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 offer. In fact, I think it is more important that we develop a general semantic framework with associated algorithms and computational techniques which offers the opportunity to define reusable theory independent semantic modules than to define 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 different applications. In this paper I shall offer 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 offer. At the same time the framework should be rich enough to specify the particular proposals of the individual approaches so that it becomes clear in formal terms exactly where the approaches differ. 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 differences in the different 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 different approaches are compatible.

∗An earlier version of this paper was presented at the Dagstuhl Seminar “Semantic Formalisms in Natural Lanugage Processing”, 22nd-26th February, 1993

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 involves abstracting out various operators such as APPLY, NEW INDEX, CONJOIN and interpreting these operators differently for different semantic theories such as Montague semantics, discourse representation theory and situation semantics.

formal specification This would involve the application of specification techniques developed in computer science as described, for example, in work on algebraic specification (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 theory^{1} and that ultimately one may wish to consider a general semantic metatheory which is more general than situation theory but includes Aczel-Lunnon abstraction.

1Certainly 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).

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 standard^{2} view of abstracts received from Montague’s semantics as functions or rather λ-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 abstracted over simultaneously. While in standard λ-notations one may have expressions such as

λx, y, z[ϕ(x, y, z)]

this is to be construed as an abbreviation for

λx[λy[λz[ϕ(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 occur^{3}, 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 defining 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 representation theory and allows its integration with Montague’s combinatory techniques.

2at least for formal semanticists 3This is important in order to achieve α-equivalence, i.e. λx[ϕ(x)] = λy[ϕ(y)]

• The feature of non-selective binding so important to classical discourse representation theory is captured by introducing quantification over simultaneous abstracts, in a manner similar to the introduction of quantification in the standard λ-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 briefly 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)  \begin{matrix}a \\ b \end{matrix}

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

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 undefined 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 r_{1} and r_{2} as

(5)  r_{1} → X, r_{2} → Y ; 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 α-equivalence.)

(6)  r_{1} → Z, r_{2} → W ; 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 significant. It is only the association of indices with the parameters which matters. Thus (7) again represents the same object.

(7)  r_{2} → Y, r_{1} → X ; a(X, Y)

In the more standard linear notation of Aczel and Lunnon an abstract is represented as in (8) where ζ is a parametric object and F is an indexed set (of the parameters in ζ).

(8) λF ζ Thus our example might be represented as

(9) λ{< r_{1}, X >, < r_{2}, Y >} 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) [\begin{matrix}r_{1} → b \\ r_{2} → c \end{matrix}]

The result of applying (5) to (10) is represented in (11).

(11)  r_{1} → X, r_{2} → Y ; a(X, Y)  [\begin{matrix}r_{1} → b \\ r_{2} → c \end{matrix}]

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.

(13)  r_{1} → X, r_{2} → Y ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix}

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:

(14)  r_{1} → X, r_{2} → Y ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix}

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).

(15)  r_{1} → X, r_{2} → Y ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix} ( r_{1} → a, r_{2} → 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).

(16)  \begin{matrix}see(a, b) \\ see(b, a) \end{matrix}

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.

(17) s ; \begin{matrix}see(a, b) \\ see(b, a) \end{matrix}

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)  \begin{matrix}see(a, b) \\ see(b, a) \end{matrix}

Here is an example of a proposition abstract.

(19) r_{1} → X, r_{2} → Y ; s ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix}

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).

(20) r_{1} → X, r_{2} → Y ; s ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix}

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.

(21) r_{1} → a, r_{2} → b ; r_{1} → X, r_{2} → Y ; s ; \begin{matrix}see(X, Y) \\ see(Y, X) \end{matrix}

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.

(22) s ; \begin{matrix}see(a, b) \\ see(b, a) \end{matrix}

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).

(23) a. S, T ; S ; smile(a,T)

b. S ; T ; 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) 1 → S, 2 → T ; S ; smile(a,T)

Note that in this abbreviatory notation the order of the S and T in the tab in (23a) is significant. We shall introduce a similar abbreviatory notation for assignments. Thus (25a) is an abbreviation for (25b).

\begin{matrix}(25) a. [S, T] \\ b. & [\begin{matrix}1 → S \\ 2 → T \end{matrix}] \end{matrix}

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 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).

(26) NP – {Anna, Claire, Maria} V_{i} – {smiles} V_{t} – {hugs}

S → NP VP VP → V_{i} VP → V_{t} NP

[[ Anna ]] = S, T ; P ; P [S, T] [a]

And similarly for the other proper names.

[[ smiles ]] = S, T ; X ; S ; smile(X, T)

[[ hugs ]] = S, T ; X, Y ; S ; hugs(X, Y, T)

[[ [NP VP]_{S} ]] = S, T ; [[ NP ]] [S, T] [[[ VP ]]]

[[ [V NP]_{VP} ]] =

S, T X

[[ NP ]] [S, T] [ S′, T′ ; Y ; [[ VP ]] [S′, T′] [X, Y] ]

Using this grammar and several applications of β-conversion we can show

[[ Anna smiles ]] = S, T ; S ; smile(a,T)

as desired.

We provide a definition 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

i → X, j → Y, k → S S

man(X) ; donkey(Y) ; own(X, Y)

The difference is that in the relation there is no role for a situation whereas this is the case in the type.

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 … beached(x)

b. s … 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.

X, E, P, Z, N, T, T′, Y, S, T′′, R

E ; beached(X)

S ; dead(Y)

E ; event  P ; place

E < N

T ; time  T′ ; time

T′

calendar-month T′, calendar-month[N]

succ

T ⊆ T′ R

whale(X) in(E, P) near(P, Z) named(Z, “San Diego”) at(E, T)

S ; state

S < N  T′′ ; time

day[T′′] = day[T]+3 S ∘ T′′ 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 offers 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 effect of non-selective existential quantification that is obtained when traditional DRSs are interpreted in a model? We cannot use interpretation in a model since our DRSs are not syntactic objects. However, abstracts are the kind of thing that can be quantified over using a variant of the same technique that is used for the introduction of quantification into the λ-calculus as represented in (31).

(31) ∃(λx[ϕ(x)])

We introduce a distinguished property of relations “instantiated” or “realized” represented by ∃. 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 different way if one wished to avoid the committment to situations.

(32) s ; ∃(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 quantified.

(33)

∃( i → X, j → Y man(X) ; donkey(Y) ; own(X, Y) )

Notice the important effect of unselective binding here. Since we are using simultaneous abstraction we have simultaneous quantification. 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 quantified 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 situation theoretic terms one such proposition is (34), though, of course, one might wish to use a different notion of proposition.

(34)

s

∃( i → X, j → Y man(X) ; donkey(Y) ; own(X, Y) )

But how, I hear you ask, are you going to achieve the effect 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 different 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 define 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 first by an example.

(35)  i → X, j → Y ; r(X, Y)  ⊕  i → W, k → Y ; r′(W, Y)

=  i → X, j → Y, k → Z ; \begin{matrix}i → X, j → Y ; r(X, Y) & [\begin{matrix}i → X \\ j → Y \\ k → Z \end{matrix}] \\ i → W, k → Y ; r′(W, Y) & [\begin{matrix}i → X \\ j → Y \\ k → Z \end{matrix}] \end{matrix}

=  i → X, j → Y, k → Z ; \begin{matrix}r(X, Y) \\ r′(X, Z) \end{matrix}

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 definition of ⊕.

(36) Definition of ⊕ If ζ is a predicate with role indices r_{ζ1}, …, r_{ζn}, ξ is a predicate with role indices r_{ξ1}, …, r_{ξm} and f is an assignment whose domain {r_{1}, …, r_{k}} = {r_{ζ1}, …, r_{ζn}} ∪ {r_{ξ1}, …, r_{ξm}} which assigns a unique parameter X_{i} (which is distinct from any free parameter in ζ or ξ) to each r_{i} in its domain then

ζ ⊕ ξ =  r_{1} → X_{1},…, r_{k} → X_{k} ; \begin{matrix}ζf \\ ξf \end{matrix}

In Appendix B I have characterized a basic DRT fragment based on part of the fragment defined in Kamp and Reyle(forthcoming) including donkey anaphora, quantified 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.

(37) A man loves a woman a man ⟹

Q ; P ; \begin{matrix}Q[X] \\ P [X] \end{matrix}  [ X ; man′(X) ]

=  P ; \begin{matrix}man′(X) \\ P [X] \end{matrix}

loves a woman ⟹ X

P ; \begin{matrix}woman′(X) \\ P [X] \end{matrix}  [ Z ; love′(X, Z) ]

=  X woman′(Y) ; love′(X, Y)

a man loves a woman ⟹

P ; \begin{matrix}man′(X) \\ P [X] \end{matrix}  [ X woman′(Y) ; love′(X, Y) ]

=  man′(X) ; woman′(Y) ; love′(X, Y)

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 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 indefinites 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 different 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 fifteenth International Conference on Computational Linguistics, coling-92, pp. 1116–1120 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 Defining 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 Specifications: 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 definition of Montague’s PTQ fragment. This definition assumes familiarity with the notational conventions of Barwise and Cooper(1991).

Notation

Parameter sorts

Parameters  Sort
S  situation
T  time
X, Y, X_{i}  individual
P  [S, T] → (([S, T] → ([X] → proposition)) → proposition)

i.e. [S, T] → type of types of individuals, a noun-phrase “intension”

M  [S, T] → proposition

i.e. a type of situations and times, a “Montague proposition”

P, Q  [S, T] → ([X] → proposition)

i.e. [S, T] → types of individuals

Combination (“Linguistic application”) α{β} = λ[S, T](α[S, T][β]) Note We give the interpretation of expressions defined in the left hand column in EKN notation (in the middle column) and linear notation (in the third column). [α_{1}, …, α_{n}] represents the assignment [ 1 → α_{1},…, n → α_{n}]

Lexicon

1. proper names

[[α]] where α is a proper name

S, T ; P ; P [S, T][α′]

λ[S, T]λ[P](P [S, T][α′])

2. pronouns

[[he_{i}]]  S, T ; P ; P [S, T][X_{i}]  λ[S, T]λ[P](P [S, T][X_{i}])

3. common nouns

[[α]] where α is a common noun

S, T ; X ; S ; α′(X, T)

λ[S, T]λ[X](S ⊧ ⟨⟨α′, X, T; 1⟩⟩)

4. determiners

[[α]] where α is a determiner

S, T ; Q ; P ; S ; α′(Q[S, T], P [S, T])

λ[S, T]λ[Q]λ[P](S ⊧ ⟨⟨α′, Q[S, T], P [S, T]; 1⟩⟩)

5. intransitive verbs

[[α]] where α is an intransitive verb

S, T ; X ; S ; α′(X, T)

λ[S, T]λ[X](S ⊧⟨⟨α′, X, T; 1⟩⟩)

6. transitive verbs

[[α]] where α is an transitive verb

S, T ; P ; X ; S ; α′(X, P, T)

λ[S, T]λ[P]λ[X](S ⊧⟨⟨α′, X, P, T; 1⟩⟩)

be′  S, T ; P ; X ; P[S, T][ Y ; X = Y ]  λ[S, T]λ[P]λ[X](P[S, T][λ[Y](X = Y)])

Where: (X = Y) abbreviates (X, Y : =)

7. verbs taking sentential complements

[[α]] where α is a VS  S, T ; M ; X ; S ; α′(X, M, T)  λ[S, T]λ[M]λ[X](S ⊧ ⟨⟨α′, X, M, T; 1⟩⟩)

8. verbs taking infinitive complements

[[α]] where α is a VVP S, T ; P ; X ; S ; α′(X, P, T)  λ[S, T]λ[P]λ[X](S ⊧ ⟨⟨α′, X, P, T; 1⟩⟩)

9. sentence adverbs

[[α]] where α is a sen-

tence adverb

S, T ; M ; α′[S, T][M]

λ[S, T]λ[P]λ[X](S ⊧ ⟨⟨α′, X, P, T; 1⟩⟩)

necessarily′  S, T ; M ; □M  λ[S, T]λ[M](□M)

10. VP adverbs

[[α]] where α is a VP adverb

S, T ; P ; X ; S ; α′(X, P, T)

λ[S, T]λ[P]λ[X](S ⊧ ⟨⟨α′, X, P, T; 1⟩⟩)

11. prepositions

[[α]] where α is a preposition

S, T ; P ; P ; X ; S ; α′(X, P, P, T)

λ[S, T]λ[P]λ[P]λ[X](S ⊧ ⟨⟨α′, X, P, P, T; 1⟩⟩)

12. Items used in Infl

[[n’t]]  S, T ; M ; ¬  M[S, T]  λ[S, T]λ[M]¬(M[S, T])

[[have]]  S, T ; M ; ∃T′  \begin{matrix}T′ < T \\ M[S, T′] \end{matrix}  λ[S, T]λ[M]∃T′(T′ < T ∧ M[S, T′])

where: (T′ < T) is (T′, T : <)

[[will]]  S, T ; M ; ∃T′  \begin{matrix}T < T′ \\ M[S, T′] \end{matrix}  λ[S, T]λ[M]∃T′(T < T′ ∧ M[S, T′])

[[Pres]]  S, T ; M ; M[S, T]  λ[S, T]λ[M](M[S, T])

13. Variables (used in quantifying in constructions)

[[x_{i}]] where x_{i} is a variable

X_{i} X_{i}

Phrase structure

Non-branching rules are assumed to yield identical interpretations for their mother constituents as for their daughters unless explicitly mentioned otherwise.

1. simple sentences

[[[S NP Infl′ VP]]]  [[Infl′]]{[[NP]]{[[VP]]}}  [[Infl′]]{[[NP]]{[[VP]]}}

2. noun phrases with determiner

[[[NP Det N′]]]  [[Det]]{[[N′]]}  [[Det]]{[[N′]]}

3. verb phrases with complements

[[[VP V_{t} NP]]]  [[V_{t}]]{[[NP]]}  [[V_{t}]]{[[NP]]}
[[[VP VS Sent]]]  [[VS]]{[[Sent]]}  [[VS]]{[[Sent]]}
[[[VP VVP VP]]]  [[VVP]]{[[VP]]}  [[VVP]]{[[VP]]}

4. sentences with modifiers

[[[S Adv Sent]]]  [[Adv]]{[[Sent]]}  [[Adv]]{[[Sent]]}

5. verb phrases with modifiers

[[[VP VP PP]]]  [[PP]]{[[VP]]}  [[PP]]{[[VP]]}
[[[VP VP AdvVP]]]  [[AdvVP]]{[[VP]]}  [[AdvVP]]{[[VP]]}

6. prepositional phrases

[[[PP Prep NP]]]  [[Prep]]{[[NP]]}  [[Prep]]{[[NP]]}

7. relative clauses [[[N_{′} N′X_{i} such-that Sent]]]

S, T ; X_{i} ; \begin{matrix}[[N′]][S, T][X_{i}] \\ [[Sent]][S, T] \end{matrix}  λ[S, T]λ[X_{i}]([[N′]][S, T][X_{i}] ∧ [[Sent]][S, T])

8. rules of conjunction and disjunction

[[[S Sent_{1} or Sent_{2}]]]

S, T
[[Sent_{1}]][S, T]∨[[Sent_{2}]][S, T]

λ[S, T]([[Sent_{1}]][S, T] ∨ [[Sent_{2}]][S, T])

[[[S Sent_{1} and Sent_{2}]]]

S, T
[[Sent_{1}]][S, T]
[[Sent_{2}]][S, T]

λ[S, T]([[Sent_{1}]][S, T] ∧ [[Sent_{2}]][S, T])

[[[NP NP_{1} or NP_{2}]]]  S, T ; P ; [[NP_{1}]][S, T][P]∨ [[NP_{2}]][S, T][P]

λ[S, T]λ[P]([[NP_{1}]][S, T][P] ∨ [[NP_{2}]][S, T][P])

[[[VP VP_{1} or VP_{2}]]]  S, T ; X ; [[VP_{1}]][S, T][X]∨[[VP_{2}]][S, T][X]

λ[S, T]λ[X]([[VP_{1}]][S, T][X] ∨ [[VP_{2}]][S, T][X])

[[[VP VP_{1} and VP_{2}]]]  S, T ; X ; \begin{matrix}[[VP_{1}]][S, T][X] \\ [[VP_{2}]][S, T][X] \end{matrix}

λ[S, T]λ[X]([[VP_{1}]][S, T][X] ∧ [[VP_{2}]][S, T][X])

9. Infl

[[[Infl Tns \{ \begin{matrix}Perf \\ M \end{matrix} \}]]]

S, T M

[[Tns]][S, T][[[\{ \begin{matrix}Perf \\ M \end{matrix} \}]]\{[M]\}

λ[S, T]λ[M]([[Tns]][S, T][[[\{ \begin{matrix}Perf \\ M \end{matrix} \}]]\{[M]\})

[[[Infl Tns ]]]  [[Tns]]

[[[Infl_{′} Infl n’t ]]]  S, T ; M ; [[n’t]][S, T][[[Infl]]{[M]}

λ[S, T]λ[M]([[n’t]][S, T][[[Infl]]{[M]})

10. rules for quantifying in

[[[S NP X_{i} Sent]]] S, T

[[NP]][S, T][ X_{i} ; [[Sent]][S,T] ]

λ[S, T]([[NP]][S, T][λ[X_{i}]([[Sent]][S, T])])

[[[VP NP X_{i} VP]]]  S, T ; Y ; [[NP]][S, T][ X_{i} ; [[VP]][S,T][Y] ]

λ[S, T]λ[Y]([[NP]][S, T][λ[X_{i}]([[VP]][S, T][Y])])

[[[N_{′} NP X_{i} N′]]]  S, T ; Y ; [[NP]][S, T] [ X_{i} ; [[N′]][S, T][Y] ]

λ[S, T]λ[Y]([[NP]][S, T][λ[X_{i}]([[N′]][S, T][Y])])

Constraints

Determiners

1. s ⊧ ⟨⟨every, τ_{1}, τ_{2}; 1⟩⟩ implies ∀x x : τ_{1} → x : τ_{2}

2. s ⊧ ⟨⟨a, τ_{1}, τ_{2}; 1⟩⟩ implies ∃x x : τ_{1} ∧ x : τ_{2}

3. s ⊧⟨⟨the, τ_{1}, τ_{2}; 1⟩⟩ implies ∃x(x : τ_{1} ∧ ∀y(y : τ_{1} → y = x)) ∧ ∀x(x : τ_{1} → x : τ_{2}) Constraint on □

We write □m for m : □.

4. (Lewis) □m iff ∀s, t((coh(s)∧compl(s)) → s, t : m). This alternative assumes that there are possible non-actual situations.

(Stalnaker/Cresswell) □m iff ∀m′, t(coh(λ[S](m′[S, t])∧compl(m′[S, t])) → λ[S](m[S, t]) ≤ λ[S](m′[S′, t])). This alternative is consistent with the assumption that there are only actual situations.

Where the following definitions hold

(a) Completeness and coherence for situations compl(s) iff ∀σ(s ⊧ σ ∨ s ⊧ \overline{σ}) coh(s) implies ¬∃σ(s ⊧ σ ∧ s ⊧ \overline{σ})

(b) Subsumption for situation types λ[S](S ⊧ σ) ≤ λ[S](S ⊧ σ ∧ τ) λ[S](S ⊧ σ ∨ τ) ≤ λ[S](S ⊧ σ)

(c) Completeness and coherence for situation types compl(τ) iff ∀σ(λ[S](S ⊧ σ) ≤ τ ∨ λ[S](S ⊧ \overline{σ}) ≤ τ) coh(τ) → ¬(∃σλ[S](S ⊧ σ) ≤ τ ∧ λ[S](S ⊧ \overline{σ}) ≤ τ)

Coherence should ultimately be made more sophisticated - relativizing to constraints at least.

Extensional verbs

5. s ⊧ ⟨⟨α′, x, P, t; 1⟩⟩ iff P[s, t][λ[Y](s ⊧ ⟨⟨α′, x, Y, t; 1⟩⟩)] is true.

B A DRT fragment

We present a basic fragment of DRT similar to that presented in the early part of Kamp and Reyle(1993).

Preliminary definitions

Conjoining DRSs

We will define an operation of predicate conjunction, ⊕, which we will use to increment a DRS with another DRS (corresponding to the next sentence in the discourse).

Example

i → X, j → Y ; r(X, Y)  ⊕  i → X, k → Y ; r′(X, Y)

=  i → X, j → Y, k → Z ; \begin{matrix}i → X, j → Y ; r(X, Y) & [\begin{matrix}i → X \\ j → Y \end{matrix}] \\ i → X, k → Y ; r′(X, Y) & [\begin{matrix}i → X \\ j → Y \\ k → Z \end{matrix}] \end{matrix}

=  i → X, j → Y, k → Z ; \begin{matrix}r(X, Y) \\ r′(X, Z) \end{matrix}

Definition of ⊕ If ζ is a relation (or type) with role indices r_{ζ1}, …, r_{ζn}, ξ is a relation (or type) with role indices r_{ξ1}, …, r_{ξm} and f is an assignment whose domain {r_{1}, …, r_{k}} = {r_{ζ1}, …, r_{ζn}}∪{r_{ξ1}, …, r_{ξm}} which assigns a unique parameter X_{i} (not a parameter of ζ or ξ) to each r_{i} in its domain then

ζ ⊕ ξ =  r_{1} → X_{1},…, r_{k} → X_{k} ; \begin{matrix}ζf \\ ξf \end{matrix}

Conditionals We introduce a distinguished binary relation between relations, ⇒. For convenience we will write the infon ⇒ (r_{1}, r_{2}) as r_{1} ⇒ r_{2} s ; r_{1} ⇒ r_{2} is true implies

for every assignment f appropriate to r_{1} such that

s ; r_{1}f is true

there is some g ⊇ f appropriate to r_{2} such that

s ; r_{2}g is true

There are a number of alternative conditional relations one could consider, e.g. one could existentially quantify over situations in the consequent or invoke the theory of channels (Barwise and Seligman) at this point.

A point of convenience

To accommodate DRSs which do not have a domain, i.e. are parametric infons, we can consider infons as 0-place relations for which the only appropriate assignment is ∅. We let λ∅ζ = ζ

Generalized quantifiers

While generalized quantifiers are not treated as such in the fragment we show here two possible definitions which show how the treatment of generalized quantifiers in the manner of DRT would interact with Aczel-Lunnon abstraction, the ⊕-operator and the relation ⇒.

We let

q(i, r_{1}, r_{2}) abbreviate

either

(weak reading)

q( i → X ; ∃(r_{1} [i → X]) , i → X ; ∃((r_{1} ⊕ r_{2}) [i → X]) )

or

(strong reading)

q( i → X ; ∃(r_{1} [i → X]) , i → X ; \begin{matrix}∃(r_{1} [i → X]) \\ r_{1} [i → X] ⇒ r_{2} [i → X] \end{matrix} )

where X is not a parameter of r_{1} or r_{2}

Information flow relative to syntax

There are two kinds of information flow relative to the syntax that have been used in deriving interpretation recursively from syntactic structure. One I shall call Fregean flow. This corresponds to Frege’s proposal as enshrined in Montague Grammar that the interpretation of a constituent should be determined in terms of the interpretation of its parts. This is often referred to as compositionality, though I want to avoid that term as it is the subject of debate. It is not supposed to matter whether you do this kind of interpretation starting at the top and working down or starting at the bottom and working up – or indeed starting in the middle and working out in both directions.

The other kind of information flow is borrowed from logic grammar. I shall call it Pereiran because to the best of my knowledge it was first introduced in Pereira’s thesis to deal with gaps and became known as gap threading.

Pereiran flow is directional. The default pattern is that information flows down from the top on the left hand side of the tree and across the daughters from left to right and then back up on the right hand side. What has become clear to us recently is that this kind of information flow corresponds to one of the senses of the word ‘dynamic’ e.g. as it is used in dynamic predicate logic. See Lewin(1992) for discussion.

If you look at the literature on DRT it seems to me that both the Fregean flow and the Pereiran flow have been proposed as alternatives at least in the computational literature. I’m not sure that the construction algorithm of Kamp and Reyle corresponds to either of them. The Johnson and Klein style of implementation uses the Pereiran flow exclusively and one finds that you have to fiddle a bit to do things like make sure that the discourse referent corresponding to the subject noun-phrase corresponds to the first argument of the condition introduced by the verb. In my implementations based on Johnson and Klein that I use for teaching computational semantics in Edinburgh I introduce an additional argument in the DCG in addition to the threaded DRSs in order to do the required bookkeeping. Johnson and Klein themselves pass the additional information around in terms of PATR features in addition to those used for encoding the DRS threading. On the other hand dynamic Montague Grammar does everything using Fregean flow as the backbone of the interpretation (abandoning the explicit threading of dynamic predicate logic) and in order to do this complicates the kind of semantic objects you need.

I want to explore here the possibility of combining both kinds of information flow in a single grammar. I think it could be advantageous to be explicit about having information flow in both ways and that this will enable us to investigate in general terms what kind of information should flow in which way. The hope is that by doing this we will be able to achieve the advantages of both approaches while avoiding the complications that arise when we try to use one exclusively. I think we have known for a long time that there is a basic compositional backbone to semantic interpretation, but that there are certain things – anaphora – which have to be dealt with in an essentially top-down left-right manner and are overlaid on the compositional backbone. I think that this was made explicit in some earlier work by Bach and Partee where they introduce, for example, local pronoun stores. In fact, their technique for dealing with locality by stores is directly reused in the information being threaded in the fragment we are about to define.

The Fragment

The basic idea is that we will interpret each constituent in a Fregean kind of way but that the interpretation will be with respect to packets of information which we will normally refer to as ‘In’ and ‘Out’.

So the basic form of an interpretation rule will be:

In[[α]]Out⟹ ϕ

Associated with these rules will be conditions on what In and Out can be which will require that information is threaded through. This kind of technique is discussed in a different context by Lewin(1992)

Threaded information

Now before you look at the following list I’d like you to think of intuitions that might underlie these Ins and Outs. You see, there are a number of different kinds of information that we have to keep track of here. In fact, let me be honest, for this fragment, each In and each Out will be a ninetuple. One way to think of these packets of information is as indices in the standard sense of e.g. Lewis’s General Semantics and this is the way to relate our threading proposal back to the classical literature on semantics. However, in those terms, it might seem a bit excessive to have to intepret everything with respect to eighteen indices. (Though I hasten to add that individual rules only make use of a small number of them.) But there’s another intuition behind these ninetuples. That is, that they are bundles of ingoing and outgoing information analyzed in terms of features. And actually we’re managing to do quite a lot of work and we’re only using nine features (or ten if you want to think of ‘In’ and ‘Out’ as features as well). So, please, when you think of actually how this threading is going to work, think in terms of something like indices. When you think of how many there are, think in terms of features.

Here for the record is a list of what they are. For any given rule most of them will not be referred to and the information will not affect the interpretation and will be passed through unchanged. Some of them will appear mysterious at this point and will get explained as we look at the relevant rules.

Gap – parameter used to interpret relative pronoun/gap Loc – local parameters used to interpret NPs within a clause NonLoc – non local parameters used to interpret NPs outside the current clause HiBind – parameters which are to be abstracted over at the level of the topmost clause (corresponding to proper names) LoBind – parameters which are to be abstracted over locally (within conditionals, corresponding to indefinite NPs) HiAnaph – parameters which may be captured in anaphoric relations and which will be abstracted over at the level of the topmost clause if not captured within the sentence LocalInds – local indices corresponding to local parameters abstracted over in cases of quantification NonLocalInds – non local indices corresponding to non local parameters abstracted over in cases of quantification FocusInd – focus index representing the focus of quantification

So: interpretation will be with respect to two nine-tuples of sets. We use e.g.

In[[α]]Out⟹ ϕ

to indicate that α can be interpreted as ϕ with respect to the tuples In and Out. Note that In[[.]]Out is not a function. Trivially because there will be a number of points where we are required to use a parameter or an index not used in the ingoing information and this will give us free choice among what we will assume to be an infinite stock of parameters or indices. (This is, of course, no different to the way things are with traditional DRT where one gets to choose a new discourse referent.)

We use e.g. In(Gap), Out(LoBind) to refer to members of the tuples.

Proper names

In[[[PN α]]]Out ⟹ \begin{matrix}P \\ P [X] & R ; named(X, α) \end{matrix}

where for any ζ, X ∉ In(ζ)

Out(Loc) = {X}∪ In(Loc) Out(HiBind) = {X}∪ In(HiBind) Out(HiAnaph) = {R}∪ In(HiAnaph) Otherwise Out(ζ) = In(ζ)

In[[[_{[ \begin{matrix}NP \\ -Quant \end{matrix} ]} PN]]]Out ⟹ In[[PN]]Out

Pronouns

In[[[Pro α]]]Out ⟹ \begin{matrix}P \\ P [X] & R ; gender(α)(X) \end{matrix}

where

X ∉ In(Loc)

Out(Loc) = {X}∪ In(Loc) Out(HiAnaph) = {X, R}∪ In(HiAnaph) Otherwise Out(ζ) = In(ζ)

In[[[_{[ \begin{matrix}NP \\ -Quant \end{matrix} ]} Pro]]]Out ⟹ In[[Pro]]Out Indefinite NPs

In[[[_{[ \begin{matrix}Det \\ -Quant \end{matrix} ]} a]]]Out ⟹ Q ; P ; \begin{matrix}Q[X] \\ P [X] \end{matrix}

where for any ζ, X ∉ In(ζ)

Out(Loc) = {X}∪ In(Loc) Out(LoBind) = {X}∪ In(LoBind) Otherwise Out(ζ) = In(ζ)

In[[[N α]]]Out ⟹ X ; α′(X)

where In = Out

The main method of combination of interpretations is, like in Montague Grammar, application, only in our case it is the application of Aczel-Lunnon abstracts to assignments rather than functional application. Recall that when we do not mention role indices in the representation of an abstract, the convention is that the role indices are the natural numbers beginning with 1. Also when we represent assignments leaving out the indices a similar convention applies. Thus [a] is used to represent [ 1 → a].

In[[[_{[ \begin{matrix}NP \\ -Quant \end{matrix} ]} [ \begin{matrix}Det \\ -Quant \end{matrix} ] N]]]Out ⟹ In[[Det]]Mid [Mid[[N]]Out]

Verb phrases with non-quantified objects

The rule for verb-phrases with non-quantified objects mimics Montague’s VP-quantification rule in PTQ.

In[[[VP V [ \begin{matrix}NP \\ -Quant \end{matrix} ]]]]Out ⟹ X In[[NP]]Out [ Y ; V′(X, Y) ]

In[[[VP_{′} VP]]]Out ⟹ In[[VP]]Out

Sentences with non-quantified subjects

We apply subject NP interpretations to (assignmentswhich assign 1 to) VP interpretations following the treatment in Montague’s PTQ.

In[[[S [ \begin{matrix}NP \\ -Quant \end{matrix} ] VP′]]]Out ⟹ In[[NP]]Mid [Mid[[VP′]]Out]

Examples

Let’s look at some examples of what we can do with the rules so far. We will give sketches of derivations since it becomes very complex to deal with all the threading in a precise way and still see what is going on. Hopefully, the implementation described in the paper by Julian Day included in this deliverable will serve the purpose of allowing us to check all the details.

A man loves a woman

a man ⟹

Q ; P ; \begin{matrix}Q[X] \\ P [X] \end{matrix}  [ X ; man′(X) ]

=  P ; \begin{matrix}man′(X) \\ P [X] \end{matrix}

loves a woman ⟹ X

P ; \begin{matrix}woman′(X) \\ P [X] \end{matrix}  [ Z ; love′(X, Z) ]

=  X woman′(Y) ; love′(X, Y)

a man loves a woman ⟹

P ; \begin{matrix}man′(X) \\ P [X] \end{matrix}  [ X woman′(Y) ; love′(X, Y) ]

=  man′(X) ; woman′(X) ; love′(X, Y)

Note that the rules so far just give us a parametric infon and not the abstract that we will actually use to model the DRS. In order to obtain the abstract we need to use the rules of discourse interpretation which will give us the abstract corresponding to a node Utt which immediately dominates the S.

John loves a woman

What we have to check here is that the restriction associated with the proper name gets handled appropriately.

John loves a woman ⟹ P

P [X] R named(X,John)

[ X woman′(Y) ; love′(X, Y) ]

=  woman′(Y) ; love′(X, Y)  R named(X,John)

Here we have β-conversion to a position within the scope of a restriction. A woman loves John Things are a little different when the restriction is introduced in object position. loves John ⟹

X P

P [X] R named(X,John)

[ Z ; love′(X, Z) ]

=  X love′(X, Y)  R named(X,John)

β-conversion to within the scope of a restriction. a woman loves John ⟹

P ; \begin{matrix}woman′(X) \\ P [X] \end{matrix}  [ \begin{matrix}X \\ love′(X, Y) & R named(X,John) \end{matrix}  ]

=  woman′(X) love′(X, Y)  R named(X,John)

=  woman′(X) ; love′(X, Y)  R named(X,John)

Distribution of restriction over conjunction.

Discourse interpretation

Let I be a set of indices, In[[S]]Out ⟹ ϕ (where In is a tuple of empty sets and F = F_{1}∪F_{2}∪F_{3} be an indexing for the parameters of ϕ such that

1. dom(F_{1}), dom(F_{2}), dom(F_{3}) are pairwise disjoint

2. F_{1} is an indexing of Out(HiBind) such that dom(F_{1})∩I = ∅

3. F_{2} is an indexing of Out(LoBind) such that dom(F_{2})∩I = ∅

4. F_{3} is an indexing of Out(HiAnaph) − (Out(HiBind) ∪ Out(LoBind)). (This may or may not overlap with I.)

Let A be an anchor (an assignment whose range contains no parameters) whose domain is included in I.

I, A[[[Utt S]]]I∪dom(F),B ⟹ < λF ϕ, B > where A ⊆ B and dom(B − A) = dom(F_{1}) ∪ (dom(F_{3}) − 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

content(r, f, s) = X_{1}, …, X_{n} ; s ; rg ; ∃

where

1. g is a total anchor for r

2. f ⊆ g

3. {X_{1}, …, X_{n}} = ran(g − f)

4. g − f is 1-1

VP negation

In[[[VP_{′} Aux not VP]]]Out ⟹ 1.

X

¬∃(λF [Mid1[[VP]]Mid2[X]])

if

F is an indexing of Mid2(LoBind) and Mid2(LoBind) ≠ ∅

2.

X

¬Mid1[[VP]]Mid2[X]

if

Mid2(LoBind) = ∅

where

Mid1(LoBind) = ∅ Mid1(HiAnaph) = ∅ Otherwise Mid1(ζ) = In(ζ)

Out(Loc) = In(Loc) ∪ (Mid2(Local) − Mid2(LoBind)) Out(LoBind) = In(LoBind) Out(HiAnaph) = In(HiAnaph) ∪ (Mid2(HiAnaph) − Mid2(LoBind)) Otherwise Out(ζ) = Mid2(ζ)

Relative clauses

In[[[N N RC]]]Out ⟹ X 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(ζ) = Mid2(ζ)

In[[[RC RelPro S]]]Out ⟹ X ; Mid[[S]]Out

where Mid(Gap) = {X} Mid(Loc) = ∅ Mid(NonLoc) = In(NonLoc) ∪ In(Loc) Otherwise Mid(ζ) = In(ζ)

In[[[NP e]]]Out ⟹ P ; P [X]

where In(Gap) = {X} Out(Gap) = ∅ Otherwise Out(ζ) = In(ζ)

Conditionals

In[[[S if S_{1} then S_{2}]]]Out ⟹ λF ϕ ⇒ λGϕ

where Mid1[[S_{1}]]Mid2 ⟹ ϕ F is an indexing for Mid2(LoBind)

Mid3[[S_{2}]]Mid4 ⟹ ψ G = G_{1} ∪ G_{2} G_{1} is an indexing for Mid4(LoBind) G_{2} is an indexing for some subset of Mid4(HiAnaph) dom(G_{2}) ⊆ dom(F) dom(G_{1}) and dom(G_{2}) are disjoint

Mid1(Loc) = ∅ Mid1(NonLoc) = In(NonLoc) ∪ In(Loc) Mid1(LoBind) = ∅ Otherwise Mid1(ζ) = In(ζ)

Mid3(Loc) = ∅ Mid3(NonLoc) = Mid2(NonLoc) ∪ Mid2(Loc) Mid3(HiBind) = (Mid2(HiBind) − Mid2(LoBind)) ∪ In(HiBind)

Mid3(LoBind) = ∅ Mid3(HiAnaph) = (Mid2(HiAnaph) − Mid2(LoBind)) ∪ In(HiAnaph) Otherwise Mid3(ζ) = Mid2(ζ)

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)

Quantified NPs

In[[[_{[ \begin{matrix}Det \\ +Quant \end{matrix} ]} every]]]Out ⟹ Q ; P ; Q ⇒ P

where In = Out

In[[[_{[ \begin{matrix}NP \\ +Quant \end{matrix} ]} [ \begin{matrix}Det \\ +Quant \end{matrix} ] N]]]Out ⟹ In[[Det]]Mid1[λF ϕ]

where Mid2[[N]]Mid3[X] ⟹ ϕ X is not a parameter of Mid2[[N]]Mid3 F is an indexing for Mid3(LoBind) ∪{X}

Mid2(LoBind) = ∅ Otherwise Mid2(ζ) = Mid1(ζ)

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) = {i|F (i) ∈ Mid3(Loc)} Out(NonLocalInds) = Mid2(LocalInds) ∪ (dom(F) − Out(LocalInds)) Out(FocusInd) = {i} such that F (i) = X

Sentences with quantified subjects

In[[[S [ \begin{matrix}NP \\ +Quant \end{matrix} ] VP′]]]Out ⟹ In[[NP]]Mid1[λF ϕ]

where Mid2[[VP′]]Mid3[X] ⟹ ϕ X is not a parameter of Mid2[[VP′]]Mid3 F = F_{1} ∪ F_{2} F_{1} is an indexing for Mid3(LoBind) ∪{X} F_{2} is an indexing for some subset of Mid3(HiAnaph) dom(F_{1}) and dom(F_{2}) are disjoint if F_{1}(i) = X, then i = Mid2(FocusInd) dom(F_{2}) ⊆ Mid2(LocalInds) ∪ Mid2(NonLocalInds) if F_{2}(i) ∈ Mid3(Loc) then i ∉ Mid2(LocalInds)

Mid2(LoBind) = ∅ Otherwise Mid2(ζ) = Mid1(ζ)

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 quantified objects

In[[In]]Out[VP V [ \begin{matrix}NP \\ +Quant \end{matrix} ]] ⟹ X Mid1[[NP]]Mid2[ i → Y ; V′(X, Y) ] where

i = Mid2(FocusInd)

Mid1(FocusInd) = ∅ Otherwise Mid1(ζ) = In(ζ)

Out(LocalInds) = In(LocalInds) Out(NonLocalInds) = In(NonLocalInds) Out(FocusInd) = In(FocusInd) Otherwise Out(ζ) = Mid2(ζ)