs [ ] where is a VP adverb
S; T P
X S
[S; T ][P ][X ](S j= hh ; X; P; T ; 1ii) 0
(X; P; T ) 0
11. prepositions
[ ] where is a preposition
S; T P P
X S
[S; T ][P ][P ][X ](S j= hh ; X; P ; P; T ; 1ii) 0
(X; P ; P; T ) 0
12. Items used in In
S; T
M [ n't]]
:
S; T
M [ have]]
9T
T where A B and dom(B ? A) = dom(F1) [ (dom(F3 ) ? I ). I; A[ [Dis Dis Utt]]]K; C =) < I; A[ Dis]]J; B 1 J; B [ Utt]]K; C 1 ,C >
Discourse content If r is a relation, f a total anchor to r and s a situation, then content(r; f; s) =
s rf
If r is a relation, f a partial (not total) anchor to r and s a situation then
X1; : : :; Xn
s
rg
content(r; f; s) =
9 where 1. 2. 3. 4.
g is a total anchor for r f g fX1; : : :; Xng = ran(g ? f ) g ? f is 1-1
VP negation In [ [ Aux not VP]]]Out =) VP 0
42
X 1.
:9(F [Mid1[ VP]]Mid2[X ]]) if
F is an indexing of Mid2(LoBind) and Mid2(LoBind) = 6 ; X 2.
:Mid1 [ VP]]Mid2[X ] if Mid2(LoBind) = ;
where Mid1(LoBind) = ; Mid1(HiAnaph) = ; Otherwise Mid1( ) = In( ) Out(Loc) = In(Loc) [ (Mid2(Local) ? Mid2(LoBind)) Out(LoBind) = In(LoBind) Out(HiAnaph) = In(HiAnaph) [ (Mid2(HiAnaph) ? Mid2(LoBind)) Otherwise Out( ) = Mid2( )
Relative clauses X In [ [ N RC]]]Out =) N
In [ N]]Mid1 [X ] Mid1 [ RC]]Mid2 [X ]
where Out(Gap) = In(Gap) Out(Loc) = In(Loc) Out(NonLoc) = Mid2(NonLoc) [ Mid2(Loc) Otherwise Out( ) = Mid2( )
43
In [ [ RelPro S]]]Out =) RC
X Mid [ S]]Out
where Mid(Gap) = fX g Mid(Loc) = ; Mid(NonLoc) = In(NonLoc) [ In(Loc) Otherwise Mid( ) = In( ) In [ [ e]]]Out =) NP
P P [X ]
where In(Gap) = fX g Out(Gap) = ; Otherwise Out( ) = In( )
Conditionals In [ [ if S1 then S2 ]]]Out =) F ) G S where Mid1 [ S1] Mid2 =) F is an indexing for Mid2(LoBind) Mid3 [ S2] Mid4 =) G = G1 [ G2 G1 is an indexing for Mid4(LoBind) G2 is an indexing for some subset of Mid4(HiAnaph) dom(G2) dom(F ) dom(G1) and dom(G2) are disjoint Mid1(Loc) = ; Mid1(NonLoc) = In(NonLoc) [ In(Loc) Mid1(LoBind) = ; Otherwise Mid1( ) = In( ) Mid3(Loc) = ; Mid3(NonLoc) = Mid2(NonLoc) [ Mid2(Loc) Mid3(HiBind) = (Mid2(HiBind) ? Mid2(LoBind)) [ In(HiBind) 44
Mid3(LoBind) = ; Mid3(HiAnaph) = (Mid2(HiAnaph) ? Mid2(LoBind)) [ In(HiAnaph) Otherwise Mid3( ) = Mid2( ) Out(Gap) = Mid4(Gap) Out(Loc) = In(Loc) Out(NonLoc) = (Mid4(NonLoc) [ Mid4(Loc)) ? In(Loc) Out(HiBind) = (Mid4(HiBind) ? ran(G)) [ In(HiBind) Out(LoBind) = In(LoBind) Out(HiAnaph) = (Mid4(HiAnaph) ? ran(G)) [ In(HiAnaph) Out(LocalInds) = Mid4(LocalInds) Out(NonLocalInds) = Mid4(NonLocalInds) Out(FocusInd) = Mid4(FocusInd)
Quanti ed NPs In [ ["
Det +Quant
#
every]]]Out =)
P
Q
Q)P
where In = Out In [ ["
NP +Quant
#
"
Det +Quant
#
N]]]Out =) In [ Det]]Mid1 [F ]
where Mid2 [ N]]Mid3 [X ] =) X is not a parameter of Mid2 [ N]]Mid3 F is an indexing for Mid3(LoBind) [fX g Mid2(LoBind) = ; Otherwise Mid2( ) = Mid1( ) Out(Gap) = In(Gap) Out(Loc) = In(Loc) Out(NonLoc) = Mid3(NonLoc) Out(HiBind) = Mid3(HiBind) ? ran(F ) Out(LoBind) = Mid1(LoBind) Out(HiAnaph) = (Mid3(HiAnaph) ? ran(F )) [ Mid1(HiAnaph) Out(LocalInds) = fijF (i) 2 Mid3(Loc)g 45
Out(NonLocalInds) = Mid2(LocalInds) [ (dom(F ) ? Out(LocalInds)) Out(FocusInd) = fig such that F (i) = X
Sentences with quanti ed subjects In [ [ S
"
NP +Quant
#
VP ]]]Out =) In [ NP]]Mid1 [F ] 0
where Mid2 [ VP ] Mid3 [X ] =) X is not a parameter of Mid2 [ VP ] Mid3 F = F 1 [ F2 F1 is an indexing for Mid3(LoBind) [fX g F2 is an indexing for some subset of Mid3(HiAnaph) dom(F1 ) and dom(F2 ) are disjoint if F1 (i) = X , then i = Mid2(FocusInd) dom(F2 ) Mid2(LocalInds) [ Mid2(NonLocalInds) if F2 (i) 2 Mid3(Loc) then i 62 Mid2(LocalInds) 0
0
Mid2(LoBind) = ; Otherwise Mid2( ) = Mid1( ) Out(Gap) = In(Gap) Out(Loc) = In(Loc) Out(NonLoc) = Mid3(NonLoc) Out(HiBind) = Mid3(HiBind) ? ran(F ) Out(LoBind) = Mid1(LoBind) Out(HiAnaph) = (Mid3(HiAnaph) ? ran(F )) [ Mid1(HiAnaph) Out(LocalInds) = ; Out(NonLocalInds) = ; Out(FocusInd) = ;
VPs with quanti ed objects X In [ In]]Out [ V VP
"
2
#
NP +Quant ] =)
6
Mid1 [ NP]]Mid2 66 4
i = Mid2(FocusInd) 46
i!Y V (X; Y ) 0
3 7 7 7 5
where
Mid1(FocusInd) = ; Otherwise Mid1( ) = In( ) Out(LocalInds) = In(LocalInds) Out(NonLocalInds) = In(NonLocalInds) Out(FocusInd) = In(FocusInd) Otherwise Out( ) = Mid2( )
47
