Partial dynamic semantics

Dag Trygve Truslew Haug

March 12, 2014

Abstract

The development of dynamic semantics was to a large extent motivated by the desire to ac-count for anaphoric expressions (such as pronominals, but also anaphoric tense), which do notin themselves have refererence, but must be resolved in the discourse context, possibly out-side the sentence in which they occur. Yet it is striking that no dynamic semantic frameworkcan give an interpretable denotation for unresolved anaphora. In this talk I present a new dy-namic framework, partial CDRT (Haug 2013) based on Reinhard Muskens’ CDRT (Muskens1996) grafted on a partial logic. In this framework, we can give an interpretable semanticsfor unresolved anaphora, thereby separating the monotonic semantics from non-monotonicprocesses such as a anaphora resolution.

1 IntroductionIt is usual in dynamic semantics to discuss sentences and discourses like (1)-(2).

(1) Every2 farmer who owns a7 donkey beats it7.(2) a. A3 cat appeared. It3 meowed.

b. No5 cat appeared. *It 5 meowed.

• What do those little numbers mean, and who put them there anyway? (Beaver 1999)• Is it possible to give a semantics for words such as every, a, and even it – without the little

number attached?

One way around this is as in (3)-(4).

(3) Every2 farmer who owns a7 donkey beats it?.(4) a. A3 cat appeared. It? meowed.

b. No5 cat appeared. *It? meowed.

This is �ne as far as it goes. But ideally we want to make sure that

• the resolution of ? is properly constrained by the semantics.• we can change the resolution non-monotonically.• we can interpret the sentences independently of resolution.

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1.1 The motivation for dynamic semanticsLet us �rst consider a static semantics for inde�nites:

(5) A cat appeared. ∃x.cat(x) ∧ appear(x)

The good news:

• The inde�nite is not referential so (5) can be true no matter which cat appeared• Because the variable is bound, its denotation does not depend on an assignment• Because the variable is bound, it doesn’t matter which variable we choose

(6) It meowed. ?

The bad news:

• Because the variable is bound, we cannot access it outside the scope of the quanti�er

We need to combine features of bound and free variables to get the right semantics in such cases.

Dynamic predicate logic: tinker with scope and variable binding (Groenendijk and Stokhof1991)

DRT: use free variables, and tinker with the truth de�nition (DRT, (Kamp 1981)).

I will follow DRT here.

1.2 DRT’s solution1.2.1 DRSs and conditions

A DRS for (5):

(7)

x

cat(x)appear(x)

A DRS is interpreted as a set of verifying assignments. Some notation:

• i, o are partial assignments of individuals to variables,• i ⊂{x} o (o extends i with {x}) means that o assigns the same individual as i to all drefs in

the domain of i, but also assigns an invidual to x,• I interprets the non-logical constants of the language, mapping n-ary predicates to sets ofn-tuples in Dn.

Veri�cation of DRSs and simple conditions:

• 〈i, o〉 veri�es the DRS 〈U,Con〉 i� i ⊂U o and for all γ ∈ Con, o veri�es γ.• o veri�es a simple condition P (x1, . . . , xn) i� 〈o(x1), . . . , o(xn)〉 ∈ I(P )

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This gives us (8) as an interpretation of (7)

(8) {〈i, o〉|i ⊂{x} o ∧ o(x) ∈ I(cat) ∧ o(x) ∈ I(appear)}

Truth relative to an input context i is de�ned as the existence of an output assignment o.Truth simpliciter is truth relative to the empty input context.

1.2.2 Connectives: negation

Negation is represented as a complex condition (embedded DRS):

(9) No cat appeared.

(10)¬

x

cat(x)appear(x)

(11) o veri�es ¬K i� there is no assignment j such that 〈o, j〉 veri�es K(12) {〈i, o〉|i = o and there is no j such that o ⊂{x} j ∧ j(x) ∈ I(cat) ∧ j(x) ∈ I(appear)}

The output assignment o is unchanged!

1.2.3 Connectives: conditionals

Conditionals introduce two layers of DRS embedding.

(13) Every cat eats a mouse.

(14)x

cat(x)⇒

y

mouse(y)eat(x, y)

(15) o veri�es K ⇒ K ′ i� for all assignments j such that 〈o, j〉 veri�es K , there is an assign-ment k such that 〈j, k〉 veri�es K ′.

(16) {〈i, o〉|i = o and for all j such that o ⊂{x} j∧j(x) ∈ I(cat), there is a k such that j ⊂{y} kand k(y) ∈ I(mouse) and 〈k(x), k(y)〉 ∈ I(eat)}

The output assignment o is again unchanged. Note that k(x) is de�ned and equal to j(x).

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1.2.4 Sequencing

Sequencing two DRSs – written K ;K ′ – is de�ned as in (17).

(17) JK ;K ′K := {〈i, o〉| there is an m such that 〈i,m〉 veri�es K and 〈m, o〉 veri�es K ′}

Intuitively, the output of the �rst DRS serves as the input to the second.

1.2.5 DRS and anaphoric accessibility

There is a truth-preserving embedding of DRT into �rst order logic. For example, (7) and (10)translate into (18) and (19) respectively.

(18) ∃x.cat(x) ∧ appear(x)

(19) ¬∃x.cat(x) ∧ appear(x)

However, while these are truth-conditionally equivalent to (8) and (11), they do not capture thedi�erence in anaphoric potential which arise from the interpretation of DRSs.

In (8), the output assignment o is extended with a dref x which is therefore available foranaphoric uptake, but in (11) (and (16)), there is no such extension. We therefore correctly predictthat (8) and (11) di�er in their anaphoric potential:

(20) A cat appeared. It meowed.(21) No cat appeared. *It meowed.

In other words, constraints on anaphoric accessibility fall out from the semantics of the connec-tives. In other words, we have a model-theoretic account of semantic accessibility. This is one ofthe main achievements of DRT.

1.3 Remaining problems1.3.1 Unresolved anaphora

Let us get back to the discourse in (22).

(22) A cat appeared. It meowed.

The second sentence gets the representation in (23).1

(23)

x

cat(x)appear(x)

⟨y

non-human(y)

,

y

meowed(y)

⟩

context representation1I follow exposition of Two Stage Bottom-Up DRS Construction in (Kamp, van Genabith, and Reyle 2011, p. 139–

144). The old notation in terms of conditions like y = ? is equivalent for our purposes.

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Such representations are not interpreted, but must be resolved. Resolution involves identifyingany drefs in the universe of a presuppositional DRS with a dref in the context and augmentingthe assertion with that identi�cation. Furthermore, given this anaphoric resolution, the contextmust entail any conditions in the presuppositional DRS (under this identi�cation), or at leastallow accommodation of these conditions. In (23), we can augment the assertion with y = x. Thecontext entails non-human(x)

(24)

x

cat(x)appear(x)non-human(x)

y

meowed(y)y = x

context representation

and we can merge the context and the representation to yield a new DRS (25).

(25)

x y

cat(x)appear(x)y = xnon-human(y)meowed(y)

DRT’s resolution strategy is explicitly designed to ensure that

1. resolution always takes place in a context which does not include information provided bythe part of the sentence which contains the presupposition trigger

2. the ordering information is preserved up to sequencing and then discarded

1 may be motivated for other kinds of presuppositions, but not for anaphora. Surely the fact that itis the subject of meowed is relevant to the resolution of the anaphor (in a context with competingantecedents). Even the following discourse is relevant, which means that 2 is problematic too.

(26) France1 is a monarchy. Every2 nation cherishes its3 king.a. So the French love King François.b. That is because his appearances in the UN have been so convincing.

We also need “backtracking” in our anaphor resolution, and 2 makes that impossible.

(27) a. Pedro1 is in a2 bar.b. Every3 woman who ever dated a4 man despises him5.c. He6 is a well-known date crasher.

Finally, in addition to the coreference problem there is a dual ‘overwrite problem’: how do wemake sure that we pick an unused discourse referent? This is connected to the use of free variables.

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2 Data structures for tracking discourse referentsThere are a variety of di�erent data structures used to track discourse referents:

DRT partial assignmentsDPL total assignmentsothers lists (Vermeulen 1993; Dekker 1994; van Eijck 2001; Nouwen 2003; Nouwen 2007; Bittner

2001; Bittner 2007; Schlenker 2005; van Eijck and Unger 2010)

A list is a data structure with two operations

^ takes a list and and individual to a new list with the individual added at the end[n] returns the nth last individual in the list

Dynamic existential quanti�cation is non-deterministic appending to the list:

(28) ∃ := λc.λc′.∃x.c^x = c′

Anaphoric expressions do not append an individual to the list but rather extract one from it:

(29) Every[ ]ˆf farmer who owns a[f ]ˆd donkey beats it1.[ ]

(30) a. A[ ]ˆc cat appeared.[c] It[1] meowed.b. No[ ]ˆc cat appeared.[ ] *It[1] meowed.

This again requires indices! To avoid that, van Eijck (2001, p. 349) suggests that, “pronouns canbe translated as invitations to pick a reference from the current context.”

(31) λc.∀c′∃x.y.c^x^y = c′ ∧ farmer(x) ∧ donkey(y) ∧ own(x, y)→ beats(x, sel(c′))

sel(c′) is in the scope of ∀c′. This falsely predicts a reading of (32) where each nation cherisheseither their own king or the king of France.

(32) France1 is a monarchy. Every2 nation cherishes its3 king.

We also run into problems with backtracking. We observed that the resolution of its is likely todepend on whether we get the continuation in (33) a or b.

(33) a. So the French love King François.b. That is because his appearances in the UN have been so convincing.

We cannot directly access past states of the list. More individuals will have been added, and thearray may have been permuted by resolution of other anaphora.

The list approach requires us to resolve anaphora in contexts, whereas we want it to be done“globally”, taking the whole discourse into account. I will argue that the best way to do this is toreturn to the original DRT intuition that even anaphors introduce discourse referents, which arethen subsequently resolved to some antecedent. This makes it possible to talk about anaphoraresolution.⇒ lists of dref occurrences, with anaphora as discourse-level identi�cation of occurrences.

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3 Partial CDRTA common complaint about DRT is that it is not compositional, because it uses uni�cation ratherthan functional application to build DRSs. Functional application has at least two advantages:

1. Using functional application instead would let us use the Curry-Howard isomorphism toprovide a syntax-semantics interface, as in Categorial Grammar (van Benthem 1986), Dy-namic Syntax (Cann, Kempson, and Marten 2005) or Glue Semantics (Dalrymple 1999).

2. Intermediate representations can be eliminated

Compositional DRT (Muskens 1996) is one attempt to equip DRT with lambdas. The basic idea isto inject assignments, which are part of the metalanguage in DRT, into the object language. Wewill do this, but within a partial rather than a classical logic.

3.1 Partial type theoryWe need to be careful in setting up our partial type theory, as there are potential lurking problems(Lapierre 1992, p. 521f.). Our logic is based on the intuition that unde�nedness is absurdity, notlack of information (as in e.g. strong Kleene logic). Here are some key features, formal details arein Haug (2013).

Unde�ned objects: We model partiality through unde�ned objects. For each type α (simpleor complex) there is an unde�ned object #α, which is the bottom of a semi-lattice 〈D′α,vα〉, wherevα is a partial order (de�nedness). ?α is a constant denoting the unde�ned object of type α.

Non-monotonicity: ?α = ?α for allα, so the identity predicate is classical and non-monotonic:?t = ?t is true, but > = ?t is false.

Weak Kleene semantics for the connectives: Unde�nedness is absurdity and propagatesthrough formulae.

Beaver’s ∂ (Beaver 1992) for presuppositions: ∂(φ) is true i� φ is true, otherwise unde�ned.Quanti�cation over de�ned objects only: ∃x.f(x) is true i� there is a de�ned object such

that f is true of it; false i� f is false of all de�ned objects; and unde�ned i� f is unde�ned for allde�ned objects. Dually, ∀x.f(x) is true if f is true of all de�ned objects; false if there is a de�nedobject such that f is false of it; and unde�ned i� f is unde�ned for all de�ned objects.

Unde�ned objects have no properties except self-identity:P (a1, . . . , an)→ ∃x1. . . .∃xn.P (x1, . . . , xn) where P is a predicate distinct from identity.

3.2 Partial CDRT3.2.1 States and registers

registers (type π) are like adresses of memory locations; at any given state of the system (here:the discourse), that memory location may hold a di�erent value.

states (type s) are the possible states of the system (discourse)the non-logical constant ν of type (s(π, e)), which in a given state s maps all registers to an

individual (possibly the unde�ned one); pairs of registers and states reconstruct DRT drefswell-ordering of registers: we can always pick out the lowest unused register in a given state

s, written xs1

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We use an abbreviation to say that two states i and j di�er at most in the inhabitants theyassign to registers δ1 . . . δn.

(34) i[δ1 . . . δn]j =abr ∀δ((δ1 6= δ ∧ · · · ∧ δn 6= δ)→ ν(i)(δ) = ν(j)(δ))

Axioms for states:

1. ∃s.∀δ.¬∃e.ν(s)(δ) = e (empty state)2. ∀s.∀e.∃s′.s[xs1 ]s′ ∧ ν(s′)(xs1) = e (expansion)3. ∀s.∀δ.∀δ′.(∃e.ν(s)(δ) = e ∧ δ′ < δ)→ ∃e′.ν(s)(δ′) = e′ (no gaps)

So in each state, ν maps a (possibly empty) initial subset of the set of registers de�ned inhabitantsand any remaining registers to the unde�ned individual.

• Axiom 2 guarantees that we never “run out of states”.• If there are n individuals that satisfy an n-place predicate in our model, then there is also a

state that has these individuals inhabiting n registers.• Quanti�cation over states works as unselective quanti�cation over individuals

The intuition is that the state grows as the discourse continues. When an expression introducesa new discourse referent, that will always be xs1 , of type (sπ) – the �rst uninhabited register inits input context. So we avoid the problems with using free variables (as in DRT) or constants(as in classical CDRT). Whenever negation, quanti�cation, conditionals or intensionality createtemporary contexts, these will “branch o�” from the main state and be closed. Notice that evenanaphoric expressions will introduce discourse referents, which are subject to special constraints.

3.2.2 DRSs and conditions

We can view DRSs as abbreviations for terms in ordinary type theory, as in (35).

(35) [oα . . . oα+n|Γ1, . . . ,Γγ] := λi.λo.∂(i[xi1 . . . xin ]o) ∧ Γ∗1(o) ∧ · · · ∧ Γ∗γ(o)

(This is a deviation from (Haug 2013). We still use the subscript index to track the relative orderingof registers in the full language, but we also use the letter to track the state in which the registeris interpreted.)

Γ∗ is the formula that results from making the same substitutions in Γ as we did in the universeof the DRS containing Γ and expanding the result according to the following rules:

(36) R(δ1, . . . , δn) λi.R(ν(i)(δ1), . . . , ν(i)(δn))δ1 is δ2 λi.ν(i)(δ1) = ν(i)(δ2)not K λi.¬∃j.K(i)(j)K or L λi.∃j.K(i)(j) ∨ L(i)(j)K ⇒ L λi.∀j.L(i)(j)→ ∃k.L(j)(k)

It can be seen that what we doing when we expand abbreviation is ‘injecting’ the DRT de�nitionof veri�cation (see e.g. (11) and (15)) into the object langauge.

(37) sequencing: K ;K ′ λi.λo.∃k.K(i)(k) ∧K ′(k)(o)

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So again, the intuition is that sequencing requires there to be an intermediate state that is theoutput of K and the input to K ′. The merging lemma guarantees that we can always get rid ofthis intermediate state and write the result as a single DRS. In the abbreviated language, we thenneed to renumber discourse referents, but only if they are declared in the second DRS:

(38) a. [o1|P (o1)] ; [j1|Q(j1)] = [o1 o2|P (o1), Q(o2)]b. [o1|P (o1)] ; [ |Q(o1)] = [o1|P (o1), Q(o1)]

A similar renumbering principle holds when we embed DRS into others via negation, disjunctionor the conditional. Unlike, (Muskens 1996), our abbreviated language is closed under sequencing,i.e. the result of sequencing two DRSs can always be written as a DRS.

Finally, truth can be de�ned analogously to the DRT de�nition, as the existence of an outputstate given the empty input context (s∅), as in (39).

(39) ∃o.∂(s∅[xs∅1 . . . xs∅n ]o) ∧ Γ∗1(o) ∧ · · · ∧ Γ∗γ(o)

3.3 A discourse exampleWe are now ready to see how we can perform compositional analysis over the abbreviated lan-guage, without having to worry about the expansions.

(40) Pedro is in a bar. Every woman who ever dated a man despises him. He is sad.

Word meanings in (41). in has been type-raised following Muskens’ treatment of transitive verbs.(This shouldn’t be necessary and depends on your overall framework.)

(41) in λP.λx.(P (λy.[ |in(x, y)]))bar λx.[ |bar(x)]Pedro λP.[o1|pedro(o1)] ;P (o1)a λP.λQ.[o1| ] ;P (o1) ;Q(o1)

(42) [o1 o2|Pedro(o1), bar(o2), in(o1, o2)]

PedroλP.[o1|pedro(o1)] ;P (o1)

λx.[o1|bar(o1), in(x, o1)]

inλP.λx.(P (λy.[ |in(x, y)]))

λQ.[o1|bar(o1)] ;Q(o1)

aλP.λQ.[o1| ] ;P (o1) ;Q(o1)

barλx.[ |bar(x)]

Observe the renumbering in the uppermost functional application! The state at this stage canbe represented as in (43), where the double circle represents the “grafting point” for the furtherdiscourse.

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(43)o_2 #o_1 ...

Lexical entries for the second sentence are as in (44). Overlining marks a register as anaphoric.(This corresponds to the predicate ant in (Haug 2013). We will return to its exact meaning). Again,there is type-raising (in transitive verbs).

(44) every λP.λQ.[ | ([o1| ] ;P (o1))⇒ Q(o1) ]woman λx.[ |woman(x) ]who λP.λQ.λx.Q(x) ;P (x)dated λP.λx.(P (λy.[ |date(x, y) ]))man λx.[ |man(x) ]despises λP.λx.(P (λy.[ |despise(x, y) ]))him λP.[o1| ∂(male(o1)) ] ;P (o1)

Straightforward composition gives us the sentence meaning in (45):

(45) [ | [k1k2|woman(k1),man(k2), date(k1, k2)]⇒[l3|despise(l1, l3), ∂(male(l3))] ]

Observe that l1 = k1, so the reference of the relative pronoun is induced by the grammar and isnot anaphoric.

Sequencing with the previous discourse in (42) involves some renumbering and yields (46).

(46) [o1 o2| Pedro(o1), bar(o2), in(o1, o2),[k3k4|woman(k3),man(k4), date(k3, k4)]⇒ [l5|despise(l3, l5),male(l5)] ]

This can be visualized as in (47), where we have grafted a branch onto (43). The octagon indicatesan anaphoric discourse referent.

(47)o_2 #

k_3

protasis...o_1

# ...k_4

l_5

apodosis

# ...

The �nal sentence of (40) straightforwardly gets the representation in (48).

(48) [o1|sad(o1), ∂(male(o1))]

We merge everything and get the DRS in (49).

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(49) o :

o1 o2 o3

Pedro(o1)bar(o2)in(o1, o2)

k :

k3 k4

woman(k3)man(k4)date(k3, k4)

⇒ l :

l5

despise(l3, l5)∂(male(l5))

∂(male(o3))sad(o3)

The drefs can be visualized as in (50).

(50)o_3 # ...o_1 o_2

k_3

protasis

# ...k_4

l_5

apodosis

# ...

• We preserve the notion of accessibility between DRSs that we have in DRT. This is a partialorder which can be read o� geometrically (leftwards and upwards) in (49).

• Register ordering (which is total) also gives us accessibility inside universes• Together, these gives us accessibility as a partial order on discourse referents (construed as

pairs of states and registers), as illustrated in (50).• Unlike DRT, we can now merge DRSs without resolving anaphora, because grafting onto

(50) does not increase the set of accessible antecedents for l5 and o3.

4 Anaphora• A holistic, discourse perspective on anaphora (rather than context-based)• Anaphoric relations as relations between words (more precisely, syntactic tokens).• Modelled as a functionR taking anaphoric expressions to their antecedents.• A discourse interpretation is a tuple 〈K,P 〉 where K is a DRS and P is a set of pragmatic

enrichments includingR.

Here is one example:

(51)Pedro1 is2 in3 a4 bar5. Every6 woman7 who8 ever9 dated10 a11 man12 despises13 him14.He15 is16 sad17.

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R is isomorphic to a functionA between drefs, i.e. pairs of states and registers. Given the indexesin (49) we get (52).

(52) A : {l5 7→ k4, o3 7→ o1}

We can now capture dref overlining as two constraints on anaphoric resolution:

• the antecedent should be coreferent with the anaphor• the antecedent should be accessible to the anaphor

Formally:

(53) ν(s)(x) = ν(s)(A(s)(x)) ∧ A(s)(x) < x

In Haug (2013), these conditions were added to the DRS in whose universe the anaphoric reg-ister is introduced. This gives the wrong result in intensional contexts, because the anaphoricrelationship ends up as part of modal context. As an alternative, we can consider overlining toabbreviate a predicate ant singling out anaphoric discourse referents (pairs of states and registers)and amend the truth de�nition as in (54) to lift the condition on anaphoricity to the main DRS.

(54) ∀δ.∀s.ant(s)(δ)→ ∂(ν(s)(x) = ν(s)(A(s)(x)) ∧ A(s)(x) < x) ∧ . . .

This gives us a model-theoretic characterization of the monotonic part of the meaning of ananaphoric expression, viz. that it should corefer with an accessible antecedent. At the same time,we keep the pragmatic resolution itself,R, out of the monotonic content.

4.1 Reference to embedded discourse referentsThere is another situation we need to capture:

(55)

Pedro1 is2 in3 a4 bar5. Every6 woman7 who8 ever9 dated10 a11 man12 despises13 him14.He15 is16 sad17.

He15/o3 can corefer with he14/k5 because he14/k5 corefers with Pedro1/o1, which belongs in themain DRS. In other words, because k5 corefers with a dref in the main DRS, it gets ‘lifted’ to themain DRS and becomes accessible there.

By the transitivity of identity, this situation is identical to one in which o3 takes o1 as itsantecedent directly:

(56)Pedro1 is2 in3 a4 bar5. Every6 woman7 who8 ever9 dated10 a11 man12 despises13 him14.He15 is16 sad17.

DRT (and list-based approaches) are forced to adopt such an analysis, but it is hard to square witha realistic account of anaphoric resolution.

To solve this, we can have pragmatics supply (55), but map that onto the truth-conditionallyequivalent (56). Formally, we map mapR ontoR∗ as follows:

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(57) R∗(s)(i) =

R(i) if i is inhabited in sR∗(s)(R(i)) ifR(i) is de�nedunde�ned otherwise

Now we can setR(15)=14, resulting inR*(15)=R(14) i�R(14) refers to some syntactic token thatintroduces a discourse referent in the main DRS.

4.2 Bridging and split antecedentsThe semantics of anaphoricity in (53)-(54) enforces identity between an anaphor and its an-tecedent. This is too strong if we want to deal with bridging (58) and split antecedence (59).

(58) The1 bus arrived. The2 driver was drunk.(59) John1 met Mary2 and they3 went for a drink.

We need a richer notion of antecedence: to capture bridging, the value of R should be a pairconsisting of an antecedent and a coreference relation (defaulting to identity) between the anaphorand the ancedent. E.g. for (58).

(60) R(2) = 〈1, λx.λy.driver(x, y)〉

If we write antcd for the antecedent – i.e. fst(A(s)(x)) – and B – i.e. snd(A(s)(x)) – for thecoreference relation, we can de�ne anaphoricity as in (61).

(61) B(ν(s)(antcd), ν(s)(x)) ∧ antcd < x

We also need to alter our de�nition ofR∗, to compose relations along the path. Again using antcd(for fst(R(s)(i))) and B (for snd(R(s)(i))), we get (62).

(62) R∗(s)(i) =

R(i) if i is inhabited in s〈fst(R∗(s)(antcd)), snd(R∗(s)(antcd)) ◦B)〉 ifR(i) is de�nedunde�ned otherwise

Here is an example:

(63) Ivan’s car1 does not start. If it’s not due to the alarm system, the2 engine must have brokendown. It3 has already malfunctioned several times before, and Ivan has been thinking ofreplacing it.

AssumeR = {2 7→ 〈1, engine〉, 3 7→ 〈2,=〉}. We computeR∗(o)(3):

(64) R∗(o)(3) ≡ 〈fst(R∗(o)(2)), snd(R∗(o)(2) ◦ =)〉 ≡ 〈1, engine ◦ =〉 ≡ 〈1, engine〉

This is not a theory of bridging, but provides a formal language that could be the basis for such atheory. Similarly PCDRT, like DRT, is no theory of anaphoric resolution, just a formal underpin-ning. For split antecedence we need the �rst element of the pair to be a set of antecedents, plus atheory of plurals.

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5 ConclusionsUnresolved anaphora receive a model-theoretic interpretation (unlike other dynamic theories)Constraints on anaphora are also captured model-theoretically (as in DRT)The overwrite problem disappears (as in array-based approaches)Separation of monotonic and non-monotonic content : updating 〈K,P 〉 with K ′ yields

〈K ;K ′, P ′〉 where ; is monotonic but P → P ′ isn’t necessarily

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