Discourse Semantics

Nicholas Asher, Department of Philosophy, University of Texas at Austin March 13, 2004

1 Outline of Course

• Why go dynamic? • Why have discourse structure? • Logical Forms with Discourse Structure • Building Logical Forms with Discourse Struc- ture, including an introduction to logical tools for defeasible inference. • Applications: the spatio temporal structure of texts, Presupposition, implicatures, agreement and disagreements in dialogue.

2 Texts and Requirements

• a course packet with basic articles on dynamic semantics (sections of Kamp and Reyle, From Discourse to Logic, Kluwer 1993, van Eijk’s and Kamp’s article in the Handbook of Lan- guage and Logic, eds. J. van Bentham and A. ter Meulen) • N. Asher and A. Lascarides, Logics of Con- versation, Cambridge, 2003. • Course Requirements: Do most of the exercises in the notes. Slides will also be mailed in pdf format to those wishing them.

3 Setting the Stage

• Semantics: traditionally ”what is said” – lexical semantics: the contribution of indi- vidual words to (discourse) content – compositional semantics: the way meanings of individual words compose together to yield sentence contents. • Pragmatics: traditionally, ”what is conveyed” over and above what is said. A Theory of the interactions between the linguistic message and other information sources. Implicatures – language users’ cognitive state (Carnap, Mor- ris, Grosz and Sidner), – the discourse context (Levinson, Vallduvi).

4 My Two Cents’ Worth

Pragmatics supplements semantic content (often underspecified) by exploiting context and informa- tion in the cognitive states of discourse partici- pants. But careful, there are many different no- tions of context: Kaplan style contexts used to fix the referents of demonstratives and indexicals, con- texts for dynamic semantics, contexts that include limited information about the cognitive states of the participants, contexts that include all the knowl- edge that participants bring to interpretation. at the lexical and comopositional level.

5 Formal Methods in Semantics and Pragmatics

• building from Tarski’s Theory of Truth for For- mal Languages (FOL), semantics establishes a hopefully compositional way of building a log- ical form in a formal language from a natural language discourse and then interprets it recur- sively relative to a model. • Compositionality (the meaning of the whole is a function of the meaning of the parts–Frege) • Different sorts of interpretations: static truth conditions, dynamic update conditions. • Pragmatics at the level of logical form: fill- ing in elements underspecified by semantics. Pragmatics also allows us to infer often defeasi- bly other information from logical forms—viz. about elements of the speaker’s or hearer’s cog- nitive state (intentions, beliefs).

6 Static Semantics-e.g., Montague Grammar

• The sentence is the unit of meaning; its seman- tic value is a set of indices (those in which the sentence or its logical form is true). The mean- ing of a discourse is the intersection of the se- mantic values of its constituent sentences. • strict compositionality–the syntax of a natural language sentence completely determines how the meaning of a sentence is built up from the meaning of its constituents. • Every (disambiguated) word has a unique syn- tactic category and by strict compositionality a unique semantic type.

Sentence Syntax ----> Logical Form -->Model

• logical methods: higher order logic together with a typed lambda calculus.

7 A Simplified Extensional Example

Every farmer owns a donkey • Every: λP λQ∀x(P x → Qx) • a: λP λQ∃y(P y ∧ Qy) • farmer: λxfarmer(x) • owns: λΦλxΦ[λyowns(x, y)] • donkey: λzdonkey(z) Construction from Bottom Up: • A donkey: λP λQ∃y(P y∧Qy){λzdonkey(z)} −→ • λQ∃y(λzdonkey(z){y}∧Qy) −→ λQ∃y(donkey(y)∧ Qy) (lambda conversion, twice) • owns a donkey: λΦλxΦ[λyowns(x, y)]{λQ∃y(donkey(y)∧ Qy)} −→ λxλQ∃y(donkey(y)∧Qy)[λyowns(x, y)] • −→ λx∃y(donkey(y) ∧ owns(x, y)) • every farmer owns a donkey: ∀x(farmer(x) → ∃y(donkey(y)∧owns(x, y)))

8 Exercise

• Add to the basic vocabulary above a transla- tion for an intransitive verb like sleeps. How does it differ from the entry for owns? Specifi- cally do we need a lambda abstracted variable over DP denotations? Why do we need it for owns? • Use your entry to give a derivation of the ”proper” logical form for a farmer sleeps. • Try to construct the ”proper” logical form for if a farmer owns a donkey, he sleeps. What goes wrong? How could you fix this?

9 Anaphora and Static Semantics

Pronominal Anaphora: (1) a. Sally likes her car. b. A farmer owns a donkey. He beats it. What are the semantic values of pronouns? Pronouns as bound variables

(2a’) ∃x1(car(x1) ∧ owns(s, x2) ∧ like(s, x2)) But the translation of (2b) is problematic: First sentence: ∃x(farmer(x) ∧ ∃y(donkey(y) ∧ owns(x, y))) Second sentence: beats(x, y) How to bind the variables in the second formula?

10 One solution: Pronouns aren’t bound variables

They’re disguised definite descriptions that con- text will tell us how to specify. Second attempt at translating 2nd sentence of (2b):

beats(ιx(farmer(x)∧∃y(owns(x, y)∧donkey(y), ??)

Still not ideal, because what do we put in for ?? —the most reasonable choice is y, but that’s still treating that variable as a bound variable. There are problems of dependency between the variables, and also problems with uniqueness. There could be several farmers that own donkeys and beat them. Exercise: what do you think about the unique- ness requirements? What if we just drop them?

11 Temporal Anaphora

(2) Un homme entra. Il fuma une cigarette. Il partit. (A man entered. He smoked a cigarette. He left.) Traditional tense logic tells us simply that the three events introduced in the text occurred in the past. But there is an anaphoric dependency of one event on the other that static approaches miss.

12 Presupposition

(3) a. A child is petting his cat. b. If a child own a cat, then he will want to pet his cat. Traditional semantics treats presuppositions sep- arately but there appears to be an important inter- action between presupposed and asserted content— e.g., the quantifier in the assertion must bind the variable introduced in the presupposition in (3b). Further, a presupposition’s ”projection” depends on the logical structure of the assertion.

13 Dynamic Semantics

Dynamic Semantics (DRT, Dynamic Predicate Logic, Dynamic Montague Grammar) • The content of a sentence is not a static ele- ment but something dynamic that alters the context—more precisely, a relation from dis- course contexts to discourse contexts. • a discourse context is either a representational structure (as in DRT) or a set of assignment functions (for a fixed model) or a set of model assignment pairs, or a set of world assignment pairs (for a fixed model) (the latter is used to handle intensional constructions).

14 DRT— A version of Dynamic Semantics

Sentence Syntax ----> DRS -->Model constr. alg. • In DRT a DRS is a semantic representation; it is a pair of sets < U, C > where U is a set of discourse referents, and C a set of conditions. • In DRT a context is modelled as a DRS. • Input Context + DRS −→ Output Context 0 • K + K = h(UK ∪ UK0), (CK ∪ CK0)i

15 DRT on Anaphora

(4) John bought a book on semantics. He is reading it now. (5) Every book John buys is about semantics. He is reading it* now. first sentence of (4): j, x

(4’) bought(j, x) book-on-semantics(x)

Sentence 2: z, u

read(z, u) (4”) z =? u =?

16 The DRS for the whole discourse:

j, x, z, u

bought(j, x) (4”’) book-on-semantics(x) read(z, u z =? u =?

With the preferred values for the pronouns: j, x, z, u

bought(j, x) book-on-semantics(x) (4+) read(z, u) z = j u = x

17 Remarks on Underspecification

• Underspecified conditions for pronouns: e.g., z =? and u =? generated by the semantics of pronouns. are incomplete semantically. • Pragmatics completes underspecified conditions (replacing ? with the appropriate discourse ref- erents) but subject to semantic constraints • Alternatively, let the semantics freely generate all possible anaphoric bindings–some ruled out by semantic constraints (accessibility), some discarded on pragmatic grounds.

18 Accessibility

A constraint stated on DRSs: Only discourse referents in a DRS universe to the left or above are accessible to conditions of the form z =? DRS for (5):

j, x

buys(j, x) ⇒ on semantics(x) (5’) book(x)

read(z, u) z =? u =?

19 Dynamic Semantics and Temporal Structure

(6) Un homme entra. Il fuma une cigarette. Il partit. (A man entered. He smoked a cigarette. He left.) x, e, n

man(x) enter(e, x) e < n Rpt := T P pt = Ept := e

20 After the second sentence:

x, e, e1, y, n

man(x) enter(e, x) e < n cigarette(y) smoke(e1, x, y) e < e1 e1 < n Rpt := T P pt = Ept := e1

21 After the third sentence

x, e, e1, e2, y, n man(x) enter(e, x) e < n cigarette(y) smoke(e1, x, y) e < e1 e1 < n leave(e2, x) e1 < e2 e2 < n Rpt := T P pt = Ept := e2

22 Summing up the Simple Past in DRT :

Introducing notation that will come in handy the DRT predictions for tense sequences in French for simple sentences are as follows:

(< α, β, λ > ∧PS(α) ∧ PS(β)) → eα < eβ Exercise: give a formal rule for the past tense in the sense that you can say what the past tense does to a DRS.

23 Formalizing Dynamic Semantics via Dynamic Pred- icate Logic:

The syntax resembles that of first order logic. • logical symbols: ¬, ∃, ∧, = • grouping indicators: (, ) • nonlogical constants: predicate symbols (P, Q, R, . . .), individual constants (a, b, c, . . .), and function symbols (f, g, h, . . .). Formulas are defined recursively as in classical logic.

24 Some Key Points about the Semantics of DPL

• The semantics of a formula in DPL is a rela- tion between states. Formulas are ”actions” or programs with an input and an output state. • States are either partial or total assignments on the set of variables into the domain of a model. • With total assignments, all formulas except ex- istential ones simply ”test” whether the input assignment classically satisfies the formula and output the assignment unchanged or else out- put nothing (crash); in classical logic ∃ “resets” the value of the bound variable but unlike clas- sical logic, this reset value is carried forward. but in so doing it changes the output state of the formula. • with partial assignments, existentials extend the assignments to a value for the existentially bound variable.

25 The formal satisfaction definition

Assume a Tarskian static notion of satisfaction |= and let ◦ be relational composition:

M • fkP t1, . . . , tnk g iff g = f∧fM, f |= P t1, . . . tniff M M M hkt1kf ... ktnkf i ∈ R M • fkt1 = t2k g iff g = f ∧ fM, f |= t1 = t2 • fkφ∧ψkM g iff ∃hfkφkM h∧hkψkM g (fkφkM ◦ kψkM g) • fk¬φkM g iff f = g ∧ ¬∃hfkφkM h a • fk∃xφkM g iff ∃h∃a ∈ D(M)f x = h∧hkφkM g

26 Exercises:

• Does ∧ have different properties in dynamic semantics from what it does in classical logic? And does this make any sense? E.g. think about Kim took off his shoes and went to bed vs. Kim went to bed and took off his shoes. • say what sort of input output contexts satisfy the formulas below; verify what happens by using the truth definition: – ∃xF x ∧ Hx – F x ∧ ∃∃¬F x – ∃¬F x ∧ F x

27 Back to DRT: Syntax

1. Discourse Referents is a set of objects denoted by x, y, z, with or without subscripts. 2. Predicates is a set of predicate constants as- sociated with various natural language nouns, verbs and adjectives. 3. Supose U ⊆ Discourse Referents; we then de- fine DRSs K and conditions γ recursively:

K := hU, 0i | K∩γ Let R ∈ Predicates be an n-ary predicate and x1, ··· , xn be discourse referents.

γ := R(x1, ··· , xn) | ¬K | K1 ⇒ K2 | K1 ∨ K2.

28 Semantics of DRT

• Model Theory for DRSs (the Standard View (Kamp and Reyle): Uses the notion of an em- bedding of one structure (A DRS) into a clas- sical first order model. • Relational Version of Model Theory for DRSs

fPM (U, O)g iff f ⊆ g ∧ dom(g) = dom(f) ∪ U

fPM (R(x1, ··· , xn))f iff RM (f(x1), ··· , f(xn)) ∩ fPM (K γ)g iff fPM (K)g ∧ gPM (γ)g

fPM (¬K)f iff ¬∃g fPM (K)g 0 0 fPM (K ⇒ K )f iff ∀g (fPM (K)g → ∃ h gPM (K )h) 0 0 fPM (K ∨ K )f iff ∃ g fPM (K)g ∨ ∃h fPM (K )h

Remark 1: Accessibility is now explained Remark 2: Note the notational variance between DRT and DPL.

29 Rapprochement with First Order Logic

Define the lifted counterpart P of Pon sets of model sequence pairs (MSP)’s for a DRS K. P(K):Pow(MSP) → Pow(MSP) is defined distributively or pointwise . That is for a context X and a given DRS K: 0 0 [P(K)](X) = {(M, g ): ∃g(M, g) ∈ X ∧ gPM (K)g )}

Let {hM, 0i : M ∈ MOD} = σ0 Characterization Lemma (Fernando 1994): • For every first order formula χ with a set of free variables U, there is a DRS (U, C) such that P(U, C)[σ0] = {(M, f): Dom(f) = U and M |= χ[f]} • Every DRS (U, C) has a characteristic formula χ where U is the set of free variables in χ and P(U, C)[σ0] = {(M, f): Dom(f) = U and M |= χ[f]} As shown in Groendijk and Stokhof’s DPL (for a discussion see e.g. van Eijck and Kamp 1996), the CCP of a sentence may also be represented as a re- lation between information states (assuming once

30 again the presence of the DRS construction proce- dure). A model-theoretic context is a set of model sequence pairs (MSPs), and so a natural candidate for the model-theoretic notion of the CCP of a sen- tence S is just the lifted function P applied to the DRS derived from S.

31 ”Un peu d’histoire”

Karttunen (1976) ”discourse referents” Webber (1976) a similar notion of discourse individual Kamp (1979) —discourse referents and DRSs for individ- uals in the analysis of anaphoric tense phenomena Kamp (1981) —Analysis of indefinites, pronouns, conditionals and universal quantifiers. ”Core Frag- ment” of DRT. (add negation) Relational Interpretation (Johnson and Klein (86), Muskens, Fernando (94), Kamp and van Eijck(96))

32 The DRS construction Procedure

Two key points about all dynamic theories: • They exploit the syntax of the sentence cur- rently being processed in defining how the in- put context gets updated to the output context (think of input and output contexts as drss). • constructing the logical form for the output context is independent of the interpretations of the input context and the new information. Only the structure or ‘form’ of the informa- tion influences this. The logical form doesn’t need to be interpreted incrementally; the drs which represents all of the discourse can be constructed before it’s evaluated against a model. • This allows dynamic semantics to be more or less compositional,and more or less dynamic (Kamp 1981 and Kamp and Reyle 1993 don’t have dynamic formulations).

33 Top Down Approach to LF (Kamp, Kamp and Reyle)

• Each drs construction rule stipulates a syntax tree fragment that must be present for the rule to apply. • The rule deletes part of the tree, replacing it with discourse referents and drs conditions. These drs conditions can refer explicitly to discourse referents that are part of the input context, making the content that’s introduced by the drs construction rule dependent on the ‘prior’ discourse referents that were introduced by a (prior) application of some other construc- tion rule. In this sense, the translation from syntax to logical form is non-compositional.

34 An Example of a Kamp and Reyle Rule

The drs construction rule cr.pro in the figure below for translating singular pronounss states that if the current sentence’s syntax has a subtree which matches either one of those given in the triggering configuration (basically, there’s a pronoun), then • one introduces a new discourse referent u to the set of discourse referents UK; • one chooses a suitable antecedent v (i.e., v must be accessible, and the gender of v must be β, to match u’s gender), and • one introduces the two conditions u = v and Gen(u) = β to the set CK of drs conditions. • Finally, one replaces the NP node in the tree and everything below it with u.

35 cr.pro Triggering Configuration: S or: VP

0 NPGen=β VP V NPGen=β

PRO PRO

α α

Choose suitable antecedent v: such that v is accessible and Gen(v) = β Introduce in UK : new discourse referent u Introduce in CK : new conditions u = v and Gen(u) = β Substitute in the triggering configuration: u for NPGen=β

PRO

α

36 Other Rules

Similar rules exploit other syntactic configura- tions: • rules for DPs: e.g.

– if [[a[φ]NP ]DP [ψVP ]]IP occurs in a DRS K, introduce a new discourse referent x into UK and replace the input syntactic config- uration with (i) the condition φ(x) and (ii) the configuration [ψ(x)]VP ]

– if [V (v)[a[ψ]NP ]DP (v)]VP occurs in a DRS K, introduce a new discourse referent x into UK and replace the input syntactic config- uration with (i) the condition ψ(x) and (ii) the condition V (v, x) • rules for VPs: e.g.

– if [[ψV ](v)]VP is the input configuration, then output V (v)

– if [V [a[ψ]NP ]DP (v)]VP is the input configu- ration, then output [V (v)[a[ψ]NP ]DP (v)]VP

37 Application

The toy rules above ensure that (7) equates the individual that walked in with the one that ordered the beer–the final form is in (70): (7) A man walked in. He ordered a beer. x, y, z

(70) man(x), walk-in(x) beer(y), order(z, y), z = x

Exercise: construct the DRS for (7) using the rules we’ve introduced. Remark This procedure for constructing drss hides essentially merges two separate tasks into one step: (i) indicate how syntax contributes to logical form; and (ii) specify how context affects meaning thus hiding the dynamics of the semantics. Composi- tional interpretation is lost here.

38 Bottom up DRS Construction

(Reyle and Frey, Wada and Asher, Asher, Zeevat) Each expression gets assigned a lambda term (we add in effect lambda calculus to dynamic predicate logic). • a : λP λQ[x|P (x); Q(x)] • man: λx[|man(x)] • walked in: λu[|walked-in(u)] • he: λP [v|v =? Use the syntactic analysis to figure out what are the arguments to which function. Rule to rule com- positionality as in Montague. • a man λP λQ[y|P (y),Q(y)](λx[|man(x)]) • λQ[y|λx[|man(x)](y),Q(y)] (lambda conver- sion) • λQ[y|man(y),Q(y)] (lambda conversion again) a man walked in: • λQ[y|man(y),Q(y)](λu[|walked-in(u)])

39 • [y|man(y), λu[|walked-in(u)](y)] • [y|man(y), walked-in(y)] Exercise: extend this lexicon to handle (7). What problems do you encounter?

40 Intensionalizing Dynamic Semantics

0 (i) (w, f)[[hU, ∅i]]M (w , g) iff w = w0 ∧ f ⊆ g ∧ dom(g) = dom(f) ∪ U ∩ 0 (ii) (w, f)[[K γ]]M (w , g) iff 00 00 00 0 ∃w ∃h(w, f)[[K]](w , h)∧(w , h)[[γ]]M (w , g) 0 (iii) (w, f)[[R(x1, ··· , xn]]M (w , g) iff 0 (w, f) = (w , g) ∧ hf(x1), ··· , f(xn)i ∈ IM (R)(w) 0 (iv) (w, f)[[¬K]]M (w , g) iff 0 00 00 (w, f) = (w , g)∧¬∃w ∃h such that (w, f)[[K]]M (w , h) 0 0 (v) (w, f)[[K ⇒ K ]]M (w , g) iff (w, f) = (w0, g)∧ ∀h∀w00(w, f)[[K]](w00, h) → ∃k∃w000(w00, h)[[K0]](w000, k) 0 0 (vi) (w, f)[[K ∨ K ]]M (w , g) iff (w, f) = (w0, g)∧ 0 0 0 (∃h(w, f)[[K]]M (w , h)∨∃k(w, f)[[K ]]M (w , k)) 0 0 (vii) (w, f)[[K > K ]]M (w , g) iff (w, f) = (w0, g)∧ 00 00 ∀w ∀h((w, f)[∗(w, [[K]]M )](w , h) → 000 00 0 000 ∃w ∃k(w , h)[[K ]]M (w , k))

41 0 (viii) (w, f)[[2K]]M (w , g) iff (w, f) = (w, g)∧ 00 00 000 0 000 ∀w (wR2w → ∃h∃w such that (w , g)[[K]]M (w , h) This truth definition for drss proceeds relative to a fixed model M. Note that although possible worlds are needed to give the right satisfaction con- ditions for the drs conditions K > K0, 2K and 3K, none of the clauses make the ‘output’ possi- ble world different from the input one (i.e., all the above clauses impose the condition w = w0). This will change when we get to questions and requests.

42 Questions

Belnap (1985), Bennett (1979), Kartunnen (1977) and Groenendij and Stokhof (1984): the meaning [[q]] of an interrogative q is a function from worlds to sets of propositions. For us, [[q]]M (w) is a set of propositions (in fact relations), corresponding to the set of all true direct answers, including the non- exhaustive ones. The denotation of Who came to the party? at w will include the proposition that John came to the party, the proposition that Mary came to the party, and the proposition that they both came to the party (if they indeed both did). Notation: p is satisfiable in a model M given a particular input context (w,f): 0 ∨ 0 ∃w ∃g(w, f)[[ p]]M (w , g).

43 Semantics for Questions

(w, f)[[?]]M ([[λx1 . . . λxnP (x1, . . . , xn)]]M = {[[p]]M :

• ∃[[α1]] ... [[αn]]M [[p]]M = [[α1]]M ... [[αn]]M ([[λx1 . . . λxnP (x1, . . . , xn))]]M ;

0 ∨ 0 • ∃w ∃g(w, f)[[ p]]M (w , g)

Minimal Semantics. Could add that answers must determine witnesses for P (x1, . . . , xn). Could specify answers to be exhaustive (contra cf. Jonathon Ginzburg’s work).

44 The Language of Requests

Requests as action terms that change the world component of the context. 1. seeing to it that p, where p is a normal formula or drs. I.e., If K is a drs, then δK is an action term;

2. If a1 and a2 are action terms, then so are a1; a2 (this stands for action sequence) and a1 + a2 (this stands for choice); 3. If a is an action term and K is a drs, then [a]K is a drs condition (this will mean that once a is performed K is true). 4. If K is a drs and a is an action term, then K ⇒ a is a drs condition (this describes guarded action).

45 Extending the Semantics

0 0 0 1. (w, f)[[δK]]M (w , g) iff (w , f)[[K]]M (w , g)

2. [[a1; a2]]M = [[a1]]M ◦ [[a2]]M 00 (i.e., (w, f)[[a1; a2]]M (w , h) iff there is a pair (w0, g) such that 0 0 00 (w, f)[[a1]]M (w , g) and (w , g)[[a2]]M (w , h).)

3. [[a1 + a2]]M = [[a1]]M ∪ [[a2]]M 0 0 (i.e,. (w, f)[[a1 + a2]]M (w , g) iff (w, f)[[a1]]M (w , g) 0 or (w, f)[[a2]]M (w , g)). 0 0 4. (w, f)[[K ⇒ a]]M (w , g) iff (w, f) = (w , g) and 00 00 for all w and h such that (w, f)[[K]]M (w , h) 000 00 000 1 there is a w and k such that (w , h)[[a]]M (w , k). 0 0 5. (w, f)[[[a]K]]M (w , g) iff (w, f) = (w , g) and 00 00 for every h and w such that (w, f)[[a]]M (w , h), 000 00 000 there exists w and k such that (w , h)[[K]]M (w , k). Note that, thanks to condition 2 above, the deno- tation of the complex action (8) is one where the individual who talks and the individual who walks is one and the same: 1Note that this semantics is exactly the same for imperative as for indicative clauses.

46 x (8) δ ; δ walk(x) talk(x)

And guarded actions can be the basis of condi- tional commands: (9) If you want to get an A, study hard.

47 Examples of Imperatives

Imperatives move the state of the world along— e.g., recipes, directions for getting to a place. But: (10) and (11): (10) Come home by 5pm and we will finish the shelves tonight. (11) Smoke a packet of cigarettes a day and you will die before you’re 30. Answer in Asher and Lascarides (2003): rhetorical relations here are crucial. Also: (12) a. A: How does one catch the 10:20 train to London? b. B: Go to platform 1. Note the difference of the meaning of (12b) in this context from its meaning in the ‘null’ context. The imperative in (12b) is not commanded. Rather, its ‘rhetorical function’ is to provide sufficient infor- mation that A can compute an answer to his ques- tion (which is an adverbial of manner, given the

48 compositional semantics of how-questions Asher and Lascarides 1998).

49 Exercise

Exercise: Extend both our analysis of impera- tives and of assertions to a fuller fragment that includes adverbials (see Dieugenio 1998) for a dis- cussion of adverbials and imperatives), capturing for example the similar meanings of (13a–b) and (13c–d): (13) a. Go into the kitchen to get the coffee urn! b. Go into the kitchen and get the coffee urn! c. Open the safe by using the key. d. Use the key to open the safe.

50