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Proceedings of Workshop on Theories of InformationDynamics and Interaction and their Application toDialogue (TIDIAD 2010), Copenhague, Danemark,

16-20 august 2010Emiliano Lorini, Laure Vieu

To cite this version:Emiliano Lorini, Laure Vieu. Proceedings of Workshop on Theories of Information Dynamics andInteraction and their Application to Dialogue (TIDIAD 2010), Copenhague, Danemark, 16-20 august2010. Lorini, Emiliano; Vieu, Laure. IRIT : Institut de recherche en Informatique de Toulouse, 2010.�hal-03672511�

TIDIAD2010 Proceedings

Workshop on Theories of Information Dynamics andInteraction and their Application to Dialogue

Workshop at ESSLLI 2010First week, 16-20 August 2010

Copenhagen, Denmark

Workshop Organizers: Emiliano Lorini, Laure Vieu

Theoretical approaches to communication and dialogue modeling are variedand often unrelated because separately focusing on di!erent aspects of dialogue(speech acts, goals, beliefs, plans, questions, conventions, roles, cooperation, dis-putes, argumentation, reference, semantics-pragmatics interface). On the otherhand, the area of foundations of multi-agent systems is inducing new devel-opments in logics of interaction and information dynamics, with a recent trendtowards comparison and integration. Analyzing the impact of this trend on com-munication and dialogue modeling is timely.

The workshop TIDIAD 2010 (Workshop on Theories of Information Dynam-ics and Interaction and their Application to Dialogue) aims at discussing formaltheories and logics of information dynamics and interaction and their applica-tions to dialogue and communication modeling. It is intended to bring togetherlogicians, linguists and computer scientists in order to provide a better under-standing of the potentialities and limitations of formal methods for the analysisof dialogue and communication. Its scope includes not only the technical aspectsof logics, but also multidisciplinary aspects from linguistics, philosophy of lan-guage, philosophy of social reality, social sciences (social psychology, economics).

The following are some examples of formal theories and logics that are rele-vant to the workshop (no order):– speech act theory,– argumentation theory,– game theory,– public announcement logic,– dynamic epistemic logic (DEL),– logics of agency and power (e.g. STIT, ATL, Coalition Logic),– theories of persuasion,– theories of commitment,– dynamic semantics,– semantic approaches to interrogative clauses,– rhetorical approaches to dialogue (e.g., SDRT).

The focus of the workshop is on recent developments, especially those that com-bine several approaches (e.g. dynamic epistemic logic and speech act theory,dynamic epistemic logic and segmented discourse representation theory, pub-lic announcement logic and commitment theories, STIT and dynamic epistemiclogic, Coalition Logic and public announcement logic) to deal with complex di-alogue and communication phenomena.

We received 18 papers, of which 10 were deemed acceptable for presentation atthe workshop by the program committee. We are grateful to the members of theProgram Committee for their service in reviewing papers and to the ESSLLI 2010organizers for their support in the organization of the TIDIAD 2010 workshop.We are glad to host two invited speakers at TIDIAD 2010, Patrick Blackburnand Jeroen Groenendijk. Our deepest thanks to them for having accepted toenhance the program of this workshop.

July 2010,Emiliano Lorini and Laure Vieu

TIDIAD2010 Organization

Programme Chairs

Emiliano Lorini, Laure Vieu

Programme Committee

Nicholas AsherGuillaume AucherAlexandru BaltagAnton BenzPhilippe BesnardPatrick BlackburnGuido BoellaJan BroersenCristiano CastelfranchiRaquel FernandezJonathan GinzburgJeroen GroenendijkDavide GrossiAndreas HerzigJoris HulstijnBarteld KooiKepa KortaAlex LascaridesDominique LonginJohn-Jules MeyerPaul PiwekHenry PrakkenHans van DitmarschWiebe van der HoekLeon van der Torre

Table of Contents

Conversational implicature in conversation (invited talk)Patrick BlackburnRadical inquisitive semantics (invited talk)Jeroen GroenendijkReasoning dynamically about what one saysNicholas Asher, Alex LascaridesExclusiveness implicatures in linguistic answersKata BaloghArgumentation games as evaluation games: The case of stableextensionsDavide GrossiDialogue and interaction : the Ludics viewAlain Lecomte, Myriam QuatriniLearning with neighboursRoland MuhlenberndMetalanguage dynamics.Tillmann ProssA bid for your approval: What does Auction Theory tell usabout linguistic behavior?Jason QuinleySplit antecedents and non-singular pronouns in dynamic se-manticsJos TellingsDynamics of implicit and explicit beliefsFernando R. Velazquez-QuesadaAn empirical model of strategic dialogue in group decisionswith uncertaintyMichael Wunder, Matthew Stone

Conversational implicature in conversation

Patrick Blackburn

INRIA Lorraine, Nancy

Radical inquisitive semantics

Abstract∗

Jeroen Groenendijk & Floris Roelofsen

ILLC/Department of Philosophy

University of Amsterdam

Inquisitive semantics: propositions as proposals

Traditionally, the meaning of a sentence is identified with its informative content.

In much recent work, this notion is given a dynamic twist, and the meaning of a

sentence is taken to be its potential to change the ‘common ground’ of a conversa-

tion. The most basic way to formalize this idea is to think of the common ground as

a set of possible worlds, and of a sentence as providing information by eliminating

some of these possible worlds.

Of course, this picture is limited in several ways. First, when exchanging infor-

mation sentences are not only used to provide information, but also—crucially—to

raise issues, that is, to indicate which kind of information is desired. Second,

the given picture does not take into account that updating the common ground is

a cooperative process. One conversational participant cannot simply change the

common ground all by herself. All she can do is propose a certain change. Other

participants may react to such a proposal in several ways. In a cooperative conver-

sation, changes of the common ground come about by mutual agreement.

In order to overcome these limitations, inquisitive semantics (Groenendijk,

2009; Mascarenhas, 2009; Groenendijk and Roelofsen, 2009; Ciardelli and Roelof-

sen, 2009, among others) starts with a different picture. It views propositions as

proposals to update the common ground. Crucially, these proposals do not always

specify just one way of updating the common ground. They may suggest alterna-

tive ways of doing so, among which the addressee is then invited to choose.

Formally, a proposition consists of one or more possibilities. Each possibility

is a set of possible worlds and embodies a possible way to update the common

ground. If a proposition consists of two or more possibilities, it is inquisitive: it

∗This abstract consists of the introductory section of a preliminary version of the full paper. The

current version of the full paper can be found at http://www.illc.uva.nl/inquisitive-semantics.

1

invites other participants to provide information in such a way that one or more ofthe proposed updates may be established. Inquisitive propositions raise an issue.They indicate which kind of information is desired. In this way, inquisitive seman-tics directly reflects the idea that information exchange consists in a cooperativedynamic process of raising and resolving issues.

A concrete implementation of inquisitive semantics for the language of propo-sitional and first-order predicate logic has been specified in (Groenendijk and Roelof-sen, 2009; Ciardelli and Roelofsen, 2009; Ciardelli, 2009a,b, among other places).Here we will argue that this system, to which we will refer as conservative inquis-itive semantics, only partially captures the central underlying conception of sen-tences as expressing proposals to update the common ground. In the full paper wepresent and illustrate an enriched implementation that we call radical inquisitivesemantics.

Logic and conversation

One of the advantages of a notion of meaning that captures both informative andinquisitive content is that it gives rise to a much wider spectrum of logical relationsbetween sentences than a notion of meaning that captures only informative content.If meaning is identified with informative content, and propositions are taken to besets of possible worlds, then there are just two logical relations that can be capturedformally: entailment (set inclusion) and consistency (set overlap). If we adopta notion of meaning that captures both informative and inquisitive content, andconceive of propositions as sets of possibilities, it becomes possible to characterizea much wider range of logical relationships between sentences.

The dominant focus of formal semantics on informative content is rooted inthe historical fact that the logical tools that it uses were originally developed toreason about the validity of arguments. For this particular purpose, informativecontent is perhaps indeed the most crucial meaning component, and entailmentand consistency are the central logical notions that need to be characterized.

However, there is no good reason why the analysis of language more generallyshould limit its notion of meaning to informative content, and its range of logicalrelations to entailment and consistency. In fact, other logical relations that onemight like to capture readily come to mind. For instance, one might be interested ina formal characterization of when one sentence is entirely directed at endorsing theproposal expressed by another sentence, or when one sentence is entirely directedat rejecting the proposal raised by another sentence. To illustrate how inquisitivesemantics facilitates such characterizations, let us zoom in on the first of theselogical relations, which we refer to as compliance.

2

Compliance

Consider the sentence in (1), and the responses in (2-a-b).

(1) Pete will play the piano or Mary will dance tonight.

(2) a. Yes, Pete will play the piano.

b. Yes, in fact Pete will play the piano and Mary will dance.

c. Yes, in fact Pete will play the piano and I ate spaghetti last night.

We take it that there is a pre-theoretical distinction between the responses in (2-a-b)

and the one in (2-c). It is appropriate to say, even in the absence of any contextual

information, that (2-a-b) are entirely directed at endorsing the proposal expressed

by (1). This is not true for (2-c). In the absence of contextual information, we

cannot conclude that (2-c) is entirely directed at endorsing the proposal expressed

by (1): the statement that I ate spaghetti last night can only be taken to serve this

purpose in very specific contexts.

Can this pre-theoretical distinction be captured formally? Consider first a se-

mantic theory whose notion of meaning only reflects informative content. In such a

theory the meaning of a sentence is taken to be the set of possible worlds in which

the sentence is true. Let’s call this the truth-set of the sentence. It is easy to see

that truth-sets are too coarse-grained to capture the distinction between (2-a-b) and

(2-c): the truth-sets of (2-a-c) are all strictly included in the truth-set of (1). There

is no way to distinguish (2-c) from the other responses on the basis of truth-sets

alone.

In inquisitive semantics, where the meaning of a sentence reflects both infor-

mative and inquisitive content, the distinction can be captured. The sentence in

(1) expresses a proposition consisting of two possibilities. One of these possibil-

ities consists of all worlds in which Pete will play the piano, the other possibility

consists of all worlds in which Mary will dance.

Now we could say that one sentence ϕ is a compliant response to another sen-

tence ψ if and only if every possibility for ϕ either directly coincides with a pos-

sibility for ψ, or can be obtained from a set of possibilities for ψ by union and/or

intersection.1

This notion of compliance captures the distinction that is relevant

here: the responses in (2-a-b) count as compliant responses to (1), but (2-c) does

not.

1In (Groenendijk and Roelofsen, 2009), we defined a slightly different notion of compliance. The

difference will be discussed below, but does not affect the present argument.

3

Counter-compliance

Compliant responses are always of a positive nature: they endorse the given pro-posal, and—ideally—specify which of the proposed updates can be established. Ofcourse, such responses play an important role in cooperative information exchange,and it is useful to have a logical characterization of their properties.

However, negative reponses, which reject a given proposal, and—ideally—specify why the proposed updates cannot be established in any way, are just asimportant in order to exchange information in an efficient, cooperative way.

In the development of inquisitive semantics so far, the focus has almost ex-clusively been on positive responses. And it turns out that the implementationpresented in earlier work does indeed not provide the necessary tools to provide aparallel characterization of positive and negative responses. To illustrate this point,consider the sentence in (3), and the possible responses in (4):

(3) Pete will play the piano and Mary will dance tonight.

(4) a. No, Pete will not play the piano.b. No, Pete will not play the piano and Mary will not dance.c. No, Pete will not play the piano and I ate spaghetti last night.

This example is completely parallel to the one we saw above. Again, there is a pre-theoretical distinction between (4-a-b) and (4-c), which we would like to capture informal terms. However, the current implementation of inquisitive semantics doesnot bring us very far in this case. The responses in (4-a-c) all express a propositionconsisting of a single possibility, and all these possibilities are disjoint from theunique possibility for (3). There is nothing that distinguishes (4-a-b) from (4-c). Itis impossible to capture the distinction between these sentences just on the basis ofthe possibilities that they are assigned.

One way to treat positive and negative responses as equal citizens is to formallyrepresent a proposal not just as a set of possibilities, but rather as a set of possi-bilities plus a set of counter-possibilities, where—roughly speaking—possibilitiescorrespond to positive responses and counter-possibilities to negative responses.Once the notion of meaning is enriched in this way, we will not only be able to de-fine a notion of compliance, but also a notion of counter-compliance, its negativecounterpart.

This approach will be explored in detail in the full paper. We will present aradical inquisitive semantics for the language of propositional logic, and illustrateits workings with a number of examples showing how it accounts for compliantand counter-compliant responses to sentences of natural language. Much of ourattention will be devoted to the treatment of conditional sentences, both condi-

4

tional assertions and conditional questions. One remarkable result is that ‘denial of

the antecedent’ responses to a conditional are characterized as counter-compliant

responses to the ‘question behind’ that conditional.

References

Ciardelli, I. (2009a). A first-order inquisitive semantics. In M. Aloni and K. Schulz,

editors, Proceedings of the Seventeenth Amsterdam Colloquium.

Ciardelli, I. (2009b). Inquisitive semantics and intermediate logics. Master Thesis,

University of Amsterdam.

Ciardelli, I. and Roelofsen, F. (2009). Inquisitive logic. To appear

in the Journal of Philosophical Logic, available via www.illc.uva.nl/

inquisitive-semantics.

Groenendijk, J. (2009). Inquisitive semantics: Two possibilities for disjunction.

In P. Bosch, D. Gabelaia, and J. Lang, editors, Seventh International TbilisiSymposium on Language, Logic, and Computation. Springer-Verlag.

Groenendijk, J. and Roelofsen, F. (2009). Inquisitive semantics and pragmatics.

In J. M. Larrazabal and L. Zubeldia, editors, Meaning, Content, and Argument:Proceedings of the ILCLI International Workshop on Semantics, Pragmatics,and Rhetoric. www.illc.uva.nl/inquisitive-semantics.

Mascarenhas, S. (2009). Inquisitive semantics and logic. Master Thesis, University

of Amsterdam.

5

Reasoning Dynamically about What One Says

Nicholas Asher

IRITUniversite Paul Sabatier, Toulouse

asher@irit.fr

Alex Lascarides

School of Informatics,University of Edinburghalex@inf.ed.ac.uk

Abstract

In this paper we make SDRT’s glue logic forcomputing logical form dynamic. This al-lows a dialogue agent to anticipate what theupdate of the semantic representation of thedialogue would be after his next contribu-tion, including the effects of the rhetoricalmoves that he is contemplating performingnext. This is a pre-requisite for planning whatto say next. We make the glue logic dynamicby extending a dynamic public announcementlogic (PAL) with the capacity to perform de-fault reasoning—an essential component ofinferring the pragmatic effects of one’s dia-logue moves. We add to the PAL languagea new type of announcement, known as ce-teris paribus announcement, and this is usedto model how an agent anticipates the defaultconsequences of his next dialogue move. Ourextended PAL validates more intuitively com-pelling patterns of default inference than ex-isting PALs for practical reaosning, and wedemonstrate via the proof of reduction axiomsthat the dynamic glue logic, like its static ver-sion, remains decidable.

1 Introduction

Speakers in dialogue anticipate their interlocutors’ in-terpretations and adjust their utterances accordingly.Researchers adopt planning (e.g., Stone (1998)), de-cision theory (e.g., Williams and Young (2007)) orgame theory (e.g., van Rooij (2001)) to model suchdecisions. But these approaches tend to use modelsof semantics that don’t capture constraints on inter-pretation stemming from logical structure (Kamp andReyle, 1993) or discourse coherence (Hobbs et al.,1993, Asher and Lascarides, 2003). On the otherhand, semantic models that do capture such con-straints either do not interpret the rules for construct-ing logical form (Poesio and Traum, 1998), or doso in a static logic (e.g., weighted abduction (Hobbs

et al., 1993), or nonmonotonic deduction (Asher andLascarides, 2003)). Since dynamic discourse updateuses the static axiomatisation but is not a part of it,a speaker cannot compare his candidate next moves,inferring how they will be interpreted—but doing thisis essential for axiomatising decisions about what tosay.

This paper aims to fill this gap. We start withSegmented Discourse Representation Theory (SDRT,Asher and Lascarides (2003)), a model of discoursewhere interpretation depends on logical and rhetoricalstructure. We will make SDRT’s existing, static gluelogic for constructing logical forms dynamic, incor-porating discourse update into the axiomatisation. Wethus achieve a pre-requisite for making strategic de-cisions during conversation—the speaker can reasonabout an interlocutor’s interpretation—but we leaveto future work the task of interfacing these expectedoutcomes with the speaker’s preferences or goals.

Section 2 describes the logical form of dialogue inSDRT and Section 3 presents its existing glue logicand the accompanying dynamic discourse update.Section 4 replaces the static glue logic with a dy-namic one that incorporates discourse update into theaxiomatisation: we extend a dynamic logic of publicannouncement with the capability to perform defaultreasoning, a necessary feature of dialogue interpreta-tion. We prove that the dynamic glue logic has thesame computational complexity as the static one; soconstructing logical form remains computable.

2 Logical Forms for Dialogue

A fundamental decision that a speaker must makeabout his next move is its effects on agreement. Las-carides and Asher (2009) argue that rhetorical re-lations (e.g., Narration, Explanation) are crucial forcapturing implicit agreement: representing the il-locutionary contribution of an agent’s utterance viarhetorical relations reflects his commitments to an-other agent’s commitments, even when this is linguis-tically implicit. For example, Karen’s utterance (1c),taken from (Sacks et al., 1974, p717), commits her to

(1b) thanks to the semantic consequences of the re-

lational speech act Explanation(1b, 1c) that she has

performed:

(1) a. Mark (to Karen and Sharon):Karen ’n’ I’re having a fight,

b. after she went out with Keith and not me.

c. Karen (to Mark and Sharon):Wul Mark, you never asked me out.

Arguably, by committing to (1b) Karen also commits

its illocutionary effects—(1b) explains (1a). These

commitments are not monotonically entailed by (1c)’s

compositional semantics nor by Karen’s asserting it.

Rather, Karen’s implicit acceptance of Mark’s contri-

bution is logically dependent on the relational speech

acts they perform and their semantics.

More generally, Lascarides and Asher (2009) pro-

pose that the commitments of each agent at a given di-

alogue turn (where a turn boundary occurs whenever

the speaker changes) is a Segmented Discourse Rep-

resentation Structure (SDRS, Asher and Lascarides

(2003)) as shown in each cell of Table 1, the pro-

posed logical form for (1). Roughly, an SDRS is a

rooted hierarchical set of discourse segments or la-bels, with each label π associated with some content

φ (written π:φ). The contents φ are expressed in a

language L of SDRS-formulae, and the hierarchical

structure occurs because R(π,π�) ∈ L where R is

a rhetorical relation—in other words, the content of

a segment can feature rhetorical connections among

sub-segments. For simplicity, we have omitted from

Table 1 the contents of the clauses (1a) to (1c), corre-

sponding to labels π1 to π3, and adopted a convention

that the root label of agent a’s SDRS for turn j is πja.

We may also refer to the content of label π as Kπ,

and to the SDRS that agent a commits to in turn j as

T a(j).The logical form of dialogue (e.g., Table 1) is

called a Dialogue SDRS (DSDRS). The agents’ SDRSs

can share labels, and each label is always associated

with the same content. Thus an agent can commit to

the the content expressed by prior speech acts, even

if they were performed by another agent. For exam-

ple, Explanation(π1,π2) is a part of Karen’s SDRS,

making her committed to her and Mark having a fight

because she went out with Keith.

Formally, Lascarides and Asher (2009) define a

dynamic semantics |=d for DSDRSs in terms of that

for SDRSs (Asher and Lascarides, 2003), where |=d

captures shared commitments. With agreement be-

ing shared public commitment, the logical form of (1)

makes the following agreed upon: Mark and Karen

were having a fight because she went out with Keith

and not Mark.

Lascarides and Asher (2009) propose a number of

default axioms for constructing these logical forms:

they predict the semantic scope of implicit and ex-

plicit endorsements and challenges, and provide the

basis for adding Explanation(π1,π2) to Karen’s SDRS

in (1). In general, the principles are designed to max-

imise one’s ongoing commitments from prior turns,

subject to them being consistent with default infer-

ences about the illocutionary contribution one intends

to make in the current turn.

This principle therefore predicts that Karen makes

different commitments if (1a) is replaced with (1a�)

(and the word after is removed from (1b)):

(1) a�. Karen is a bitch.

In this case, as before, Karen is committed to

Explanation(b, c) and therefore is committed to the

content of (1b). However, this time, the inference

that (1c) explains (1b) supports a further default in-

ference about the speech act that Karen performed in

uttering (1c): namely, the Explanation segment pro-

vides Counterevidence to the content of (1a�). This

captures the intuition that Karen uses (1c) to jus-tify her choices as conveyed in (1b), making those

choices reasonable rather than vindictive and thereby

undermining (1b) as an explanation of (1a�)—the con-

tent that Mark committed to. Thus, since default in-

ferences about the illocutionary contribution of (1c)

makes Karen committed, via Counterevidence, to the

denial of (1a�), and this is inconsistent with a commit-

ment to Explanation(a�, b), the logic for constructing

logical form does not add Mark’s commitments from

the first turn to Karen’s commitments for the second

turn, even though Karen is committed to a part of

what Mark committed to: namely, (1b).

3 SDRT’s Glue Logic

Asher and Lascarides (2003) argue that constructing

logical form should be decidable, so as to provide

a competence model of language users who largely

agree on what was said if not its cognitive effects

(Lewis, 1969). This glue logic must involve non-

monotonic reasoning (see, for instance, the above dis-

cussion of (1)), and hence consistency tests: agents

have only partial information about the context, in-

cluding the speech acts that they intended to perform

or commit to. So to remain decidable, the glue logic

must be separate from but related to the logic |=d for

interpreting logical form. That is, computing what

was said doesn’t require evaluating whether what was

Turn Mark’s SDRS Karen’s SDRS

1 π1M : Explanation(π1,π2) ∅2 π1M : Explanation(π1,π2) π2K : Explanation(π1,π2) ∧ Explanation(π2,π3)

Table 1: A representation of dialogue (1).

said is true. SDRT achieves this separation throughunderspecified semantics (e.g., Egg et al. (2001)).

An underspecified logical form (ULF) is a partialdescription of the form of the intended logical form,here a DSDRS. SDRT’s glue logic builds ULFs, and soits language Lulf describes DSDRSs: each n-ary con-structor inL corresponds inLulf to an n+1-ary predi-cate symbol that takes labels as arguments. Labels de-note scopal positions in the DSDRS being described;so the first n arguments to the predicate P in Lulf —where P corresponds to the n-place constructor P inL—denotes the scopal positions of the arguments toP in the described DSDRS(s), and the (n + 1)th labelis the scopal position of P itself. This label is writ-ten to the left of a colon with its predicate and otherarguments to the right. So l:dog(l�) ∧ l�:d in Lulf

describes dog(d) in L. Lulf also includes variableslike ?, which indicate that the value of a constructorin the described DSDRS(s) is unknown. For instance,the compositional semantics of a pronoun introducesinto the ULF the formula l0:=(l1, l2)∧ l1:x∧ l2 :?—anequality between x and some unknown variable. Thelanguage Lulf also features Boolean connectives like∧, and a weak conditional > that’s used to formalisedefaults (A > B means “If A then normally B”).

The glue logic derives a logical form (or, in fact,a ULF) via schemata that predict pragmatically pre-ferred values for underspecified semantic elements.They in particular predict rhetorical connections, andhave the form given in (2a):

(λ:?(α, β) ∧ Info(α, β, λ)) > λ:R(α, β, λ) (2a)(λ:?(α, β) ∧ α:int) > λ:IQAP(α, β) (2b)

In words, if segment β is rhetorically connected to αas part of a segment λ but the relation is unknown, andmoreover Info(α, β, λ) holds of the content labelledby λ, α and β, then normally the rhetorical relation isR. The conjunct Info(α, β, λ) is proxy for particularULF formulae and the axioms are justified on the basisof linguistic knowledge, world knowledge, or knowl-edge of cognitive states. Rule (2b) is an example fromAsher and Lascarides (2003) where int stands for in-terrogative mood; so (2b) states that a response to aquestion is normally an indirect answer (IQAP standsfor Indirect Question Answer Pair).

Definition 1 gives the model theory for the gluelanguage Lulf : intuitively, M, s |=g φ means thatφ is a (perhaps partial) description of the DSDRS s.As we’ve said, not all |=d-consequences from the dy-namic semantics of DSDRSs are transferred into theglue logic: those arising from substitution of equali-ties, ∧- and ∨-elimination and ∃-introduction are val-idated by �g but ∃-elimination is not and so �g losesthe logical equivalence between the SDRS-formulae¬∃x¬φ and ∀xφ. This model theory is static; in Sec-tion 4 we make it dynamic.

Definition 1 Static Glue Model TheoryA model M = �S, ∗, V � for Lulf consists of:

• A a set of states S where each s ∈ S is a uniqueDSDRS,1

• A function ∗ from a state and a set of states to aset of states (for interpreting >), and

• A function V for interpreting Lulf ’s non-logicalconstants (so V is constrained by the partialtransfer of |=d-entailments from L describedabove).

Then the truth definitions are:

• M, s |=g φ iff s ∈ V (φ) for atomic φM, s |=g φ ∧ ψ iff M, s |=g φ and M, s |=g ψM, s |=g ¬φ iff M, s �|=g φM, s |=g φ > ψ iff ∗M(s, [[φ]]M) ⊆ [[ψ]]M,

where [[φ]]M = {s� : M, s� |=g φ}

The glue logic also has monotonic and nonmono-tonic relations �g and |∼g (Asher and Morreau, 1991,Asher, 1995), with |∼g yielding default inferences, viaaxioms like (2b), about discourse interpretation, in-cluding particular resolutions of anaphora. �g abidesby axioms of classical logic plus axioms and rules on>-formulae such as those in (3) (corresponding con-

1In fact, it is a finite constructor tree (Egg et al., 2001), whichis a function from a tree domain (i.e., a subset of N∗ which isclosed under prefix and sibling) to constructors in L. Each finiteconstructor tree thus corresponds to a unique DSDRS.

straints on ∗ are omitted here):

�g A → B

�g A > B(3a)

�g A → B

�g A > B(3b)

�g B → C

A > B �g A > C(3c)

�g A → B

A > C,B > ¬C �g B > ¬A(3d)

|∼g-consequences are computed via �g by converting>-formulae into →-ones proviso the result being �g-consistent. |∼g validates many intuitively compellinginferences such as those below, and the logic is sound,complete and decidable.

Defeasible Modus Ponens: φ,φ > ψ |∼ ψ

Penguin Principle: If φ �g ψ then φ,φ > χ, ψ >¬χ|∼gχ

Nixon Diamond: If φ ��g ψ and ψ ��g φ thenφ,ψ,φ > χ,ψ > ¬χ|∼/gχ (and |∼/g¬χ)

Weak Deduction: if (a) Γ,φ|∼gψ, (b) Γ|∼/gψ and (c)Γ|∼/g¬(φ > ψ), then (d) Γ|∼g(φ > ψ)

The Penguin Principle is valid because (3d) is.Definition 2 makes the updated ULF include all the

|∼g-consequences of the old and the new information(see Simple Update). So update always adds con-straints to what the dialogue means. If there is morethan one choice of labels that the new content attachesto, then update is conservative and generalises over allthe possibilities (see Discourse Update).

Definition 2 Discourse Update for DSDRSsSimple Update of a context with new content β,

given a particular attachment site α.Let T (d, m,λ) ∈ Lulf mean that the label λ is

a part of the SDRS T d(m) in the DSDRS being de-scribed. So the ULF-formula λ:?(α, β) ∧ T (d, m,λ)specifies that the new information β attaches to theDSDRS as a part of the SDRS T d(m). Let σ be aset of (fully-specified) DSDRSs, and let Th(σ) be theset of all ULFs that partially describe the DSDRSs inσ. Let ψ be either (a) a ULF Kβ , or (b) a formulaλ:?(α, β) ∧ T (d, m,λ), where Th(σ) �g Kβ . Then:

σ + ψ = {τ : if Th(σ),ψ|∼gφ then τ �g φ},provided this is not ∅;

σ + ψ = σ otherwise

Discourse Update. Suppose that A is the set ofavailable attachment points in the old information σ.

update SDRT(σ,Kβ) is the union of DSDRSs that resultsfrom a sequence of +-operations for each member ofthe power set P(A) together with a stipulation thatthe last element of the updated DSDRS is β.

The power set P(A) represents all possible choicesfor what labels in σ the new label β is attached to, soupdate SDRT is neutral about which member of P(A) isthe ‘right’ choice.

Discourse update typically doesn’t yield a specificenough ULF to identify a unique logical form or DS-DRS. But intuitively, some DSDRSs that satisfy the|∼g-consequences are ‘preferred’ because they aremore coherent. SDRT makes degree of coherence in-fluence interpretation by ranking the DSDRSs in theupdate into a partial order. This partial order ad-heres to some very conservative assumptions aboutwhat contributes to coherence, as stated in Defini-tion 3 from Asher and Lascarides (2003).

Definition 3 Maximise Discourse Coherence(MDC) Discourse is interpreted so as to maximisediscourse coherence, where the (partial) ranking �among interpretations adheres to the following:

1. All else being equal, if DSDRS φ has morerhetorical connections between two labels thanDSDRS ψ, then φ � ψ.

2. All else being equal, φ � ψ if φ features moresemantic values that support |∼g-inferences forparticular rhetorical relations.

3. Some rhetorical relations are inherently scalar.For example, the quality of a Narration is de-pendent on the specificity of its common topic.All else being equal, φ � ψ if φ features higherquality rhetorical relations.

4. All else being equal, φ � ψ if φ hasfewer labels but no semantic anomalies: e.g.,π0:Contrast(π1,π2) ∧ Condition(π2,π3) isanomalous because the first speech act ‘as-serts’ Kπ2 and the second doesn’t, butπ0:Contrast(π1,π), π:Condition(π2,π3) isn’tanomalous.

4 Dynamic Commitments in SDRT

Definition 2 uses the entailment relation |∼g but itis external to it; so is MDC. It is impossible to rea-son about dynamic updates within a static glue logicand so we need to make it dynamic so as to sup-port strategic decisions about what to say. We dothis with a public announcement logic (PAL) (Baltag

et al., 1999). A PAL features the action of announc-ing a formula, which changes the model by restrict-ing the states in the output model to those in whichthe announced formula is true. SDRT’s glue logic canthus be recast in terms of the effects of announcinga formula: the states of the model are still DSDRSs(see Definition 1) with announcements eliminatingDSDRSs from the input model that fail to satisfy theannouncement.

As Definition 2 suggests, we need to specify theeffects of three sorts of announcements:

1. Kβ—the ULF of an utterance or segment.

2. λ:?(α, β) ∧ T (d, j, λ)—a choice of where to at-tach a new segment.

3. last = β—β is the last entered element.

If all consequences of one’s announcements weremonotonic, then simple PAL would do. But Section 3makes plain that nonmonotonic consequences of an-nouncements determine the DSDRSs, since discourseinterpretation is generally a product of commonsensereasoning.

Extensions to PAL that support nonmonotonic rea-soning exist. For instance, van Benthem (2007)and Baltag and Smets (2006) propose dynamic PALsfor modelling belief revision; they incorporate into astandard PAL conditional doxastic models, with log-ics equivalent to AGM belief revision theory (Al-chourron et al. , 1985). Like their PALs, ours isextended by introducing a weak conditional connec-tive. However, our logic differs from theirs in thatour extension to PAL, being based on the connectivefrom Commonsense Entailment (Asher and Morreau,1991, Asher, 1995), validates the monotonic axiom(3d) and hence also validates the Penguin Principle.The Penguin Principle incorporates an important andintuitively compelling principle of nonmonotonic in-ference that we have shown extensively elsewhere isvital for accurately predicting the logical form of co-herent discourse (Lascarides and Asher , 1993, Asherand Lascarides, 2003). So it is essential that our dy-namic PAL version of the glue logic continue to sup-port this type of inference.

Our strategy, then, is to introduce another sortof announcement—not simple announcement but an-nouncement ceteris paribus or ACP—and we will de-fine ACPs in terms of the conditional >, so that ACPssupport similar inference patterns to those supportedin the static version of the Glue Logic.

We will convert the static model theory from Def-inition 1 into a dynamic one for interpreting ACPs.

This involves (a) extending the language Lulf to ex-press announcements; and (b) defining how modelsare transformed by such announcements in interpre-tation. As is standard in PAL, we add a modality [!φ]to Lulf , to express the announcement that φ. Theformula [!φ]ψ means that ψ follows from announc-ing φ. The above three values of φ are all >-free(although Kβ may contain a predicate symbol cor-responding to the distinct constructor > in L). Sowe make announcements >-free. We extend standardPAL by introducing a new modality [!φ]cp for ACPs,where [!φ]cpψ means that ψ normally follows fromannouncing φ.

Definition 4 assigns this extended language Lulf

a dynamic model theory, with announcements trans-forming models. Observe that [!φ]cpψ is defined interms of the conditional connective >.Definition 4 Dynamic Glue Model Theory

Let M = �S, ∗, V � be a model as in Definition 1.We defineMφ in the standard way andMcp(φ) usingthe nonmonotonic closure of φ given the backgroundtruths of glue logic GL that characterise SDRT’s con-straints on attachment of new constituents in a givenSDRS and on what relations can be inferred betweentwo given points of attachment (e.g., see axiom (2b)).We assume that this background theory holds in allmodels and, crucially, is finite. For such finite theo-ries, there exists a prime implicate or strongest finiteformula that is a nonmonotonic consequence of thetheory and from which all other nonmonotonic con-sequences follow (Asher, 1995). The prime implicateof the background theory together with φ, which weshall write as Iφ, characterises the nonmonotonic clo-sure of φ. In other words:

∀ψ such that φ|∼ψ, Iφ → ψ

Because |∼ is supra-classical (see (3a)), we have asan axiom Iφ → φ. Furthermore, Iφ incorporates theconsequences of axiom (3d) and it abides by the Pen-guin Principle: in other words, it entails the nonmono-tonic consequences of more specific information thatfollows from Iφ when it conflicts with the nonmono-tonic consequences of less specific information in Iφ.

We now define:Mφ = �Sφ, ∗M|Sφ, V �, where

Sφ = SM ∩ [[φ]]M

Mcp(φ) = �Scp(φ), ∗M|Scp(φ), V �, whereScp(φ) = SM ∩ [[Iφ]]M

Iφ → ψ, ∀ψ such that φ |∼ ψ

The dynamic interpretations of [!φ]ψ and [!φ]cpψ are:

M, s |= [!φ]ψ iff M, s |= φ →Mφ, s |= ψ

M, s |= [!φ]cpψ iff M, s |= Iφ →Mcp(φ), s |= ψ

In words, model Mφ is formed from M by eliminat-ing all states that don’t satisfy the monotonic conse-quences of announcing φ; and Mcp(φ) is formed byeliminating all states that don’t satisfy the nonmono-tonic consequences of announcing φ. Note that be-cause > is supra-classical (see (3a)), ACPs, like ‘sim-ple’ announcements, presuppose that the announce-ment is true; i.e. Scp(φ) ⊆ [[φ]]. This means that theULF of an utterance is always a ULF for the entire di-alogue; it does not mean that the the utterance is trueor even that the speaker is committed to it.

Glue logic axioms like (2b) make the conse-quences of ACPs express information about rhetor-ical connections or specific values for other under-specified elements introduced by linguistic syntax.The axioms from Lascarides and Asher (2009) (butomitted here) also ensure that ACPs predict whichcommitments from prior turns are current commit-ments. For instance, for dialogue (1), the axiomsensure that M, s |= [!Kπ3 ]cp([!(π2K :?(π2,π3) ∧T (K, 2,π2K))]cpπ2K :Explanation(π1,π2)), whereM is the model constructed by updating with utter-ances π1 and π2 in that order and s ∈M.

It is now simple to define discourse update withinthe logic. We imagine that the set of DSDRSs σ issimply the set of states of a model Mσ:

Definition 5 Dynamic Simple Update:σ + φ |= ψ iff Mσ |= [!φ]cpψ

To define full DSDRS update, we take Boolean combi-nations of ACP updates so as to match the DiscourseUpdate process from Definition 2 (see its second para-graph).

Definition 6 Dynamic Discourse Update:LetMσ be ‘old’ information and the ULF Kβ be new

information. Let Σ1, . . . Σn be all the jointly compos-sible attachment sites of β, chosen from the set A ofall possible attachment sites for each DSDRS in σ. Letki be an enumeration of the compossible attachmentsites in Σi, 1 ≤ i ≤ n. And let ki be the sequence ofassumptions about attachment provided by the enu-meration ki of sites in Σi:

ki = λi1:?(αi

1,β) ∧ T (d, j, λi1) ∧ . . .

∧λiki

:?(αiki

,β) ∧ T (d, j, λiki

)Then

Update(Mσ,Kβ) |= ψ iff ∀s ∈ SMσ ,Mσ, s |= [!(Kβ ∧ last = β)](

�ni=1[!ki]cpψ)

We now also axiomtise MDC from Definition 3within the glue logic. We will take MDC as impos-ing a coherence order on what is announced—φ � ψmeans in words that ceteris paribus announcing φ

(in this particular context) is less coherent than ce-teris paribus announcing ψ. We assume that the φand ψ that get ordered are essentially states that ver-ify formulae in the background description of the dis-course: in other words, if the (partial) description ofthe logical form of the discourse context is Th(σ) andKβ is the ULF for the new information, then φ andψ resolve some underspecified semantic elements inTh(σ) ∧ Kβ . Without loss of generality, we can as-sume that φ and ψ are conjunctive formulae, where atleast one of the conjuncts is an assumption λ:?(α, β)about attachment of the new information. This is be-cause if φ and ψ differ only in how some other as-pect of underspecified content is resolved—such asthe antecedent to a pronoun, for instance—then φ andψ can include the same conjunct λ:?(α, β) and differonly in the conjunct that fixes the antecedent. Sincediscourse must be coherent, we know that β must at-tach to something, and hence there is no loss of gen-erality in including that attachment in all ACPs. Wealso assume a partial ordering ≤ on rhetorical rela-tions: R ≤ R� means that R is a less coherent relationthan R� (see clause 3. from Definition 3). For exam-ple, Background ≤ Explanation would make MDCprefer interpreting new information as an Explanationrather than as a Background, all else being equal.

The principles that govern degree of coherence arethen stipulated in Definition 7. Clause 1. says that anACP φ that yields a less coherent rhetorical connec-tion compared with the ACP ψ is normally less coher-ent. Clause 2. says that an ACP φ that resolves fewerunderspecified elements than ACP ψ is normally lesscoherent. Finally, clause 3. says that an ACP φ that re-sults in a logical form with more segments than ACPψ is normally less coherent. Stipulating clauses 2. and3. calls for an extension to our object language Lulf ,where all variables—including higher order ones—are typed as variables and distinguished from typescorresponding to constant symbols. We also need ex-istential quantification over labels and variables (seeclauses 3. and 2. respectively).

Definition 7 Coding up MDC

1. (Th(σ) ∧Kβ) > φ � ψ ifthere’s a permutation f on ΠTh(σ) ∪ΠKβ stR� ≤ R ∧ ∀π1π2([!ψ]cpR(π1,π2) →

[!ψ]cpR�(f(π1), f(π2)))

2. (Th(σ) ∧Kβ) > φ � ψ if[!φ]cp∃≥n?1, . . .?n → [!ψ]cp∃≥n?1, . . .?n

3. (Th(σ) ∧Kβ) > φ � ψ if[!ψ]∃≥nπ1, . . . πn → [!φ]∃≥nπ1, . . . πn

This is obviously only an approximation of the prin-

ciples described in Definition 3. We have not, for in-

stance, encoded the principle that interpretations with

more rhetorical connections are more coherent than

those without (see clause 1. from Definition 3). But

given that the number of rhetorical relation symbols

in L is finite, it would be very straightforward to ex-

press this coherence factor via the ordering relation≤on predicate symbols in Lulf .

The axioms in Definition 7 can be used to in-

fluence interpretation. Our existing update func-

tion Update(M,φβ) abstracts away entirely from

how interpretation is influenced by degree of co-

herence. But we add to it a new update function

Best-update(M,φβ), whose definition is exactly like

that of Update, save that the consequences of the an-

nouncement [!Kβ ∧ last = β] are restricted to those

conjuncts about attachment that are maximal on the

partial ordering � given by Definition 7. Thus a

speaker can anticipate what the most coherent inter-

pretation of his announcement will be, as well as the

range of possible coherent interpretations, as given by

Update.

The logic will support an inference that φ � ψ only

if one or all the axioms in Definition 7 are verified—

so just like Definition 3 of MDC, ψ must be at least as

coherent as φ in all three respects, and more coherent

in at least one of them. If, for example, φ is a logi-

cal form with more segments than ψ, but ψ features

lower quality rhetorical connections, then the default

axioms whose antecedents are satisfied will conflict,

with one having consequent φ � ψ and the the other

having conflicting consequent ψ � φ. This results in

a Nixon Diamond and no inferences about the relative

coherence of the announcements φ and ψ.

We have extended the language to include existen-

tial quantification over a finite number of labels and

variables, and permutations over a finite number of

labels. Because the quantification and permutations

are over finite domains we can do without quantifi-

cation. So |∼ remains decidable even if the premises

and conclusions are Σ1-formulae. But we must now

extend the models to include a fixed set of labels that

remains constant as we move from M to Mφ.

The computational complexity of PAL is demon-

strated by proving reduction axioms and rules (Bal-

biani et al., 2007). The reduction axioms for the [!φ]operator are quite standard—axiom IV is equivalent

to one given in van Benthem (2007), for instance,

though we offer a proof for this reduction axiom here.

I [!φ]p ↔ (φ → p)

II [!φ](ψ ∧ χ) ↔ ([!φ]ψ ∧ [!φ]χ)

III [!φ]¬ψ ↔ (φ → ¬[!φ]ψ)

IV [!φ](ψ > χ) ↔ (φ → ((φ ∧ [!φ]ψ) > [!φ]χ))

As regards axiom IV, note that [[φ ∧ [!φ]ψ]]M =

[[ψ]]Mφ

, since s ∈ [[φ ∧ [!φ]ψ]]M

iff s ∈[[φ]]

M ∩ [[[!φ]ψ]]M

iff s ∈ [[ψ]]Mφ

. To prove

axiom IV, we observe ∗Mφ([[φ ∧ [!φ]ψ]]M , s) ⊆

∗M ([[φ ∧ [!φ]ψ]]M , s) by definition. Since ∗

is reflexive in the model theory of GL and

common sense entailment (i.e., ∗(p, s) ⊆ p),

∀s� ∈ ∗M ([[φ ∧ [!φ]ψ]]M , s), s� ∈ [[φ]], and this means

∗Mφ([[φ ∧ [!φ]ψ]]M , s) = ∗M ([[φ ∧ [!φ]ψ]]

M , s).This suffices to prove axiom IV, since

Mφ, s |= ψ > χ iff ∗Mφ([[ψ]]Mφ

, s) ⊆ [[χ]]Mφ

iff ∗M ([[φ ∧ [!φ]ψ]]M , s) ⊆ [[χ]]

Mφiff

∗M ([[φ ∧ [!φ]ψ]]M , s) ⊆ [[φ ∧ [!φ]χ]]

Miff

∗M ([[φ ∧ [!φ]ψ]]M , s) ⊆ [[[!φ]χ]]

M(the last step

follows again because of the reflexivity of ∗ in GL).

✷.

The more interesting question concerns ACPs. Sim-

ply using the definition of ACPs, we have the follow-

ing additional reduction axioms:

V Γ � [!φ]cpψΓ,φ|∼ψ

VI Γ,φ|∼ψΓ � [!φ]cpψ

Rules V and VI follow directly from Definition 4. A

strengthened reduction rule like (4) using defeasibleinference with ACPs is not valid, however, because

like many nonmonotonic logics ours suffers from the

Drowning Problem (Benferhat et al., 1993)—defaults

from φ ‘drown out’ those from Γ when they are mixed

together.

(4) Γ|∼[!φ]cpψΓ,φ|∼ψ

The problem comes from nested conditionals, such as

those in (5).

(5) a. Γ : {A,A > D, A > ((B ∧ E) > C)}b. φ : B ∧ E ∧ ((A ∧ E) > ¬D)c. ψ : C

In (5), Γ|∼(B ∧ E) > C and therefore Γ|∼[!φ]cpψ.

But Γ,φ|∼/ψ since Γ,φ|∼/(B ∧ E) > C—the defaults

from φ ‘drown out’ those from Γ when they are mixed

together.

Using the prime implicate Iφ, we can get proper

reduction axioms for [!φ]cpψ. This is because the

prime implicate encapsulates the nonmonotonic rea-

soning inherent in |∼. The connection between Mand Mcp(φ)

is this: [[ψ]]Mcp(φ)

= [[Iφ ∧ [!φ]cpψ]]M

.

VII [!φ]cpp ↔ (Iφ → p)

IX [!φ]cp¬ψ ↔ (Iφ → ¬[!φ]cpψ)

X [!φ]cp(ψ ∧ χ) ↔ ([!φ]cpψ ∧ [!φ]cpχ)

XI [!φ]cp(ψ > χ) ↔(Iφ → ((Iφ ∧ [!φ]cpψ) > [!φ]cpχ))

Given the restrictions that hold of GL’s backgroundtheory, prime implicates exist and can be decidablycomputed (Asher, 1995). Furthermore, the base logicof GL is decidable (Asher and Lascarides, 2003).These reduction axioms thus ensure that our exten-sion of PAL is decidable as well.Fact 1 Dynamic GL is decidable

As to the proofs of (VII-XI), they pattern closelywith the proofs for (I-IV). We sketch here proofs for(VII) and (XI). To prove the base case, assume forthe left to right direction that M, s |= [!φ]cpp. Ifs �∈ [[Iφ]]M , then we are done. So assume s ∈ [[Iφ]]M .So s ∈ Scp(φ), and by the satisfaction definitionM cp(φ), s |= p. The right to left direction followsstraightforwardly from the definitions.

To prove axiom XI, start by observ-ing that ∗Mφ([[(Iφ ∧ [!φ]cpψ]]M , s) ⊆∗M ([[Iφ ∧ [!φ]cpψ]]M , s) by definition. Since ∗ is re-flexive in the model theory of GL and common senseentailment, ∀s� ∈ ∗M ([[Iφ ∧ [!φ]cpψ]]M , s), s� ∈ [[φ]],and this means ∗Mcp(φ)([[Iφ ∧ [!φ]cpψ]]M , s) =∗M ([[Iφ ∧ [!φ]cpψ]]M , s). This suffices toprove (XI), since M cp(φ), s |= ψ > χ

iff ∗Mcp(φ)([[ψ]]Mcp(φ)

, s) ⊆ [[χ]]Mcp(φ)

iff∗M ([[Iφ ∧ [!φ]cpψ]]M , s) ⊆ [[χ]]M

cp(φ)iff

∗M ([[Iφ ∧ [!φ]cpψ]]M , s) ⊆ [[Iφ ∧ [!φ]cpχ]]M iff∗M ([[Iφ ∧ [!φ]cpψ]]M , s) ⊆ [[[!φ]cpχ]]M (the last stepfollows again because of the reflexivity of ∗ in GL).

The reduction axioms VII to XI go beyond thosegiven in Baltag and Smets (2006) and van Benthem(2007) since they don’t include in their logics the dis-tinct ceteris paribus type of announcement. None ofthe reduction axioms VII to XI are particular to the>-axiom of (3d) or to the Penguin Principle, however.These latter properties of our logic are reflected in thenature of the prime implicates and by the particularinferences our PAL logic will license. We note that itis due to the particular characteristics of SDRT’s gluelogic that such prime implicates exist.

5 Conclusion

In this paper we have made SDRT’s glue logic forcomputing logical form dynamic. This allows a di-alogue agent to reason about what the update of the

DSDRS will be after his contribution, including the ef-fects of his candidate rhetorical moves. This is a pre-requisite for planning one’s next move, but so is rea-soning about attitudes like preferences. The next stepis to examine SDRT’s other shallow logic, the logic ofcognitive modeling, so as to optimise the trade offsbetween expected interpretations and speaker prefer-ences.

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Exclusiveness Implicatures in Linguistic Answers

Kata Balogh

University of Amsterdam

In this paper I discuss the the specific issue of the exhaustive interpretation of answers, concentrating on themain phenomenon of exclusiveness of free focus constructions. It is widely agreed that narrow focus constructionsin sentential answers, as well as short answers, given to a wh-question are interpreted exhaustively. Consider, e.g.,the question Who came to the concert?. Both the short answer Amy and Ben. and the long answer [AMY and BEN]Fcame to the concert. with narrow focus on the subject mean that Amy and Ben came to the concert, and in addition,both cases are interpreted as implying that besides them nobody else came. Hence, both the short answer and thenarrow focus in the long answer provide an exhaustive listing of the individuals of whom the question predicateholds. I investigate here the interpretation of linguisitic answers and provide an analysis of exhaustification in theframework of Inquisitive Semantics, developed by Groenendijk (2008, 2009). I present my proposal of exhaustiveinterpretation derived as a pragmatic inference in question-answer relations. My analysis is mainly based on thegeneral principles of Gricean pragmatics, applied to and restated according to the logical language of InquisitiveSemantics.

The main aim behind Inquisitive Semantics is to create a logical system that models the flow of coherentdialogue. The principal goal is to provide a model of information exchange as a cooperative process of raisingand resolving issues. The main source of inquisitiveness in the system is disjunction (Groenendijk 2008). Thesemantics of the system is defined as an update semantics: the interpretation of a sentence is defined as a contextchange potential, a function from (information) states to states. A state determines a subset of the set of indices(i, j, k, ...) where each index is a function from atomic sentences to truth values. The set of indices models theinformation contained in the information state. Two indices in a state can be connected or disconnected. Whentwo indices are connected, they are considered to be related by indifference, so we are not interested in the actualdifference between the two. In a state we can single out maximal sets of indices where all of them are connected.These maximal sets are called the possibilities (ρ1 , ρ2 , ...). Possibilities correspond to propositions and are formallydefined as sets of indices. A special characteristic of the logical language of InqS is, that possibilities can overlp,where the overlapping parts are special as containing indices that belong to two (or more) possibilities. In case astate has more than two possibilities, then it raises an issue.

1 Exhaustivity and scalar implicatures

The pragmatic analyses of exhaustive interpretation claim that it is closely related to the phenomenon of scalarimplicatures, hence the main aim of such pragmatic approaches is to provide a uniform mechanism, that calculatesboth the exhaustive listing and the scalar implicature. One of the core issues in this respect is disjunction. Repre-sentatives of the globalist view such as Schulz and van Rooij (2006) and Spector (2007) follow the reasoning ofthe Gricean analysis in the calculation of exhaustive interpretations. Schulz and van Rooij take exhaustivity as thebasis and claim that scalar implicatures can be derived as a subclass. Spector takes a different position and claimsthat on the basis of Gricean reasoning both exhaustivity and scalar implicatures can be derived.

My proposal takes the position of the above analyses in the sense that the mechanism for the computation ofinferences operates after the semantic content is computed. In my analysis I introduce the issues that have recentlybeen raised around the exhaustive interpretation of answers in relation to the phenomenon of scalar implicaturein the ongoing debate between the global and local views. In my analysis, exhaustive interpretation is due to the

so-called secondary uptake of the utterance and is carried out technically by the pragmatic operation of alternativeexclusion, which is an alteration of the original idea of Groenendijk (2008). My definition captures formally the

essence of the Quantity maxim, in excluding all strictly stronger possibilities from the actual context. By my def-

inition I obtain the intended interpretation for exhaustive answers and the scalar implicature of disjunctions by a

uniform mechanism. My analysis is in the narrow sense a globalist one, since I first calculate the semantic contri-

bution of the utterance and the implicature calculation follows as a separate step. Nevertheless, the mechanisms in

Inquisitive Semantics are developed in such a way that semantics and pragmatics has not a sharp division.

There are several interesting problems that occur in the discussion in connection with the matters of exhaustivity

and scalar implicatures. In my proposal I emphasize problems that have a direct connection with focusing. I will

discuss in detail the phenomenon of exhaustivity, as well as focus on disjunction and its interpretational effects.

There are two special issues in relation to the implicature calculation of disjunctions. In case we have an answer

where the disjunction of constituents is focused, we get not only the exhaustive interpretation, but also the scalar

implicature. Moreover, next to the scalar implicature, also an ignorance (or clausal) implicature is drawn.

(1) a. Who came to the concert yesterday?

b. [AMY or BEN]F came. [A ∨B]❀ exhaustivity: and nobody else came [¬C]❀ scalar implicature: and not both came [¬(A ∧B)]❀ ignorance implicature: speaker does not know if A or B [✸A ∧✸B]

In example (1) the answer is interpreted exhaustively as besides Amy and Ben nobody else came. Furthermore,

the answer leads to the scalar implicature that it is not the case that both Amy and Ben came, as well as to the

ignorance implicature, that the speaker does not know that Amy came and she does not know that Ben came, hence

it is possible that Amy came and it is possible that Ben came. On the basis of these phenomena, I claim that an

analysis is desirable that provides a mechanism that captures all three inferences: exhaustivity, scalar implicature

and ignorance implicature.

1.1 Choosing the alternatives

The first problem that occurs is how to choose the alternative set, hence what count as alternatives of the given

utterance. The core problem here, as pointed out by Sauerland (2004), Spector (2007) and Fox (2007), is that in

case of a disjunction A ∨B, if A and B are both members of the alternative set, this leads to a problem. If we take

both A and B as alternatives of A ∨B, then exhaustivity applied to A ∨B will derive that (A ∨B) ∧ ¬A ∧ ¬B is

the case, hence it excludes both alternatives, A and B, which leads to a contradiction. There are several proposals

in the recent literature to solve this problem. Sauerland (2004) modifies the alternative set by a technical trick: he

introduces two special connectives L, R and replaces A and B with A L B and A R B respectively; A L B is

semantically equivalent to A and A R B is semantically equivalent to B, but by using the connectives they remain

distinct objects.

To avoid the same problem concerning the alternative set, Fox (2007) – similarly to Gazdar (1979) – introduces

the notion of innocently excludable alternatives, which takes care that A and B are not excluded from the alternative

set. Fox (2007) assumes a covert exhaustivity operator that is responsible not only for the exhaustive listing but for

the scalar implicatures as well. This covert exhaustivity operator takes the utterance p and the alternative set A, and

provides the interpretation, that it is asserted that p is true and every member of the alternative set that is entailed

by p is false:

(2) [[exh]](A<s,t>)(pst)(w) ⇔ p(w) ∧ ∀q ∈ NW (p, A) : ¬q(w)

The alternative set A is determined by the placement of focus. There is another set introduced in the definition,

NW (p, A): the set of no-weaker alternatives of p from the alternative set A. This set contains the propositions in

A that are not entailed by p, hence the real alternatives of p. According to the definition, it takes the alternative

set and the proposition p and gives the worlds where p is true and all non-weaker alternative propositions are

false. However, as Fox points out, this definition still faces a problem, coming basically from the formulationof the alternative set. If we assume that the above definition of the exhaustivity operator is right, the answer‘AMY or BEN came’ after the wh-question ‘Who came?’ would give the wrong result. The alternative set of thisanswer is the set of propositions of the form x came. According to the definition of exhaustivity, the propositionp =‘Amy or Ben came’ should entail the propositions ‘Amy did not come’ and ‘Ben did not come’, which isclearly not the case. Let us see this latter example in detail. The final goal is to get the inference that eitherAmy or Ben came, but it is not the case that they both came. In our example the proposition p is the disjunction‘Amy or Ben came’ translated as C(a) ∨ C(b). Take a domain of two individuals D = {amy, ben}. Then, thealternative set is derived from the answer as the set of propositions of the form x came, where x is replaced by‘amy or ben’, ‘amy’, ‘ben’, ‘amy and ben’ based on the Horn set of the scalar item ‘or’. Thus our alternative setis A = {(C(a) ∨ C(b)), C(a), C(b), (C(a) ∧ C(b))}. The set of non-weaker alternatives is the set of alternativesin A that are not entailed by p. In our case NW (p, A) = {C(a), C(b), C(a) ∧ C(b)}, the elements from A thatare not entailed by C(a) ∨ C(b). Then, according to the definition, applying the exhaustivity operator we get thefollowing, which is clearly a wrong result:

(3) [[exh]](A)(p)(w) = (C(a) ∨ C(b))(w) ∧ ¬C(a)(w) ∧ ¬C(b)(w)

One possible solution to this problem is introducing and adding to the definition the notion of minimal worldsMIN(w).1 This step solves the actual problem, however, Fox (2007) still disregards it. He claims that free-choiceinterpretations should be derived by the same computational system as scalar implicatures and this latter modifi-cation contradicts free-choice. He suggests another solution and introduces the notion of innocently excludablealternatives. Given the alternative set A, the alternative q is innocently excludable if there is no other alternative q�

in A not entailed by p such that if p ∧ ¬q holds than q� holds as well.

(4) Definition I-E(p, A): q is innocently excludable given A if ¬∃q� ∈ NW (p, A)[p ∧ ¬q ⇒ q�]

On the basis of the definition of innocently excludable alternatives Fox proposes a different modification of theoriginal definition ‘exh’ that is claimed to handle correctly both scalar implicatures and exhaustive interpretation.

(5) [[exh]](A<s,t>)(pst)(w) ⇔ p(w) ∧ ∀q ∈ NW (p, A)[q is innocently excludable given A → ¬q(w)]

Applying this definition to our example we get the interpretation (C(a) ∨ C(b))(w) ∧ ¬(C(a) ∧ C(b))(w),which says that either Amy came or Ben came but not both of them. In our example above the only innocentlyexcludable element from the set NW (p, A) is the alternative C(a) ∧ C(b). C(a) and C(b) are not innocentlyexcludable, because for both of them there is another alternative q� in NW (p, A) of which it holds that if p ∧ ¬qthen q�. C(a) ∧ C(b) is innocently excludable, since none of C(a) and C(b) is a logical consequence of (C(a) ∨C(b)) ∧ ¬(C(a) ∧ C(b)).

(C(a) ∨ C(b)) ∧ ¬C(a) ⇒ C(b)(C(a) ∨ C(b)) ∧ ¬C(b) ⇒ C(a)(C(a) ∨ C(b)) ∧ ¬(C(a) ∧ C(b)) �⇒ C(a)(C(a) ∨ C(b)) ∧ ¬(C(a) ∧ C(b)) �⇒ C(b)

The other solution by interpretation in minimal models is most prominently represented by the approach ofSchulz and van Rooij (2006).2 Schulz and van Rooij propose a uniform analysis based on interpretation in minimalmodels, that are selected by a certain ordering on the set of all models (possible worlds).3 They take the definition

1See, for example, Schulz and van Rooij (2006) and Spector (2007).2This paper is closely related to their earlier paper: van Rooij and Schulz (2004).3The uniformity of their approach is the use of minimal models, however, they provide three independent interpretation functions with

three independent notions of ordering: < P , < Prel and � P,A that all minimize the set of models in different ways: based on the

interpretation of the predicate P in different worlds (< P ), or based on the notion of relevance (< Prel ) or on the notion of knowledge over

P in a given world (� P,A).

of exhaustive interpretation by Groenendijk and Stokhof (1984) as a starting point and intend to provide a modifi-cation that overcomes its shortcomings. They first define the standard operation of exhaustive interpretation by theoperator exhstd

W that makes use of an order on the set of models (worlds), providing an interpretation in minimalmodels as the model-theoretic version of predicate circumscription from artifical intelligence (McCarthy 1980).The ordering on the worlds in W is defined as v < Pw, relative to a question predicate P , that says that v is moreminimal than w relative to P if they are exactly the same except for the interpretation of P , and [P ](v) is a propersubset of [P ](w). The definition of the operation exhstd

W provides the set of P -minimal models of the answerA: exhstd

W (A, P ) = {w ∈ A | ¬∃v ∈ [A]W : v < Pw}. The operator takes an answer A to a questionwith question-predicate P and provides the set of P−minimal worlds from A. This modified definition is almostthe same as the exhaustivity operator of Groenendijk and Stokhof, but a crucial difference is that the definition ofexhstd

W is sensitive to certain restrictions of the context (e.g. meaning postulates), since W is not necessarily theset of all models (worlds), but a set provided by the context.

Yet another mechanism to determine the alternative set is proposed by Alonso-Ovalle (2008) in the frameworkof alternative semantics. Alonso-Ovalle investigates the puzzle by McCawley (1981) and Simons (1998), that pointsout another problem disjunction leads to regarding how we determine the alternative set. The puzzle concernsdisjunctions with more than two disjuncts, as in example 6.

(6) Sandy is reading Moby Dick or Huckleberry Finn or Treasure Island.

According to the standard mechanisms based on binary disjunctions, for M ∨ H ∨ T above the alternativesare derived as the set: {(M ∧ H) ∨ T, (M ∨ H) ∧ T, (M ∧ H) ∧ T} from which we cannot infer that Sandyis not reading more than one book. Based on this particular problem, Alonso-Ovalle (2008) proposes a differentmechanism to determine the alternatives of disjunctions. Generating the alternatives for disjunctions Alonso-Ovalletakes the intersection of their meanings. The alternative set of a disjunction S is [[S]]ALT∩ = {p | ∃B[B ∈℘([[S]]) ∧B �= ∅ ∧ p = ∩B]}. With this definition Alonso-Ovalle generates the alternative set for M ∨H ∨ T in(6) as {M, H, T, (M ∧H), (H ∧ T ), (M ∧ T ), (M ∧H ∧ T )} that does not face the problem of the McCawley-Simons puzzle. However, Alonso-Ovalle uses the mechanism of Innocent Exclusion of Fox (2007) to prevent theexclusion of the atomic disjuncts that are elements of the alternative set.

My proposal is in certain respects on the same track as the analysis of Alonso-Ovalle. In my approach thealternatives of a disjunction are determined by the underlying wh-question — explicitly or implicitly by the theme— that provides or determines several possibilities. This way I can provide a non-stipulative solution to the problemof choosing the right alternatives. A crucial property of my analysis is that in our system possibilities can overlap,so indices (valuations) can belong to two or more possibilities simultaneously. On the basis of these possibilitiesand their overlaps I define possible propositions that correspond to the alternatives (or competitors) in the standardapproaches in terms of alternative semantics. In my proposal, overlapping parts of the possibilities, hence theirintersections, count as possible propositions/alternatives. Note, however, that in my approach intersections arealready in the picture of the state being introduced by the question, thus involving them in the analysis is not an adhoc step.

1.2 The epistemic step

Another problem I want to address is the problem of the epistemic step introduced by Sauerland (2004). Accordingto Gricean reasoning, if the speaker utters ‘Peter ate some pancakes.’ then the hearer takes this as the optimallyinformative utterance the speaker could have chosen, hence she concludes that the speaker does not know that Peterate all of the pancakes. Unfortunately, this inference — called the ‘primary implicature’ — is not enough, sincewe want to infer that the speaker knows that Peter did not eat all the pancakes. Deriving this latter, secondaryimplicature needs an extra step called the epistemic step. This phenomenon is crucial in approaches that provide anepistemic analysis for scalar implicatures. In case a disjunction A ∨ B is uttered, according to Gricean reasoningwe can only infer the weak implicature that the speaker does not know that A∧B, while we want to infer the strongimplicature that the speaker knows that not A ∧B.

Example 1.1 (Epistemic step)following Grice: A ∨B ⇒ ¬KA;¬KB ⇒ ¬K(A ∧B)epistemic step: from ¬K(A ∧B) to K¬(A ∧B)

Sauerland (2004) claims that the secondary implicature can be derived from the primary implicature and its

logical consequences. First, the set of the primary implicatures is extended with their logical consequences and

then the secondary implicature can be inferred in case it does not contradict the elements of this derived set.

Accordingly, given the primary implicature ¬Kϕ from which we cannot derive ¬K¬ϕ, we can infer the secondary

implicature as K¬ϕ.

Schulz and van Rooij (2006) derives the strong implicature by adding a competence order on top of their

pragmatic interpretation function griceC (A, p). First of all they provide the function griceC (A, p), an extension

of their basic definition of exhaustivity. In a question-answer relation it captures the assumption that a cooperative

speaker, given the knowledge she has, does not withhold information that helps resolving the question. Hence, the

new definition of the pragmatic interpretation, griceC (A, P ), makes reference to the knowledge of the speaker:

griceC (A, P ) = {w ∈ [KA]C | ∀w� ∈ [KA]C : w � P ,Aw�}. The definition griceC (A, P ) is mainly based upon

the ordering � P ,A that captures the concept of how much the speaker knows about the predicate in a given model.

The interpretation function griceC (A, P ) works as follows: from all models where the speaker knows the answer

A ([KA]C ) it selects the ones where she knows the least about (� P ,A) the question predicate P , that is, she knows

of the least number of individuals that they have property P . The definition captures that if the speaker had known

more about the question predicate, she would have said so, as the Gricean maxim states it. To capture the secondary

implicature, Schulz and van Rooij introduce an additional ordering, that compares the speakers competence.4

With

this new ordering relation (� P ,A) they propose a strengthened version of the pragmatic interpretation function:

epsC (A, P ) = {w ∈ griceC (A, P ) | ∀w� ∈ griceC (A, P ) : w �❁ P ,Aw�}. Based on the new competence

ordering, the function epsC (A, P ) further selects from the set of models given by griceC (A, P ) the ones where

the speaker is maximally competent. Notice that this selection comes on top of the interpretation by griceC (A, P ),so it is very hard to compute what it does as long as it is not so clear intuitively how � P ,A and � P ,A relate.

In Fox’s (2007) analysis the problem of the epistemic step does not arise, since the mechanism works without

belief operators. My proposal is also a non-epistemic approach, where the strong inference of the scalar implicature

of disjunctions is directly derived. Hence, my proposal also does not raise the issue of the epistemic step.

As for the ignorance implicature, the approach by Fox (2007) needs an extra rule. Following Gazdar (1979), Fox

introduces the extra rule to capture that the speaker does not know which one of the disjuncts is true (¬KA∧¬KBor ✸A ∧ ✸B). My analysis supports the ignorance implicature without any special mechanisms, so it can be

incorporated it in a natural way.

2 Groenendijk (2008) on exclusiveness

My analysis id based on the inquisitive version of the exclusiveness implicature that is introduced in (Groenendijk

2008). First I show Groenendijk’s reasoning on the pragmatic inference carried by disjunction, and I argue that

his reasoning needs some reconsideration. However, I believe that the core idea behind his definition is correct.

Groenendijk (2008) claims that the expression p ∨ q comes with the pragmatic inference of exclusion of (p ∧ q).He derives this implicature using the notion of Comparative Compliance based on the dialogue principle Be ascompliant as you can!, which is regarded as the inquisitive version of the Gricean Maxim of Quantity. The full,

formal definition of Comparative Compliance can be found in (Groenendijk 2008) and in (Balogh 2009), I only

introduce here the essence of it. An expression φ is compliant to the state s in case it holds that (1) every possibility

in s∗[φ] (where s∗is the same as s without the issue in s) are possibilities or unions of possibilities in s, and (2)

the state s∗[φ] is equally or less inquisitive than s. In Groenendijk’s version the reasoning process of the responder,

hence the calculation of the implicature ¬(p∧q), goes as follows. If we take the expression (p∨q)5, then the uptake

4The idea of maximizing the competence at the interpretation of answers is already introduced in van Rooij and Schulz (2004).

5As dialogue initial starting at the initial state ���, ι�.

of its semantic content, the disjunction (p ∨ q), leads to the state s = ω[p ∨ q] with two overlapping possibilitiesρ1 and ρ2 , where ρ1 corresponds to the proposition p and ρ2 to q6. Relative to this state s there are three possiblecompliant responses: !(p ∨ q), p and q. Updating s∗ with any of these possible responses will result in a state thatis not less informative and not more inquisitive than s — hence these responses are compliant to the given state s.

Example 2.1 (Compliant responses)s = ω[p ∨ q] :

•pq •p¬q

•¬pq ◦¬p¬q

s∗[!(p ∨ q)] :

•pq •p¬q

•¬pq◦¬p¬q��

�s∗[p] :

•pq •p¬q

◦¬pq ◦¬p¬q

s∗[q] :

•pq ◦p¬q

•¬pq ◦¬p¬q

From the three expressions p and q are both more compliant to s than !(p∨q), since p and q eliminate more indices,thus they are more informative. The propositions p and q are equally compliant here. The expression (p ∧ q) isnot a compliant response, since s∗[(p ∧ q)] is not related to s, because relatedness requires that each possibility ins∗[(p ∧ q)] is the union of a subset of the set of possibilities in s and that is not the case. Because of this fact theinitiator who uttered (p∨ q) has made a suggestion, that (p∧ q) does not count as a response to his utterance. Thenthe responder utters p; thereby he accepts the suggestion of the initiator, that (p ∧ q) is excluded. The dialoguemanagement in the system of InqS is built up in such a way that all uptakes (primary, secondary) are first provisionalupdates, that get either accepted or canceled by the response. If the actual update conflicts with the responder’s owninformation state, she has to cancel some (or all) of the provisional updates, and this cancellation must be explicitlysignaled. In case she does not cancel, she accepts the updates including the pragmatic inferences (if any).

Example 2.2 (“Suggestion”)ω[p ∨ q]:• •• ◦

suggestion:◦ •• ◦

p uttered:• •◦ ◦

results in:◦ •◦ ◦

According to Groenendijk (2008) the responder can utter (p ∧ q) in case she explicitly signals that she is aware ofthe fact that her response is not compliant to the immediate context. She signals it by uttering, for example, ‘Well,actually, p and q’. In case the response goes against the suggestion, and this is explicitly signaled, the suggestionwill be discarded and the response gets interpreted in the original state without the exclusion.

Example 2.3 (Against the suggestion)ω[p ∨ q]:• •• ◦

(p ∧ q) :

• ◦◦ ◦

According to the semantics of InqS, the corresponding first-order formula ∃x.P (x) relative to a domain of twoindividuals leads to a corresponding picture as we see for p∨ q. Hence, in the same way as above we can concludethat ∃x.P (x) (over D = {a, b}) has the implicature that P (a) ∧ P (b) does not hold. Just as the disjunction p ∨ qin examples 2.1 and 2.2 ∃x.P (x) has two overlapping possibilities, corresponding to P (a), P (b). The compliantresponses are: !∃x.P (x), P (a) and P (b). Here as well the conjunction P (a) ∧ P (b) is not a compliant response,thus it is excluded. But of course if we take a bigger domain that says more. Consider a domain of three individualsD = {a, b, c}. Then we intend to conclude that the expression ∃x.P (x) means that either only a is P or only b isP or only c is P , hence we exclude P (a)∧P (b), P (a)∧P (c), P (b)∧P (c), as well as P (a)∧P (b)∧P (c). Thenthe picture of ω[∃x.P (x)] has three overlapping possibilities, corresponding to the propositions P (a), P (b) andP (c), that are the most compliant responses as well. Other, less compliant, responses are !∃x.P (x), (P (a)∨P (b)),!(P (a)∨P (b)) etc. Again, updating ω[∃x.P (x)]∗ with P (a)∧P (b), P (a)∧P (c), P (b)∧P (c) or P (a)∧P (b)∧P (c)would lead to a non-compliant state, hence these propositions are out.

6For an easier read • stands for indices present in the state and ◦ stands for indices that are eliminated.

Example 2.4 (Suggestion)ω[∃x.P (x)] :

a

ab

c ac abc bc b

∅

suggestion:

a

ab

c ac abc bc b

∅

In this context, with the suggestion, if P (a) is uttered we mean that only a is P . The suggestion or implicature

excluded the indices that belong to the overlapping area of the possibilities in the state ω[∃x.P (x)]. In fact, these

overlapping parts single out further propositions, that are not compliant responses in this state, but can be derived

from the possibilities that refer to (the most) compliant propositions. Again, the three possibilities are the following:

ρ1 that corresponds to the proposition P (a), ρ2 that corresponds to P (b) and ρ3 that corresponds to P (c). The

area where ρ1 and ρ2 are overlapping refers to the proposition (P (a) ∧ P (b)), and similarly the overlap of ρ1

and ρ3 to (P (a) ∧ P (c)), the overlap of ρ2 and ρ3 to (P (b) ∧ P (c)) and where all three possibilities overlap to

(P (b) ∧ P (b) ∧ P (c)).However, although the intuition behind the approach (that the overlapping parts of the possibilities are prag-

matically excluded) is correct, the above reasoning has its shortcomings and needs some reconsideration. First

of all, the reasoning is counter-intuitive, since it suggests that existential expressions such as ‘someone came’ —

∃x.C(x) — are themselves interpreted as pragmatically meaning that only one individual came. That is, the expres-

sion carries the suggestion by itself instead of looking at the relation between it and the response to it. Secondly,

Groenendijk (2008) argues that in case of p ∨ q the response p ∧ q is not compliant — and by this not relevant —,

thus it should not count. However, the original Gricean reasoning says that relevant propositions that are strictly

stronger are pragmatically excluded. I agree with Groenendijk that overlapping possibilities are special, and claim

furthermore that all overlapping parts correspond to a proposition that counts as a legitimate response. In the next

section I will define these overlapping parts as the set of possible propositions in a state, and redefine the operation

of ‘alternative exclusion’ as the new operation of exhaustification ([EXH ]) that is responsible for certain quantity

implicatures.

3 Exhaustification

To formulate the new definition of exhaustification I first define the notion of possible propositions in a state.

In our semantics the context or common ground is defined as a stack of states, where all states consist of one

or more possibilities. Possibilities are defined as maximal sets of indices, such that all indices are connected to

each other. In some cases the possibilities can overlap, where the overlapping part is special, belonging to two

or more possibilities at the same time. By the definition of possible propositions we can refer to these special

overlapping parts that correspond to propositions different from the ones formed by the possibilities. The set of

possible propositions of the state σ is the set of possibilities in σ closed under intersection. By means of possible

propositions in a state we can redefine the rule of exhaustification in a way that provides exhaustivity and the

implicature of scalar expressions such as A or B as well.

Definition 3.1 (Possible propositions)Let P s be the set of possibilities in s. Πs is the set of possible propositions in s that is defined as follows:Πs is the biggest set of possibilities such that

if ρ1 , . . . , ρn ∈ P s and ρ1 ∩ . . . ∩ ρn �= ∅ then ρ1 ∩ . . . ∩ ρn ∈ Πs

Definition 3.2 (Exhaustification)Let P t be the set of possibilities in t and Πs be the set of possible propositions in s.��σ, s�, t�[EXH ] = ���σ, s�, t�, u�, where u = {�i, j�|∃ρ ∈ P t : i, j ∈ ρ ∧ ¬∃α ∈ Πs : α ⊂ ρ ∧ i ∈ α or j ∈ α}

Definition 3.1 defines the set of possible propositions in a state, such that it contains all the possibilities in

the state and all the possible propositions (set of indices) determined by their overlapping parts. For example,

in case we have three overlapping possibilities ρ1 , ρ2 and ρ3 , the set of possible propositions is the following:

{ρ1 , ρ2 , ρ3 , ρ1 ∩ ρ2 , ρ1 ∩ ρ3 , ρ2 ∩ ρ3 , ρ1 ∩ ρ2 ∩ ρ3}.

The operation of exhaustification [EXH ] on the current common ground stack ��σ, s�, t� adds a state u on the

top, where u contains all indices i from t that do not belong to any possible proposition α in s that is strictly

stronger than any possibility ρ in t. Both possible propositions and possibilities refer to propositions, and they are

both defined as sets of indices. As such, entailment is defined on sets in terms of the subset relation: α entails

ρ if it is a subset of it, α ⊆ ρ. A possible proposition is strictly stronger than a possibility if it asymmetrically

entails it, hence if α is a proper subset of ρ, α ⊂ ρ. The definition of [EXH ] captures formally the essence of the

gricean quantity maxim, since it says that every strictly stronger possible proposition is ruled out. Exhaustification

looks at the relation between the last two states in the common ground stack, where the state s is considered as the

actual context for the utterance φ, and t is the state as the result of updating s with the semantic content of φ. The

state s contains the possible propositions that can be singled out from the overlapping possibilities of s and each

of them corresponds to a strictly stronger proposition as the ones determined by the possibilities. According to the

definition of [EXH ] , after the uptake of the semantic content of φ, the indices that belong to a possible proposition

in s which is stronger than φ will be excluded.

Singling out and making use of the possible propositions in the operation of exhaustification is motivated by the

notion of true answer at an index. The core of the idea is to assume that the responder who is expected to answer

the question of the initiator is indeed an expert, hence the answer she gives is the true answer. I assume that in order

to achieve a successful conversation the speaker poses a question to somebody she believes to know the answer.

Of course, it may be that the speaker is wrong and the responder is not an expert, which she may correctly signal

with a response such as ‘I don’t know’. In case the responder gives an answer, the speaker believes that she plays

according to the rules of a coherent dialogue and her answer is indeed the true answer. Then the speaker concludes

that in case the responder gave an answer — taken to be the true answer — the other possible answers are out. This

reasoning is captured formally by my new definition of exhaustification in definition 3.2. To make the motivation

complete, first of all I define the notion of true answer at an index relative to the state determined by the underlying

question.

Definition 3.3 (True answer at an index)Let s be the state determined by the question (s = ω[?ϕ]) and ρ a possibility in s (ρ ∈ P s ). The true answer atindex i (Ansi ) is defined as: Ansi = ∩{ρ ∈ P s | i ∈ ρ}

The true answer at index i after ?ϕ is the intersection of the possibilities in s (=ω[?ϕ) that contain the index

i. Let me give some examples for illustration. Consider the question ?(p ∨ q) that provides three possibilities

ρ1 , ρ2 , ρ3 .

Example 3.1 (True answer at indices)s:

•k •l

•i •j

Ansi :

•k •l

•i •j

Ansj :

•k •l

•i •j

Ansk :

•k •l

•i •j

Ansl :

•k •l

•i •j

The possible true answers in the state s are p ∧ q at index i, p at index j, q at index k and ¬p ∧ ¬q at index l.Index i is in ρ1 and ρ2 that have an overlap (intersection) as illustrated above. Indices j, k and l are included in the

possibilities ρ1 , ρ2 and ρ3 respectively. In case the information state of the responder consists of the pair �k, k�(or the index k), then she provides the true answer by uttering q. The speaker will then infer that q ∧ ¬p is the case

on basis of assuming the responder is an expert. In case responder knows that the actual index is i, then she has to

utter (p∧q) to provide the true answer. Motivated by the notion of true answers, I define the set of propositions that

count as an answer to the actual question. The set of possible true answers in a state is exactly the same as the set

of possible propositions in the state.7 By the notion of (true) answer and possible answers I can offer a solution tothe problem of plural answers that InqS ran into, since it filtered the responses by the logical notion of compliance.As I noted before the answer p ∧ q is not compliant after ?(p ∨ q), hence it should be ruled out. Similarly, on thebasis of the notion of compliance ‘AMY and BEN came.’ should be ruled out as an answer after the question ‘Whocame?’ that we translate as ?∃x.C(x); nevertheless this answer is perfectly in order linguistically.

Let us look at some examples in detail. In this section I will illustrate that by my definition of exhaustification([EXH ]) we get the right results for the exhaustive interpretation of (7a-b), as well as for the scalar implicature ofthe disjunction in (7c).

(7) Who came yesterday?a. AMY (came). ❀ impl nobody elseb. AMY and BEN (came). ❀ impl nobody elsec. AMY or BEN (came). ❀ impl not both; nobody else

All three answers are interpreted in the context of the same wh-question translated as ?∃x.C(x), that providesthe common ground ����, ι�, ω[?∃x.C(x)]�; the state on the top contains three possibilities ρ1 , . . . , ρ4 :

Example 3.2 (Picture of ω[?∃x.C(x)])•k

•j

•m •l •i •n •o

•p

ρ1 ❀ P (a)

ρ2 ❀ P (b)

ρ3 ❀ P (c)

ρ4 ❀ ¬∃x.P (x)

i(C) = {a, b, c}j(C) = {a, b}k(C) = {a}

l(C) = {a, c}m(C) = {c}n(C) = {b, c}

o(C) = {b}p(C) = {}

According to my definition of possible propositions, these four overlapping possibilities provided by the se-mantic content determine eight possible propositions α1 , . . . ,α8 :

Example 3.3 (Possible propositions in ω[?∃x.C(x)])α1 = ρ1 ❀ C(a)α2 = ρ2 ❀ C(b)α3 = ρ3 ❀ C(c)α4 = ρ1 ∩ ρ2 ❀ C(a) ∧ C(b)α5 = ρ1 ∩ ρ3 ❀ C(a) ∧ C(c)α6 = ρ2 ∩ ρ3 ❀ C(b) ∧ C(c)α7 = ρ1 ∩ ρ2 ∩ ρ3 ❀ C(a) ∧ C(b) ∧ C(c)α8 = ρ4 ❀ ¬∃x.C(x)

After the primary uptake of the semantic content of the answers in (1) is completed the operation of exhaus-tification ([EXH ]) applies. This operation belongs to the secondary uptake of the utterance, which instantiates thepragmatic inferences. The operation of [EXH ] applies in all cases blindly after the primary uptake, however theactual effect of it depends on the relation of the top states in the common ground stack. Exhaustification has aneffect only in certain cases where there is a special relation between the top states. It does not do anything, forexample, when there are no overlapping possibilites in s in the stack ��σ, s�, t�. For the examples in (1) we have toapply exhaustification relative to the following common ground stacks respectively:

7The definition of true answer is also interesting, for example, in cases of conditional questions such as p→?q.

Example 3.4 (Contexts for exclusion)(a) . . .�, ω[?∃x.C(x)]�, ω[C(a)]�[EXH ](b) . . .�, ω[?∃x.C(x)]�, ω[C(a) ∧ C(b)]�[EXH ](c) . . .�, ω[?∃x.C(x)]�, ω[C(a) ∨ C(b)]�[EXH ]

The operation of exhaustification will provide us with the right results for all three cases in example 3.4. In(3.4a) we get as a result a single possibility containing a single index where only Amy came. In this examplethe primary uptake, hence the semantic content of the answer, provides a state with a single possibility ρ thatcorresponds to the proposition C(a). This proposition consists of four indices (relative to our domain), of whichafter [EXH ] the indices survive that do not belong to any of the possible propositions α1 ...8 in ω[?∃x.C(x)] thatare strictly stronger than the proposition corresponding to the possibility ρ. Hence, we have to consider hereα4 , . . . ,α8 , since these propositions are all strictly stronger than ρ (they all entail ρ but not the other way around).After excluding the indices from ω[C(a)] that belong to any of α4 ...8 in ω[?∃x.C(x)] we end up with the singlepossibility containing the single index where only Amy came. By the definition of [EXH ] we excluded the threeother indices where besides Amy, Ben and/or Claire came as well.

Example 3.5 (Exhaustification: singular term). . .�, ω[?∃x.C(x)]�, ω[C(a)]�[EXH ] =. . .�, ω[?∃x.C(x)]�, ω[C(a)]�, ω[∀x.C(x) ↔ x = a]�

ω[?∃x.C(x)] : ω[C(a)] : ω[∀x.C(x) ↔ x = a] :

•k

•j

•m •l •i •n •o

•p

•k

•j

◦m •l •i ◦n ◦o

◦p

•k

◦j

◦m ◦l ◦i ◦n ◦o

◦p

The operation in (3.4b) goes similarly: in this case the indices from ω[C(a) ∧ C(b)] which do not belong toα5 , . . . ,α8 survive. We do not have to consider α1 , . . . ,α4 , since they are possibilities in one of the two states.The result is again a single index, namely the one where only Amy and Ben came, hence we excluded the indexfrom ω[C(a)∧C(b)] where besides Amy and Ben, Claire came as well. The resulting common ground consists ofthe following states:

Example 3.6 (Exhaustification: conjunction). . . , ω[C(a) ∧ C(b)]�, ω[∀x.C(x) ↔ (x = a ∨ x = b)]�

ω[?∃x.C(x)] : ω[C(a) ∧ C(b)] : ω[∀x.C(x) ↔ (x = a ∨ x = b)] :

•k

•j

•m •l •i •n •o

•p

◦k

•j

◦m ◦l •i ◦n ◦o

◦p

◦k

•j

◦m ◦l ◦i ◦n ◦o

◦p

The operation in (3.4c) on the answer by disjunction will give us the intended result: we end up with two possibil-ities, each of them consisting of a single index. One possibility contains the index where only Amy came, and theother possibility contains the index where only Ben came. Hence, we get the interpretation that either only Amycame or only Ben came, with the right inference that not both of them came. Here, the indices from ω[C(a)∨C(b)]that are not in α4 , . . . α8 survive. The resulting common ground is as follows:

Example 3.7 (Exhaustification: disjunction). . .�, ω[C(a) ∨ C(b)]�, s�

ω[?∃x.C(x)] : ω[C(a) ∨ C(b)] : s :

•k

•j

•m •l •i •n •o

•p

•k

•j

◦m •l •i •n •o

◦p

•k

◦j

◦m ◦l ◦i ◦n •o

◦p

The above example illustrates an important result of my analysis. With the naive notion of disjunction applying the

gricean reasoning results in the empty set, because C(a) is stronger than C(a) ∨ C(b) as well as C(b) is stronger

than C(a)∨C(b). By our – independently motivated – richer notion of disjunction this problem does not occur and

by applying ‘Grice’ we get the right result of the exhaustive interpretation without any extra rules like, for example,

the notion of ‘innocently excludable’ (Fox 2007) proposition.

4 Summary

In this paper I provided an analysis of the exhaustive interpretation of answers in the framework of Inquisitive

Semantics, based on the original idea of alternative exclusion of Groenendijk (2008). In the framework of In-

quisitive Semantics, exhaustive interpretation of answers is due to a pragmatic implicature and technically carried

out by the operation of alternative exclusion. Keeping the orginal intuititon I provided a new definition of this

operation, [EXH ], that fixes some shortcomings that are faced by the proposal of Groenendijk (2008) and better

fits the Gricean reasoning (Grice 1975). My operation [EXH ] gives the right results not only for the exhaustive

interpretation but also for the scalar implicature of disjunctions in a uniform way. The alternative exclusion refers

to the possible propositions that are singled out from the possibilities in the context. Each overlapping part of

two or more possibilities determines a proposition. The definition of alternative exclusion captures formally the

essence of the Quantity maxim, since it excludes all strictly stronger answers from the actual context. My anal-

ysis is among the neo-Gricean global analyses, that calculate implicatures at the sentential level. In the dialogue

management system, the exclusive implicature is calculated after the uptake of the semantic content. Different

from the classical Gricean reasoning is that in my analysis the operation of [EXH ] is applied in all cases right after

the uptake of the semantic content of the utterance. However, [EXH ] does not have an effect in all cases, only

in case of special relations between the states on the top of the common ground stack. These special states are

the ones that have one or more overlapping possibilities, where the overlapping parts are considered to be special.

These areas make an important contribution determining the possible propositions that count as an answer and as

such can be seen as the alternative set on which the inference of exclusiveness is carried out. Recent analyses of

Chierchia (2004), Fox (2007) and Spector (2007) provide different solutions. All of them give an analysis of the

scalar implicatures in terms of exhaustification. In this respect my analysis is as effective as the others, however my

analysis has important advantages. First of all, regarding the problem of the definition of the alternative set, which

is investigated by Spector (as well as by Fox): in my system I do not need to stipulate what counts as the alternative

set, as it is directly determined by the underlying issue that can be — and often is — an explicit wh-question or

the theme of the utterance itself. Furthermore, I do not need to assume a special notion like innocently excludable(Fox 2007), or minimal models. According my proposal we can infer the intended interpretation for exhaustive

answers and the scalar implicature of disjunctions by a single mechanism, viz. the operation [EXH ] based on the

possible propositions given by the context.

References

Alonso-Ovalle, L.: 2008, Innocent exclusion in an alternative semantics, Natural Language Semantics 16.

Balogh, K.: 2009, Theme with Variations. A Context-based Analysis of Focus, PhD thesis, ILLC, University ofAmsterdam, Amsterdam.

Chierchia, G.: 2004, Scalar implicatures, polarity phenomena, and the syntax/pragmatics interface, in A. Belletti(ed.), Structures and Beyond, Oxford University Press, Oxford.

Fox, D.: 2007, Free choice disjunction and the theory of scalar implicatures, in U. Sauerland and P. Stateva (eds),Presupposition and Implicature in Compositional Semantics, Palgrave Macmillan, New York.

Gazdar, G.: 1979, Pragmatics, Academic Press, London.

Grice, H. P.: 1975, Logic and conversation, in P. Cole and J. Morgan (eds), Syntax and semantics, Vol. 3, AcademicPress, New York.

Groenendijk, J.: 2008, Inquisitive Semantics and Dialogue Pragmatics, Universiteit van Amsterdam, Amsterdam.

Groenendijk, J.: 2009, Inquisitive semantics: Two possibilities for disjunction, in P. Bosch, D. Gabelaia and J. Lang(eds), Logic, Language and Computation. 7th International Tbilisi Symposium on Logic, Language and Com-putation. Revised Selected Papers., Springer-Verlag, Berlin-Heidelberg.

Groenendijk, J. and Stokhof, M.: 1984, Studies on the Semantics of Questions and the Pragmatics of Answers, PhDthesis, University of Amsterdam, Amsterdam.

McCarthy, J.: 1980, Circumscription – a form of non-monotonic reasoning, Artificial Intelligence 13.

McCawley, J.: 1981, Everything that Linguists Have Always Wanted to Know about Logic but Were Ashamed toAsk, University of Chicago Press, Chicago.

Sauerland, U.: 2004, On embedded implicatures, Journal of Cognitive Science 5.

Schulz, K. and van Rooij, R.: 2006, Pragmatic meaning and non-monotonic reasoning: the case of exhaustiveinterpretation, Linguistics and Philosophy 29.

Simons, M.: 1998, Or: Issues in the Semantics and Pragmatics of Disjunction, PhD thesis, Cornell University,Ithaca, NY.

Spector, B.: 2007, Scalar implicatures: Exhaustivity and gricean reasoning, in M. A. B. Aloni and P. Dekker (eds),Questions in Dynamic Semantics, Elsevier, Amsterdam.

van Rooij, R. and Schulz, K.: 2004, Exhaustive interpretation of complex sentences, Journal of Logic, Languageand Information 13.

D. Grossi Argumentation Games as Evaluation Games

Argumentation Games as Evaluation Games:The Case of Stable Extensions ∗

Davide GrossiILLC, University of Amsterdam

d.grossi@uva.nl

1 Introduction

The paper proposes an application of modal logic to argumentation theorybuilding on results presented in [5]. Its line of argument is summarized inthe following points, which also outline the structure of the paper:

� Dung’s argumentation frameworks [2] can be studied as Kripke frames.

� Several argumentation-theoretic notions have a modal logic formula-tion. As a consequence, techniques and results can be imported frommodal logic to argumentation theory in a direct way.

� In particular—and this is the specific contribution of this paper—Dung’s argumentation theory can be given a game-theoretic proof-theory by means of evaluation games (aka model-checking games).With only one notable exception [3], all argumentation games havebeen developed and studied independently from one another givingrise to a quite heterogeneous family of games (cf. [6]). Using theevaluation game abstraction allows us to derive all kinds of games forargumentation simply by instantiating the same evaluation game withdifferent logical formulae expressing the various types of extensions.

� As an illustration of such application the paper defines adequategames (credulous and skeptic) for stable extensions. This answersan open problem in the literature of argumentation theory concern-ing the definition of games for stable extensions that are adequatewith respect to the existence of a winning strategy for a player—theproponent (cf. [1, 6]).

The paper presupposes some familiarity with both modal logic and abstractargumentation theory.

∗eg4arg.tex, Tuesday 15th June, 2010, 17:05.

1

D. Grossi Argumentation Games as Evaluation Games

cA is the char. function ofA iff cA : 2A −→ 2A s.t.cA(X) = {a | ∀b : [b � a⇒ ∃c ∈ X : c � b]}

X is conflict-free inA iff �a, b ∈ X s.t. a � bX is a complete extension ofA iff X is conflict-free and X = cA(X)

(X is a conflict-free fixpoint of cA)X is a stable extension ofA iff X is a complete extension ofA

and ∀b � X,∃a ∈ X : a � b

Table 1: Some basic notions of argumentation theory. X is a set of arguments.

2 Arguments in modal logic

An argumentation framework is a relational structure A = (A,�) whereA is a set (of arguments), and �⊆ A2 a binary (attack) relation. Table 2recapitulates some of the key notions developed in [2] which are consideredin this abstract.1

Now, if an argumentation framework can be viewed as a Kripke frame,then an argumentation framework plus a function assigning names from aset P to sets of arguments can be viewed as a Kripke model.

Definition 1 (Argumentation models). Let P be a set of propositional atoms.An argumentation modelM = (A,I) is a structure such that: A = (A,�) is anargumentation framework; I : P −→ 2A is an assignment from P to subsets of A.The set of all argumentation models is called A. A pointed argumentation model isa pair (M, a).

So, the fact that an argument a belongs to I(p) in a given model M,which in logical notation reads (A,I), a |= p, can be interpreted as statingthat “argument a has property p” , or that “p is true of a”. Generalizing thisintuition to formulae ϕ of a suitable language we can develop a logic forargumentation frameworks, which is the topic of the next section.

2.1 Logic K∀This section introduces logic K∀, an extension of the minimal modal logic Kwith the universal modality.

2.1.1 Syntax

Language L∀ is a standard modal language with two modalities: ��� and�∀� (the universal modality). It is built on the set of atoms P by the followingBNF:

L∀(P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ���ϕ | �∀�ϕ1We refer to [5] for a more comprehensive exposition.

2

D. Grossi Argumentation Games as Evaluation Games

where p ranges over P. We assume the standard definitions of the remaining

Boolean and modal operators. In the following we will often work with

formulae of L∀ in positive normal form, i.e., formulae where negation

occurs only in front of atoms.

2.1.2 Semantics

Definition 2 (Satisfaction). Let ϕ ∈ L∀. The satisfaction of ϕ by a pointedargumentation model (M, a) is inductively defined as follows:

M, a |= ���ϕ ⇐⇒ ∃b ∈ A : (a, b) ∈ �−1 andM, b |= ϕM, a |= �∀�ϕ ⇐⇒ ∃b ∈ A :M, b |= ϕ

Boolean clauses are omitted. As usual, ϕ is valid in an argumentation modelMiff it is satisfied in all pointed models ofM, i.e.,M |= ϕ; ϕ is valid in a classM ofargumentation models iff it is valid in all its models, i.e.,M |= ϕ. The truth-set ofa formula ϕ is denoted |ϕ|M. The set of L∀-formulae which are true in the class Aof all argumentation models is called (logic) K∀.

Logic K∀ is therefore endowed with modal operators of the type “there

exists an argument attacking the current one such that”, i.e., ���, and “there

exists an argument such that”, i.e., �∀�, together with their duals. Given

an argumentation model M we can thereby express statements such as

the ones adverted to above: “a is attacked by an argument in a set called

ϕ” corresponds to ���ϕ being true in the pointed model (M, a) and “a is

defended by the set ϕ” corresponds to ������ϕ being true in the pointed

model (M, a).

Logic K∀ has a sound and strongly complete axiomatization, and the

complexity of checking whether a formula of L∀ is satisfied by a pointed

modelM is known to be P-complete.

2.2 Argumentation theory in K∀: examples

Perhaps surprisingly, logic K∀ is already expressive enough to capture sev-

eral basic notions of argumentation theory. Here are two illustrative exam-

ples:

Compl(ϕ) := [∀]((ϕ→ [�]¬ϕ) ∧ (ϕ↔ [�]���ϕ)) (1)

Stable(ϕ) := [∀](ϕ↔ [�]¬ϕ) (2)

The adequacy of these definitions with respect to Table 2 is easily checked.

Example 1 (Nixon diamond or 2-cycle). Consider the following arguments:a) Nixon is a pacifist because he is a quaker; b) Nixon is not a pacifist becausehe is republican. These arguments and their respective attack relations can be

3

D. Grossi Argumentation Games as Evaluation Games

represented by the following Dung framework: A = ({a, b}, {(a, b), (b, a)}). On the

top of A consider also a labeling I : {1, 0} −→ 2{a,b}

such that I(1) = {a} and

I(0) = {b}. It is easy to check that the resulting argumentation modelM = (A,I)

(right-hand side of Figure 1) is such that:

M, a |= Compl(1) ∧ Stable(1) ∧ 1M, b |= Compl(0) ∧ Stable(0) ∧ 0

That is to say, argument a belongs to the complete and stable extension labeled 1,

and argument b belongs to the complete and stable extension labeled 0.

The following examples illustrates an argumentation-theoretic perspec-

tive on cooperative games.

Example 2 (Dominance in cooperative games as attack relation). A cooper-

ative n-person game with transferable utility is a tuple T = (N,V) where N is

a finite set of agents such that |N| = n and V : sN −→ R is a function assigning a

‘value’ to each coalition of agents. Function V is taken to be superadditive, i.e.: for

all C,D ⊆ N such that C ∩D = ∅, V(C ∪D) ≥ V(C) + V(D). An imputation is

a tuple ι = (v1, . . . , vn) such that v1 + . . . + vn ≤ V({1, . . . ,n}). The coalition of a

given imputation i is denoted c(ι) and its i-th element ιi. Finally, an imputation ιdominates an imputation ι�, in symbols ι � ι� if and only if there exists a coalition

C ⊆ c(ι�) such that for all agents i ∈ c(ι), ιi ≥ ι�i . Given a game T , we say that a set

I of imputations is a Von Neumann-Morgenstern solution of T , in short a stable

set, if and only if [9]:

i) ∀ι ∈ I,�ι� ∈ I s.t. ι� � ι;

ii) ∀ι � I,∃ι� ∈ I s.t. ι� � ι.

As [2, p. 336] observes, given a game T , each Von Neumann-Morgenstern

solution for T corresponds to a stable extension of the argumentation framework

AT = (AT ,→) for T , where A

Tis the set of all possible imputations for T . As a

direct consequence, given a labeling I : P −→ 2AT

, we have that:

“I(p) is a stable set of T ”⇐⇒ (AT ,I) |= Stable(p).

3 Evaluation games for argumentation

Using formulae such as Formulae 1-2 it is possible to check whether a given

ϕ enjoys a specific property (e.g., stability) by means of a game.

3.1 Game-theoretic semantics for K∀In evaluation games, a proponent or verifier (∃ve) tries to prove that a given

formula ϕ holds at one point a of a modelM, while an opponent or falsifier

(∀dam) tries to disprove it. Here is the formal definition.

4

D. Grossi Argumentation Games as Evaluation Games

Definition 3 (Evaluation game for K∀). Let ϕ ∈ L∀ be in positive normal form,andM = (A,�,I) be an argumentation model. The evaluation game of ϕ onMis a tuple E(ϕ,M) = (N,S, turn, move, win) where:

� N := {∃,∀}.

� S := L∀ × A, that is, the set of positions consists of the set of formula-argument pairs. Sequences of elements of S are denoted s.

� turn : S −→ N is a partial function assigning players to positions accordingto Table 2 (second column).

� move : S −→ 2S assigns to each position a set of available moves (or accessiblepositions) according to Table 2 (third column). The set Play(E(ϕ,M))denotes the set of all finite sequences s—which we call plays—consistentwith function move.

� win : Play(E(ϕ,M)) −→ N assigns a winning player to each play accordingto Table 3.

A game E(ϕ,M) is instantiated by pairing it with an initial position (ψ, a) withψ ∈ L∀ and a ∈ A, in symbols: E(ϕ,M)@(ψ, a).

So an evaluation game is a finite two-player zero-sum extensive gamewith perfect information. The important thing to notice is that positions ofthe game are pairs of a formula (in positive normal form) and an argument,and that the type of formula in the position determines which player has toplay: ∃ if the formula is a disjunction, a box, a false atom or ⊥, and ∀ in theremaining cases. Since at each stage of the game the syntactic complexityof the formula gets smaller (cf. Table 2), the game is finite. Last thing tonotice is that a play is won by an agent if and only if its opponent has runout of moves (Table 3).

Now, take a strategy for the instantiated game E(ϕ,M)@(ψ, a) to be afunction σ : {(ξ, a) | ξ ∈ SubF(ψ) and a ∈ A} −→ S telling the player which

Position Turn Available moves

(ϕ1 ∨ ϕ2, a) ∃ {(ϕ1, a), (ϕ2, a)}(ϕ1 ∧ ϕ2, a) ∀ {(ϕ1, a), (ϕ2, a)}

(���ϕ, a) ∃ {(ϕ, b) | (a, b) ∈�−1}([�]ϕ, a) ∀ {(ϕ, b) | (a, b) ∈�−1}(�∀�ϕ, a) ∃ {(ϕ, b) | b ∈ A}([∀]ϕ, a) ∀ {(ϕ, b) | b ∈ A}

Position Turn Available moves

(⊥, a) ∃ ∅(�, a) ∀ ∅

(p, a) and a � I(p) ∃ ∅(p, a) and a ∈ I(p) ∀ ∅

(¬p, a) and a ∈ I(p) ∃ ∅(¬p, a) and a � I(p) ∀ ∅

Table 2: Rules of the model-checking game for K∀.

5

D. Grossi Argumentation Games as Evaluation Games

Matches ∃ wins ∀ winss ∈ Play(E(ϕ,M)) turn(s) = ∀ and move(s) = ∅ turn(s) = ∃ and move(s) = ∅

Table 3: Winning conditions for the K∀-evaluation game.

move to make at any possible position consisting of a subformula ξ of ψ(ξ ∈ SubF(ψ)) and an argument.

Definition 4 (Winning strategies and positions). LetE(ϕ,M) be a K∀-evaluationgame, (ϕ, a) ∈ S. Strategy σ is winning for ∃ in E(ϕ,M)@(ψ, a) if and only iffor all s ∈ σ(G) s.t. s0 = (ψ, a) it is the case that win(s) = ∃. A position (ψ, a) iswinning for ∃ if and only if ∃ has a winning strategy in E(ϕ,M)@(ψ, a). The setof winning positions of E(ϕ,M) for ∃ is denoted Win∃(E(ϕ,M)).

We can now state the adequacy of E(ϕ,M) with respect to the semanticsof K∀ (Definition 5) for any ϕ andM.

Theorem 1 (Adequacy). Let ϕ ∈ L∀, and letM = (A,I) be an argumentationmodel. Then, for all a ∈ A:

(ϕ, a) ∈Win∃(E(ϕ,M))⇐⇒M, a |= ϕ.

Proof. The inductive proof is standard. For details we refer to [5]. �

An immediate consequence of Theorem 1 is the existence of adequategames for all argumentation-theoretic properties which are expressible inK∀ such as, for instance, the above properties of completeness (Formula 1)and stability (Formula 2):

Complete : E(ϕ ∧ Complete(ϕ),M)Stable : E(ϕ ∧ Stable(ϕ)),M).

It is worth noting that the architecture of the evaluation game (Definition3) remains the same while what changes is simply the formula whose truthis evaluated by the game.

Example 3 (Evaluation games for stable on the Nixon diamond). Consider thegame depicted on the right-hand side of Figure 1, the so-called Nixon diamond. Wewant to run an evaluation game for checking whether a belongs to a stable extensioncorresponding to the truth-set of 1. Such game is the game E(1∧Stable(1), (A,I))initialized at position (1 ∧ Stable(1), a). Spelling out the definition of Stable(1):E(1∧ [∀](1↔ ¬���1))@(1∧ [∀](1↔ ¬���1), a). Such a game, played accordingto the rules in Definitions 3 and 4, gives rise to the tree in Figure 1.

6

D. Grossi Argumentation Games as Evaluation Games

(1 ∧ [∀](1↔ ¬���1), a)

([∀](1 ↔ ¬���1), a)

(1↔ ¬���1, a) (1↔ ¬���1, b)

(¬1 ∨ ¬���1, a) (1 ∨ ���1, a)

(¬���1, a) (¬1, a)

(1, b)

(1, a)

∃ wins!

∀ wins!

∀

∀

∃ wins!

∃ ∃

∀

∀

∀

a

b

1

0

Labeled

Nixon Diamond

Figure 1: K∀-game for a stable extension in the 2-cycle.

4 Model-checking games as dialogue games

There is an essential difference between the game presented above, andthe sort of games studied in the literature on argumentation theory (theso-called dialogue games). In the model-checking game for K∀ you aregiven a model M = (A,I), a formula ϕ and an argument a, and ∃ve isasked to prove thatM, a |= ϕ. In dialogue games, the check appointed to∃ve—the proponent—is inherently more complex since the input consists,in that case, only of an argumentation framework A, a formula ϕ and anargument a. ∃ve is then asked to prove that there exists a labeling I suchthat (A,I), a |= ϕ. This is a satisfiability problem in a pointed frame which,in turn, is essentially a model-checking problem in monadic second-orderlogic: “A |= ∀p1, . . . , pn¬STa(ϕ)?” where p1, . . . , pn are the atoms occurringin ϕ and STa(ϕ) is the standard translation of ϕ realized in state a. This is,however, no inherent limitation for the thesis we are supporting here—i.e.,that model-checking games are a sound game-theoretic proof-theory forargumentation—and the present section supports this claim.

4.1 A Σ11 logic for argumentation

4.1.1 Syntax

We expand languageL∀(P) by allowing quantification over predicates (i.e.,propositional variables). At the same time we also restrict the quantifierpattern to have alternation depth 0. The resulting second-order languageis built by the following BNF:

L2∀(P) : ϕ ::= p | ⊥ | ¬ϕ | ϕ ∧ ϕ | ���ϕ | �∀�ϕ | ∃p.ψ(p)

where ψ(p) ∈ L∀ and p occurs free in ψ.

7

D. Grossi Argumentation Games as Evaluation Games

4.1.2 Semantics

Definition 5 (Satisfaction). Let ϕ ∈ L2∀. The satisfaction of ϕ by a pointed

argumentation model (M, a) is defined as follows (the clauses for L-formulae areomitted):

M, a |= ∃p.ϕ(p) ⇐⇒ ∃X ⊆ A :Mp:=X, a |= ϕ(p)

whereMp:=X denotes modelMwhereI(p) is set to be X. The set of allL2∀ formulae

which are satisfied by all argumentation model is called (logic) K2∀.

Intuitively, we are simply adding to K∀ the necessary expressivity to talkabout properties involving a quantification over one set of arguments.

Remark 1 (K2∀ and SOPML). Logic K2

∀ is a sub-logic of second-order propositionalmodal logic [4] extended with the universal modality—let us denote this logicSOPML

∀. More exactly, it is the fragment of SOPML∀ where second-order

quantification occurs only with alternation depth 0. In turn, SOPML∀ is a strict

fragment of monadic second-order logic (MSO). The inclusion is strict as, forinstance, SOPML

∀ does not enable counting of successors while MSO does. Itmight be finally instructive to notice that SOPML

∀ strictly includes theµ-calculus.In this case the inclusion is strict as there are SOPML

∀-formulae which are notbisimulation-invariant (cf. [8]).

4.2 Argumentation theory in K2∀: examples

We can now talk about the existence of complete and stable sets of argu-ments, as well as the property of belonging to at least one extension (theso-called credulous semantics), or to all of them (skeptical semantics):

ExistsStable := ∃p.Stable(p) (3)CredulousStable := ∃p.Stable(p) ∧ p (4)

SkepticalStable := ∀p.Stable(p) ∧ p. (5)

4.2.1 Game-theoretic semantics

All we need to do is to adapt Definition 3 to logic K2∀.

Definition 6 (K2∀-evaluation game). Let ϕ ∈ L2

∀, and M = (A,�,I) be anargumentation model. The evaluation game of ϕ on M is a tuple E(ϕ,M) =(N,S, turn, move, win) where:

� N := {∃,∀}.� S := (L2

∀ × A) ∪ {E(ϕ,M)@(ϕ, a) | ϕ ∈ L∀ and a ∈ A}, that is, the setof positions consists of the set of formula-argument pairs plus the set ofinstantiated evaluation games forL∀ formulae onM. Sequences of elementsof S are denoted s.

8

D. Grossi Argumentation Games as Evaluation Games

Position Turn Available moves

(ϕ1 ∨ ϕ2, a) ∃ {(ϕ1, a), (ϕ2, a)}(ϕ1 ∧ ϕ2, a) ∀ {(ϕ1, a), (ϕ2, a)}

(���ϕ, a) ∃ {(ϕ, b) | (a, b) ∈�−1}([�]ϕ, a) ∀ {(ϕ, b) | (a, b) ∈�−1}(�∀�ϕ, a) ∃ {(ϕ, b) | b ∈ A}([∀]ϕ, a) ∀ {(ϕ, b) | b ∈ A}

(⊥, a) ∃ ∅(�, a) ∀ ∅

(p, a) and a � I(p) ∃ ∅(p, a) and a ∈ I(p) ∀ ∅

(¬p, a) and a ∈ I(p) ∃ ∅(¬p, a) and a � I(p) ∀ ∅

(∃p.ϕ(p), a) ∃ {E(ϕ(p),Mp:=X)@(ϕ(p), a) | X ⊆ A}(∀p.ϕ(p), a) ∀ {E(ϕ(p),Mp:=X)@(ϕ(p), a) | X ⊆ A}

E(ϕ(p),Mp:=X)@(ϕ(p), a) − {(ϕ(p), a)}

Table 4: Rules of the model-checking game for K2

∀.

� turn : S −→ N is a function assigning players to positions according toTable 4 (second column).

� move : S −→ 2S assigns to each position a set of available moves (or accessiblepositions) according to Table 4 (third column). The set Play(E(ϕ,M))

denotes the set of all finite sequences s, i.e., the plays, consistent with functionmove.

� win : Play(E(ϕ,M)) −→ N assigns a winning player to each play accordingto theTable 3.

A gameE(ϕ,M) is instantiated by pairing it with an initial position s: E(ϕ,M)@s.

Note that Definition 6 is structurally identical to Definition 3. The

essential difference consists in the three rules for {∃,∀} given at the bottom

of Table 4. These rules state which player, in the presence of a 2nd order

quantifier, can choose which K∀-evaluation game to play. Once the choice

is made, the last rule in the Table determines univocally the next position

from which the K∀ evaluation game can be played. Note that such position

is univocally determined, as it belongs to a singleton (last line of Table 4).

The winning conditions are the same as for the K∀-evaluation game, as are

the notions of winning strategy and winning position (Definition 4).

Remark 2. It is worth observing that the game specified in Definition 6 could beformulated otherwise by making explicit at all positions which is the relevant model

9

D. Grossi Argumentation Games as Evaluation Games

on which the K∀-evaluation game is played. Positions would then become triplesconsisting of a formula, an argument and a model (or simply a labeling function asthe underlying argumentation framework is constant). In the current formulation,along each branch of the game tree the relevant model is kept track of by allowingpositions consisting of full instantiated K∀-evaluation games. These positions arethen reduced to standard positions by an automatic move not belonging to any ofthe two players (see Table 4).

It can easily be proven that the K2∀-evaluation game is adequate with

respect to the semantics of K2∀ (Definition 5).

Theorem 2 (Adequacy). Let ϕ ∈ L2∀, and letA = (A,I) be an argumentation

model. Then, for all a ∈ A:

(ϕ, a) ∈Win∃(E(ϕ,M))⇐⇒M, a |= ϕ.

Proof. The proof is direct by Theorem 1, Definition 6 and Definition 5. �

Again, this gives us a range of adequate games for the sort of argumentation-theoretic properties expressible in K2

∀, such as the ones concerning stableextensions in Formulae 3-5:

Existence of Stable : E(ExistsStable,M)Credulous for Stable : E(CredulousStable,M)

Skeptical for Stable : E(SkepticalStable,M).

Again, it is worth stressing that the architecture of the evaluation game(Definition 6) remains the same while what changes is simply the formulawhose truth is evaluated by the game.

Example 4 (Evaluation game for credulous stable in the Nixon Diamond).To follow up on the Nixon diamond example, a K2

∀-game for checking whetherargument a belongs to at least one stable extension would start with ∃ve choosinga valuation forA on which to play then a K1

∀ model-checking game. In the example∃ve’s winning strategy is namely to choose the game depicted in Figure 1 on themodel such that I(1) = {a}. All other ∃ve’s choices would lead to a winningposition for ∀dam. The first stages of the K2

∀-game for the Nixon diamond areshown in Figure 2.

The following example expands on Example 2 and shows how to applythe evaluation games presented here to cooperative games.

Example 5 (A game for stable sets in majority games). Consider the 3-playercooperative game with transferable utility (see Example 2) T = ({1, 2, 3}, v) with

v such that, for any C ⊆ {1, 2, 3}: v(C) =�

1 if |C| ≥ 20 otherwise . This game—known

as majority game—has an infinity of stable sets (cf. [7, Ch. 14]), and therefore

10

D. Grossi Argumentation Games as Evaluation Games

∃ wins! ∀ wins!

∀

(CredulousStable, a) ∃

(E(Stable(p) ∧ p),Mp:={b}@(Stable(p) ∧ p), a))

(E(Stable(p) ∧ p),Mp:={a}@(Stable(p) ∧ p), a))

(E(Stable(p) ∧ p),Mp:=∅@(Stable(p) ∧ p), a))

(E(Stable(p) ∧ p),Mp:={a,b}@(Stable(p) ∧ p), a))

(Stable(p) ∧ p), a)

(Stable(p) ∧ p), a)

(Stable(p) ∧ p), a)

(Stable(p) ∧ p), a)

∀

∀

∀

∀ wins!∀ wins!

Figure 2: K2∀-game for a stable extension in the 2-cycle.

its associated argumentation framework AT has an infinity of stable extensions(see Example 2). Now, suppose to run the K2

∀-evaluation game for credulousstable on the argumentation model (AT ,I), with I arbitrary, starting at theargument/imputation ι = ( 1

2 ,12 , 0). More concisely, suppose to play:

E(CredulousStable, (AT ,I))@(CredulousStable, (12,

12, 0)).

where recall that CredulousStable = ∃p.Stable(p) ∧ p (Formula 4). A winningstrategy for ∃ve in this game consists in choosing to play this evaluation game:

E(Stable(p) ∧ p, (AT ,I)p:={( 12 ,

12 ,0),(0, 12 ,

12 ),( 1

2 ,0,12 })@(Stable(p) ∧ p, (

12,

12, 0)).

As the set {( 12 ,

12 , 0), (0, 1

2 ,12 ), ( 1

2 , 0,12 } is a stable set ofT , it is also a stable extension

ofAT . So, by Theorem 1, ∃ve has a winning strategy in the above game.

5 Conclusions and future work

The paper has illustrated an application of evaluation games to argumen-tation theory inspired by the modal logic view of argumentation put forthin [5]. Most concretely, the paper has defined adequate games (Theorem2) for the existence of stable extensions, as well as for the skeptical andcredulous stable semantics for argumentation.

11

D. Grossi Argumentation Games as Evaluation Games

Future work will focus on an a comparison between dialogue gamesand evaluation games. For certain argumentation semantics (e.g. groundedextensions) there are adequate games of both kind. The interesting questionthen arises about the structural differences and similarities of these games.

Acknowledgements.

Davide Grossi is supported by the Nederlandse Organisatie voor Wetenschap-

pelijk Onderzoek (NWO) under the VENI grant 639.021.816.

References

[1] M. Caminada and Y. Wu. An argument game for stable semantics.Journal of the IGPL, 17(1), 2009.

[2] P. M. Dung. On the acceptability of arguments and its fundamental rolein nonmonotonic reasoning, logic programming and n-person games.Artificial Intelligence, 77(2):321–358, 1995.

[3] P. M. Dung and P. M. Thang. A unified framework for representationand development of dialectical proof procedures in argumentation. InProceedings of the Twenty-First International Joint Conference on Artificial

Intelligence (IJCAI-09), pages 746–751, 2009.

[4] K. Fine. Propositional quantifiers in modal logic. Theoria, 36:336–346,1936.

[5] D. Grossi. On the logic of argumentation theory. In W. van der Hoek,G. Kaminka, Y. Lesperance, and S. Sen, editors, Proceedings of the 9th

International Conference on Autonomous Agents and Multiagent Systems

(AAMAS 2010), pages 409–416, Toronto, Canada, May, 10–14 2010.

[6] S. Modgil and M. Caminada. Proof theories and algorithms for ab-stract argumentation frameworks. In Y. Rahwan and G. Simari, editors,Argumentation in AI. Springer, 2009.

[7] M. J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press,1994.

[8] B. Ten Cate. Expressivity of second order propositional modal logic.Journal of Philosophical Logic, 35:209–223, 2006.

[9] J. von Neumann and O. Morgenstern. Theory of Games and Economic

Behavior. Princeton University Press, 1944.

12

Dialogue and Interaction : the Ludics view

Alain Lecomte∗and Myriam Quatrini

†

Abstract

In this paper, we study dialogue as a game, but not in the sense in which there would exist winning strate-

gies and a priori rules. On the contrary, it is possible to play with rules, provided that some geometrical

constraints are respected (orthogonality). We owe much to Ludics, a frame developed by J-Y Girard,

while remaining close to the approach of discourse and dialogue in N. Asher’s tradition. We interpret

the ludical notion of locus in rhetoric terms, as a location in a discourse from which a particular theme

is developed. A set of loci subordinated to a same initial locus is a topic. We explain the processes of

narration and elaboration, and also the speech acts of assertion, denegation and interrogation in ludical

terms.

1 Introduction

In what follows, we shall refer to Ludics, a theory elaborated by J-Y. Girard (Girard 01, 03, 06) in the goal of

reconstructing logic starting from the notion of interaction. We think that this frame is particularly suitable

for representing dialogues. In a word, it starts from the observation that proofs in a polarized logic (as Linear

Logic may be seen) can be presented as processes which make alternate negative and positive steps. This

observation opens the field to the concept of duality between abstract processes generally called designs.

These objects have in fact two readings: one is as proofs, and the other as strategies in a game. We may

think that the proof aspect is convenient for dialogues in that it represents the argumentative content of a

statement. The strategy aspect is convenient also in that it involves goals and directions towards which a

dialogue is oriented. With regards to other game theories, in Ludics, rules are not a priori given, interaction

itself determines them. Otherwise, each step in a play records all the previous ones, thus allowing more

flexibility, in particular for backtracking during a discussion.

The semantic which can be given, via Ludics to utterances is not simply truth-conditional. We may start

from our intuitive notion of what it is for a piece of dialogue (or of discourse) to be ”well formed”, to

give rise to elementary situations of interaction, thus suggesting another (empirical) view on Semantics

[Lecomte-Quatrini10]. In this presentation we shall make a link between Ludics and the well known notions

of SDRT (Asher & Lascarides) applied to dialogue as an example of the expressivity of Ludics.

2 Interaction as a basis...

0 � 1k � ∆1k . . .

.

.

.

0 � nk � ∆nk

� 0, ∆

.

.

.

� 0 � I1, Γ . . .

.

.

.

� 0 � Ik, Γ . . .

.

.

.

� 0 � In, Γ

0 � Γ

∗Laboratoire : “Structures formelles du langage”, Paris 8 Universite/CNRS

†Laboratoire : “Institut de Mathematiques de Luminy”, Aix-Marseille Universite/CNRS

1

.

.

.

� 0 � ik � 1, . . . , 0 � ik � n, ∆ik

0 � ik � ∆ik

.

.

.

0 � ik � 1 � Γ1 . . .

.

.

.

0 � ik � n � Γn

� 0 � Ik, Γ

Figure 1: Continuation right and left

The archetypal figure of interaction is provided by two intertwined processes the successive times of which,

alternatively positive and negative, are opposed by pairs. On the left, we have a process starting from a set

of loci1 that we assume to be ”positive”, among which there is one which is focused, here denoted by ’0’,

and at the first step, this locus is made to vary across the various manners a given theme may be addressed.

Each of these issues selects a subset of loci among the remaining (not focused) ones, thus showing that,

according to the way the theme is addressed, various subthemes may be discussed later.

On the right, we have a second process, for which a locus has been already chosen, and therefore put in a

negative position (the left hand side of the so-called ”fork”, representing in fact a sequent with at most one

locus on the left hand side). This represents a receptive attitude : the locus is the one which has been selected

in the other process. The first step of this process consists in a survey of all the various ways it could be

possible to decline this locus (to address this theme). Among them, if things are going well, there is the one

taken in the first process. Such a configuration may be associated with cooperation: both processes have a

dialogue together, and we may imagine it lasts some time. Let us admit for instance that the right process be

continued, it records the positive action of the left one and it is now its turn to perform a positive action, but

the left process must have planned this action. This gives the figure 12. This situation could be illustrated

by the following dialogue :

- This year, for my holidays, I will go to the Alps with friends and by walking,- well, in the Alps, there are a lot of winter entertainmentsThe first speaker extracts from a set of loci, a locus ’0’ associated with the topic of her holidays, that she

may topicalize by asserting something on where she goes, with whom, by what means and so on. The second

speaker records the theme of holidays and is ready to accept various ways of addressing it (a set of sets of locithat we shall call a set of thematic variations). After accepting the way her interlocutor addresses it, she may

focalize on one of the aspects the first speaker introduced, for instance here on the where, thus introducing

various ways of topicalizing it. The first speaker has already in mind a whole directory of possible thematic

variations concerning the point developed by the second one.

In Ludics, such an interaction may continue in various ways. One way is the acceptance by one of the two

participants that the ”game” is over (for instance she acknowledges that she got enough information, or she

accepts the argument of the other participant), this is marked by the positive rule †. If such a case occurs and

if all the loci introduced in the dialogue have been visited by the normalization process3

(in this case, the

set of both processes is said to form a closed net), the two processes are said to be orthogonal or that they

converge. An alternative issue is provided by the case where a positive action introduces a thematic variation

which is not included into the expectations of the other participant: the interaction is said to diverge. We

shall see in a future paper that such a breakdown may be repaired in a dynamic way, by (litterally) changingthe rules the players are using (for instance by extending the span of the expected thematic variations).

1a locus is a mere address like a memory cell or like a specific position occupied by a statement in a conversational network.

2i � J , where J is a set of indexes {j1, j2, ..., jm} means {i � j1, i � j2, ..., i � jm}.3That is, at each step of the confrontation, a positive locus is put in correspondance with a negative one, and all the negative loci

are finally ”cancelled” by their positive counterparts

2

3 Dialogical relations

3.1 Topicalization

We will mainly consider designs developing from positive forks with an arbitrary finite number of loci :

� Λ. The first action in this case is necessarily positive and consists in choosing a focus. Let ξ be this focus,

then the basis may be written � ξ, Λ0. Λ0 is said to be the context in which the theme ξ is developed. Of

course, we may have always a single locus, by exploring the design in the top-down direction (instead of the

bottom-up one). That involves to introduce pre-steps to the current one. This is always possible provided

that the loci ξ,Λ0 may be rewritten as ξ = τ �0�0, τ �0�1, · · ·�τ �0�n. Going downward like this unifies

the focus and its context in a topic. This can be seen as a primitive operation of discourse, which deserves

to be named topicalization. The operation consists in:

� τ � 0 � 0., . . . τ � 0 � n = Λ(−, τ � 0, {I})

τ � 0 �(<>,+, {0})

� τ

where we consider the first step as virtual (as if the constitution of the topic came from an expectation from

the Other Speaker).

EXAMPLE

Suppose that you decide to describe your next holiday : “this year, for my holiday, I will go to the Alps,

with friends, by walking . . . ” and that your addressee asks you: “when are you on holiday ?”. If you do not

wish to abandon at once, that is, if you wish to go on interacting with him/her, you have just to enlarge the

interaction.

1. First step you decide to describe your next holiday : “this year, for my holiday, I will go to the Alps . . . ”

ξ � 1 �� ξ

(Y)our intervention

2. Instead of anchoring her intervention on ξ �1, asking you more questions on the description of your holiday, your addressee

asks you: “when are you on holiday ?”

Then, you have to manage a locus from which you may answer to him/her in the same interaction as the one you started

with ; you have not only to put a locus ξ from which you can tell the description of your holidays but also another one, ρ,

from which you can tell the date of your holidays. More precisely you have to replace the fork � ξ by the fork � ξ, ρ and

then to unify the loci in the same context: “the topics of your holiday”, simply by setting:

� ξ, ρ = � τ � 0 � 0, τ � 0 � 1.

Then, the following design is used. It may be understood as two successive virtual utterances “I can tell you something

about my holiday” (corresponding to the action (+, τ, {0}) and “I am then ready to answer any questions about the dates

and its description” (corresponding to the action (−, τ � 0, {0, 1})).

� τ � 0 � 0, τ � 0 � 1

τ.0 �� τ

Y

3. you can then answer to the question on the dates (with the action (+, τ � 0 � 1, {6}).

3

τ � 0 � 0 � 1 �� τ � 0 � 0

∅τ � 0 � 1 � 6 � τ � 0 � 0

� τ � 0 � 0, τ � 0 � 1

τ.0 �� τ

Y

τ � 0 � 0 �

∅� τ0 � 0 � 1 � i

i∈Nτ � 0 � 1 �

� τ � 0

τ �A

4. The foregoing interaction reduces to:τ � 0 � 0 � 1 �� τ � 0 � 0

Y

� τ � 0 � 0 � 1

τ � 0 � 0 �A

The situation is hence the same as the one which resulted from your first intervention “this year, for my holiday . . . ”. Youare now waiting for your addressee’s questions about the description of your holiday.

3.2 Elaboration

We may imagine a locus be chosen initially in the positive process (that is the one the first rule of whichis positive) and then each positive step consists in elaborating on that locus or on its sub-loci (a sub-locusof ξ where ξ is an address, that is a sequence of biases (integers) is simply a locus a prefix of which isξ), without the facing negative process forcing it to go to a disjoint locus4. In this case, we say that thediscursive relation between all the utterances made is elaboration. An example is provided by:- I go to the mountain.(a1)- I like skiing.(a2)- above all cross-country skiing- it’s not dangerous if you are careful- you have nevertheless to plan the avalanches- but fortunately I have a NARVAThe formal representation of that discourse is provided by the design on the left hand side of figure 2.Theright hand side provides an orthogonal process. This is a virtual process. The important point to notice hereis that the virtual process to which the Speaker is confronted when producing her discourse is as important asher own process of speaking. It is because she has in mind this process that she continues her discourse thisway. For example she adds some successive virtual questions as “Why going to the mountain ?” ; “Whichstyle of skiing ?” ; “Is it not dangerous ?” . . . That would be a different thing if the virtual process in one ofits positive actions had selected a locus in ∆ (for instance at the second step the question could have been :“To the Alsp or to other mountains ?”).

...

ξ0i0 � ξ01, ..., ξ0n,∆a2

� ξ01, ..., ξ0i, ..., ξ0n,∆

ξ0 � ∆a1

� ξ,∆

TOP : holidays

ξ01 � ∆1, ...

...

� ξ0i0, ..., ξ0im,∆i

ξ0i � ∆i ξ0n � ∆n

� ξ0,∆

ξ � ∆

4a disjoint locus should be a locus which has no common prefix with the current one, in fact when staying inside a same topic,they have at least the locus of this topic as a common prefix, but we may assume it is the only common prefix they have.

4

Figure 2: Elaboration

3.3 Narration

When a theme is given up, this translates into a new locus being selected in the so-called context. When all

the loci inside the initial context are explored, this results in a narration. Let us see an example:

- I went to the mountain (a1)- then I took the plane to Frisco (a2)- from there I visited California- and then I went back to EuropeThe formal representation is given by figure 3,

...{∅}

ξ20 � ξ3, ..., ξna2

� ξ2, ..., ξn{∅}

ξ10 � ξ2, ..., ξna1

� ξ1, ξ2, ..., ξn

∅� ξ10

ξ1 �...

∅� ξi0

ξi �...

Figure 3: Narration

where {∅} denotes a particular case of the negative rule. The dual (family of) design(s) is given on the right,

where the only positive steps are labelled by the rule ∅. Again, the other (virtual) speaker has a fundamental

role : she determines the first speaker not to develop a theme, and to select another one until the range of

themes be exhausted.

3.4 Assertion, denegation and interrogation

The previous remarks concerning the necessity of a two-faces process for a representation of discursive

relations opens the field to a deeper reflection on elementary speech acts. Discourse is above all action

and commitment : action of Myself on Yourself and reciprocally [Beyssade & Marandin 06]. Like it is said

by Walton [Walton 00], “asserting” is “willing to defend the proposition that makes up the content of the

assertion, if challenged to do so” . This results in the fact that when Myself asserts P , I must have in mind

all justifications for predictable objections. That is, “I” have a design like in figure 4

D1

� ξ0.I1 ...

Dn

� ξ0.InN

ξ0 �

� ξ

Figure 4 : Assertion

where N is a set of predictable thematic variations on the theme “I” introduce, and where every Di is a

design which never ends up by a †.Denegation is slightly different. We can refer to works by O. Ducrot in the eighties [Ducrot 1984] according

to whom discursive negation (that we shall name denegation in the present paper in order to avoid confusion

5

with standard logical negation) is necessarily polyphonic. We may for instance have the following utterance

in a dialogue:

- Mary is not nice, on the contrary, she is execrablewhere the second part of the utterance may be understood only if we think that it denegates not the first part

of the sentence but a dual utterance, the one which the present one is confronted with and which could be

Mary is nice. We are therefore led to conclude that a (de)negation like Mary is not nice is always opposed

to a virtual positive statement like Mary is nice. A possible modeling of such a case consists in having a

positive action by Myself which compels the other speaker to accept my denegation by playing her †. If not,

she enters into a diverging process5. The only way for Myself to force the other speaker to play her † is to

use the ∅ positive rule. On the other hand, the denegation assumes that the virtual other speaker produced

a statement which is now denied by Myself. This statement is in fact a paradoxal assertion since the set Nis reduced to {∅}! (The virtual speaker has no plan to sustain the claim she makes). Denegation therefore

supposes we make a step downward, to the fictitious claim (see figure 5)

∅� ξ,Λ vs

†� Γ

ξ � Γ

↓

∅� ξ0,Λ

{{0}}ξ � Λ

vs

†� Γ

ξ0 � Γ{0}

� ξ,Γ

Figure 5 : Denegation

Interrogation is still another game. If other speech acts can always be represented as anchored at a single

locus (modulo some “shift” which makes us going downward, searching the topic or the basis for a denega-

tion), we assume questions always starting from two loci, among which one is called the locus of the answer.

The design of a question has therefore a basis � ξ,σ with σ devoted to an answer, and ends up by a Faxσ,

so that, in interaction with a dual design E , the answer to the question is moved to σ. Let us now take

as examples two elementary dialogues consisting of sequences of Questions-Answers, where one is well

formed and the other ill formed.

The first one is : - YOU : Have you a car?- I : Yes,- YOU : Of what mark?It is represented on figure 6

†� σ

Faxξ010,σ

ξ010 � σY ou3

� ξ01, σ{∅,{1}}

ξ0 � σY ou1

� ξ,σ vs

.

.

.

� ξ010ξ01 �� ξ0ξ �

5that coud be repaired in a dynamic way.

6

Figure 6 : First dialogue

The answer ”yes” is represented by by {1}, creating hence a locus from which the speaker may continue the

interaction on the car’s topic and for example may ask which is its mark.

The answer ”no” is represented by ∅ (there is no more to say about this car) .

The second dialogue is :

- Have you a car?- No, I have no car- ∗ Of what mark?and it may be represented either on figure 7 where the dialogue fails since Y OU did not planified a negative

answer, or on figure 8 where the dialogue also fails since Y OU can only play on “your” left branch, thus

confusing the locus σ (which is a place for recording the answer) and the locus ξ.0 which corresponds to the

fact that the answer would have been “yes”.

Faxξ010,σ

ξ010 � ξ01, σY ou3

� ξ01, σ{{1}}

ξ0 � σY ou1

� ξ,σ vs

∅� ξ0ξ �

Figure 7 : Second dialogue-1

ξ010 �� σ

Faxξ010,σ

ξ010 � σY ou3

� ξ01, σ{∅,{1}}

ξ0 � σY ou1

� ξ,σ vs

∅� ξ0ξ �

Figure 8 : Second dialogue-2

4 Conclusion

Ludics provides a frame in which we can explore the speech acts realized in discourse as really two-faces.

This is mainly because Ludics, as a locative framework, makes it possible to make interact two parallel

processes, thus generalizing the well known dynamics of proofs (that we have already in Gentzen’s sequent

calculus, by means of the procedure of cut-elimination) to the dynamics of paraproofs (see the Annex 5.3).

In such a framework, there is no truth properly speaking but only ways for a proof-candidate to pass testswhich are themselves other proof-candidates. In a concrete dialogue situation, our utterance is a proof-

candidate : it has necessarily to cope with counter proof-candidates, which are either the reactions of the

other speaker or some kind of virtual reaction that we have in mind. This way, our interventions are doubly

driven: once by our positive acts, and secondly by the positive actions of the other speaker or of such a virtual

partner and by the way we record these reactions, that is by negative acts. Of course, while in a dialogue

7

each participant has to take into consideration the expectations and reactions of the other, in monologues,utterances are co-determined by the speaker herself, and by her virtual interlocutor. It is this interactionwhich drives the speech until a tacit agreement occurs either coming directly from the speaker or indirectlyvia the image she has of her speech.We also assume that concrete dialogues are always concrete manifestations of potential ones, that is we thinkour view is plainly coherent with one which takes for granted that any utterance commits its speaker to givereasons and justifications for saying it. This entail a conception of semantics which is different from theusual referential framework and seems to involve an inferential semantics of the kind R. Brandom arguesfor [Brandom 00]. This articulation between inferential semantics and dialogue pragmatics will be the topicof our future works.

5 Annex: a very short presentation of Ludics

Ludics is a recent theory of Logic introduced by J.-Y. Girard in [Girard 01]. We introduce below some of itsmains notions.

5.1 Proofs as processes

Let us start from a particular formulation of Linear Logic : Hypersequentialized Linear Sequent Calculus.This formulation elaborates on the fact that Linear Logic may be polarized, that is, we have positive con-nectives (⊗ and ⊕), which are also said active, in the sense that they make non reversible choices in theconstruction of a proof, and negative ones (& and ℘) which don’t. By grouping together successive positive(resp. negative) steps, it is possible to present a proof as an alternation of positive and negative steps. Alogic results, which has only two “logical” rules: one positive and the other negative. It also has a cut ruleand axioms.

� A11, . . . , A1n1 ,Γ . . . � Ap1, . . . , Apnp ,Γ

(A⊥11 ⊗ · · ·⊗A⊥1n1)⊕ · · ·⊕ (A⊥p1 ⊗ · · ·⊗A⊥pnp

) � Γ

Ai1 � Γ1 . . . Aini � Γp

� (A⊥11 ⊗ · · ·⊗A⊥1n1)⊕ · · ·⊕ (A⊥p1 ⊗ · · ·⊗A⊥pnp

),Γ

ou ∪Γk ⊂ Γ6 and, for k, l ∈ {1, . . . p}, Γk ∩ Γl = ∅.The first of these two rules is the negative one. A negative formula (left-hand side of the sequent) all thesubformulae of which are combined by positive connectives (if it were on the right hand side, the connectiveswould be replaced by ℘ et &, which are negative) happens to be decomposed in a canonical way whenapplying this rule: there is no particular choice to make.The second one is the positive rule. A positive formula all the sub-formulae of which are combined bypositive connectives happens to be decomposed according to some possible choices.We also have, as usual, the cut-rule :

A � B, ∆ B � ΓA � ∆,Γ

6The fact that ∪kΓk can be strictly included into Γ allows to retrieve weakening.

8

Unary operators called shifts allow changes of polarities of formulae, thus permitting to break a big stepinto several smaller ones. As may be easily seen on another hand, it is possible to present this calculus byusing only “forks ”, that is sequents of the form A � ∆ with the left hand side possibly empty, where all theformulae are positive (negative ones being transferred on the left after they have made positive).

5.2 Locativity

A remarkable property of Linear Logic resides in the ability it provides to represent a proof by a graph, ornet, simply called proofnet. Starting from a sequent to demonstrate, we decompose its formulae until wereach atoms and according to their polarity and the types of links by which their two main subformulaeare combined, then, positive and negative instances of the same atoms are connected (axiom links). Ifsome geometrical criterion is satisfied, we are sure that the sequent is provable. Girard noticed that in fact,locations in the net, and links between them are sufficient to identify a proof, exactly as if we were gettingrid of formulae! (“everything is at work without logic!”). This provides a basis for locativity, which opensthe way to Ludics. If we ignore formulae, we can only reason on locations (loci).

5.3 Designs

By getting rid of formulae, we may formulate the previous rules entirely in terms of addresses (the loci)where these formulae were anchored. Loci are sequences of biases (or integers). Then emerge two mainrules:

- Positive rule

· · · ξ.i � ∆i · · ·(ξ, I)

� ∆, ξ

where I may be empty and for every indexes i, j ∈ I (i �= j), ∆i and ∆j are disconnected and every ∆i is included in ∆.

- Negative rule

· · · � ξ.I, ∆I · · ·(ξ,N )

ξ � ∆

whereN is a possibly empty or infinite set of ramifications such that for all I ∈ N , ∆I is included in ∆.

Of course, the removal of formulae apparently deprives us of rules which make explicit use of formulae,that is mainly the axiom rule and the cut rule. Actually, the cut rule is externalized : the cut is simply acoincidence of loci with opposite polarities and identity will be expressed by the so called Fax (see 5.5).In Ludics, we make the assumption that a proof attempt may be stopped at any arbitrary stage. That corre-sponds to the use of the daimon rule:

†� ∆

This rule is of course a paralogism when compared with axiom rules in Hypersequentialized Sequent LinearLogic since taken litterally it would say that every positive sequent is derivable. But its interpretation is fromnow on different, it only means that we don’t go further in an argumentation. Moreover, the admission ofthis rule shows the need to embed proofs inside a more general class of objects, some of which are simply

9

not proofs at all, and hence often called paraproofs).

A design is a tree of forks built by means of these three rules. Its basis is the fork at its bottom. But there is

another way to see a design, since a proof process may be also seen as a sequence of negative and positive

steps in a game. It is as a set of possible plays. These plays are called chronicles. A chronicle may be built

from a design according to the previous definition. Starting from the bottom, we record all the branches and

their sub-branches. In order to correspond to a true design, these chronicles must satisfy some conditions

(coherence, propagation, positivity, totality).

5.4 Interaction

Considering two designs of bases of different polarities, interaction consists in a coincidence of two loci in

dual position in these bases. This creates a dynamics of rewriting of the cut-net made by the designs, called,

as usual, normalization. We sum up this process as follows: the cut link is duplicated and propagates over

all immediate subloci of the initial cut-locus as long as the action anchored on the positive fork containing

the cut-locus corresponds to one of the actions anchored on the negative one. The process terminates either

when the positive action anchored on the positive cut-fork is the daımon, in which case we obtain a design

with the same basis as the initial cut-net, or when it happens that in fact, no negative action corresponds

to the positive one. In the later case, the process fails (or diverges). The process may not terminate since

designs are not necessarily finite objects.

When the normalization between two designs D and E (respectively based on � ξ and ξ �) succeeds, the

designs are said to be orthogonal, and we note: D ⊥ E . In this case, normalization ends up on the particular

design :

†�

Let D be a design, D⊥denotes the set of all its orthogonal designs. It is then possible to compare two

designs according to their counter-designs. Moreover the separation theorem [Girard 01] ensures that a

design is exactly defined by its orthogonal: if D⊥ = E⊥ then D = E .

5.5 Infinite designs

Infinite designs are useful. Some of them may be recursively defined. It is the case of Fax, which is defined

as follows:

Faxξ,ξ� =...

...

Faxξ�i,ξi

ξ� � i � ξ � i ...(+, ξ�, J)

� ξ � J, ξ� ...(−, ξ,Pf (N))

ξ � ξ�

At the first (negative) step, the negative locus is distributed over all the finite subsets of N, then for each set

of addresses (relative to some J), the positive locus ξ� is chosen and gives rise to a subaddress ξ� � i for each

i ∈ Jk, and the same machinery is relaunched for the new loci obtained.

5.6 Behaviours

One of the main virtues of this ”deconstruction” is to help us rebuilding Logic.

10

• Formulae are now some sets of designs. They are exactly those which are closed (or stable) by inter-action, that is those which are equal to their bi-orthogonal. Technically, they are called behaviours.

• The usual connectives of Linear Logic are then recoverable, with the very nice property of internal

completeness. That is : the bi-closure is useless for all linear connectives. For instance, every designin a behaviour C⊕D may be obtained by taking either a design in C or a design in D.

• Finally, proofs will be now designs satisfiying some properties, in particular that of not using thedaımon rule.

References

[Andreoli 92] J.-M. Andreoli Logic Programming with Focusing Proofs in Linear Logic, The Journal ofLogic and Computation, 2, 3, pp. 297-347, 1992,

[Asher & Lascarides 98] N. Asher & A. Lascarides Questions in dialogue, Linguistics and Philosophy, vol21, 1998, pp 237–309,

[Beyssade & Marandin 06] C. Beyssade and J-M. Marandin The speech act assignment problem revisited,CSSP proceedings, http://www.cssp.cnrs.fr, 2006,

[Brandom 00] R. Brandom Articulating reasons. An introduction to Inferentialism, The President and Fel-lows of Harvard College, 2000,

[Bras et al. 02] M. Bras, P. Denis, P. Muller, L. Prevot, L. Vieu Une approche semantique et rhetorique du

dialogue, Traitement Automatique des Langues, vol 43, no 2, 2002, pp 43–71,

[Curien 04] Pierre-Louis Curien Introduction to linear logic and ludics, part I and II, to appear, download-able from http://www.pps.jussieu.fr/ curien/LL-ludintroI.pdf,

[Ducrot 1984] O. Ducrot, Le dire et le dit, Editions de Minuit, Paris, 1984.

[Girard 99] J.-Y. Girard On the Meaning of Logical Rules-I in Computational Logic, U. Berger and H.Schwichtenberg eds. Springer-Verlag, 1999,

[Girard 01] J.-Y. Girard Locus Solum Mathematical Structures in Computer Science 11, pp. 301-506, 2001,

[Girard 03] J.-Y. Girard From Foundations to Ludics Bulletin of Symbolic Logic 09, pp. 131-168, 2003,

[Girard 06] J.-Y. Girard Le Point Aveugle, vol. I , II, Hermann, Paris, 2006,

[Lecomte-Quatrini10] A. Lecomte and M. Quatrini Pour une etude du langage via l’interaction : dialogues

et semantique en Ludique, Mathematique et Sciences Humaines, no189, 2010,

[Walton 00] D. Walton The place of dialogue theory in logic, computer science and communication studies

Synthese 123: pp 327-346, 2000

11

Learning with Neighbours

Roland MühlenberndUniversity of Tübingen

Abstract

In this paper I will present a game theoretic multi-agent system to simulate evolutionary processesresponsible for the existence of pragmatic phenomena. Each agent behaves in a rational sense and cancommunicate with his direct neighbours via signaling games. Doing so he can learn the behaviour of hisinterlocutors by observation and these learnt data have an influence on his own communication behaviour.The complete system simulates an evolutionary process of communication strategies, which agents canlearn in a structured spatial society. The object of study in this paper is the Division of pragmatic labour.The questions of research are i) the emergence and arrangement of different strategy players and theirstability and ii) the influence of signaling game parameters on the course of evolution.

1 IntroductionDuring a students’ party you accidentally eavesdrop on a group of collegians talking about suspicious soundingtopics. By chance you pick up the expression “John went to (the) jail“, and you’re not certain if the speakerhas really used the word “the“. What difference does this word make? Obviously a deciding one! With thisword you would rather infer the literal interpretation that John really went to the jail building, maybe tovisit a prisoner, but without it you would infer the prototypical interpretation that John himself has beenjailed. This example is an instance of the general pragmatic rule division of pragmatic labour, originated byHorn (1984). This rule says that an simple (unmarked) expression describes a prototypical case, wherebya complex (marked) expression describes a rare (less prototypical) case. Horn presented a lot of examplesobeying his rule and depicting a conventional language use. Horn himself affirmed that this rule traces backto evolutionary forces1 and van Rooy (2004) gave “a game-theoretical explanation of how Horn’s rule couldhave become conventionalized through the forces of evolution.“

In the last years game-theoretical accounts characteristically based on Lewis’ (1969) signaling games

gained popularity as a means to examine pragmatic phenomena and language conventions. But standardsolution concepts like the Nash Equilibrium may lead to multitude equilibria, of which only a subset depictsthe phenomenon under discussion. Like van Rooy, several scientists like Jäger (2006, 2007) set this accountin an evolutionary framework by using evolutionary game theory (EGT) to consider interactions betweenindividuals in a population and therefore to examine in a much more realistic framework regarding phenomenasupposed to be evolved by evolutionary forces. It could be shown that in such a framework behaving accordingto Horn’s rule (playing Horn strategy) is a evolutionary stable strategy (ESS)2. But also the Anti-Horn

strategy3 is an ESS, but only the Horn strategy is stochastically stable.4 And van Rooy (2004) mentioned

1The force of unification and the force of Diversification (Zipf, 1949)

2A ESS is a strategy that once it is fixed in a population, natural selection alone is sufficient to prevent alternative strategies

from successfully invading.3Inverted Horn strategy: Using the complex expression for the prototypical case and the simple expression for the rare case

4ESS with maximal invasion barrier, which is the amount of mutations necessary to push the system into the basin of

attraction of another ESS

1

that “communities that use the Anti-Horn strategy will die out leaving only communities that use the Horn

strategy.“

The aim of this paper is to extend this line of research to a more realistic evolutionary setting taking into

account (i) the structure of social interaction, (ii) learning processes and (iii) spatial arrangement in form of

neighbourhood communication. Zollmann (2008) uses a similar setting: Just like my account he conceives a

multi-agent system where agents communicate in form of signaling games and play against each of his eight

neighbours. But while my agents choose their strategy regarding beliefs about their interlocutors obtained

by a learning process, his agents choose their strategy by adaptation. However he could show that for a

simple signaling game with two equivalent good strategies for successful communication the population of

his system is divided in two types of stable regions of agents playing in each case one of both strategies. By

adopting the phenomenon the division of pragmatic labour for my model, I will show that small regions of

Anti-Horn players among a majority of Horn players are stable over time, but their emergence depends on

how prototypical the prototypical case is. Furthermore the emergence of perfect communication in form of

Horn-strategies seems to depend on a fixed value regarding the difference of complexity of the simple and

the complex message.

This paper is structured in the following way: In section 2, I will introduce the conditions for a typ-

ical communicative situation between two rational agents related to a decision problem for assignment of

proposition and expression, both for a sender and a receiver. A central issue here is that such a decision

depends on a belief function, which reflects an estimation about the dialogue partner. Such a belief is the

result of fictitious play learning, a learning process by observation of the interlocutor’s behaviour in previous

communicative situations. In section 3, I extend the communicative behaviour of an agent from the situation

between two agents to a situation where an agent is faced with a group of different agents. The central issue

in this section is for the agent not to communicate with each group member separately, but to find an unique

communication strategy regarding all group members to arrive at a distinct strategy regarding a group. In

section 4, I will present an iteration of my simulation which represents the structure of an evolution step.

Each step consists i) of the realization of a communicative situation with neighbours and ii) of a completing

learning step by processing the observed behaviour of interlocutors. Consecutive evolution steps constitute

an evolutionary process, containing a learning process and possibly a strategy stabilization performance. In

section 5, I use this model to examine i) an example of a simple signaling game to compare my results with

Zollmann’s (2008) and ii) the division of pragmatic labour as my main topic in question. The final section

6 contains a short conclusion of my model and my results.

2 Learning by conversation

The way we use language and the act of collecting information about how interlocutors use language are

two intertwined processes during a conversation. Thus when we talk to people, we learn information about

their behaviour via language. And their way to use language affects our usage. E.g. think about a situation

at a students party standing next to the food buffet and talking to a collegian you’ve just met. Among

other things there are chips and french fries on the buffet table. Now your conversation partner is saying:

“These chips are awesome! You should try!“ But you have no idea if she’s using American English (AE)

and means These chips are fantastic. or British English (BE) and means These french fries are terrific.Thus you need information if she’s an AE or BE speaker. But now she’s going on and says: “Especially in

combination with fish!“ Now you know that she is most probably British and therefore she’s using BE and

particularly “chips“ for french fries and “crisps“ for chips. Thus to avoid misunderstanding you’ll use BE for

further conversations with her. All in all, her way to use language affects your belief about her usage, which

in turn affects your own language use. And the same can be the case the other way around. This kind of

communication behaviour is au natural as a strategy to avoid dialogue failure without taking to much effort in

extra communication.5

Therefore I want to introduce a model that conjoins a communicative performance as

5Regarding the aforementioned example you would have the opportunity to ask, if she’s an AE or BE speaker, but thiswould lead to extra communication, non-relevant to the given context but bound with extra costs.

2

strategic behaviour of language use and a learning process about the interlocutor’s communication strategy,

which takes place during this communicative performance.

To model strategic language use in a general way I’ll use signaling games, first introduced by Lewis

(1969). According to his general idea a common communicative situation between two agents z1 and z2,

whereby the former is the sender, the latter is the receiver, is defined in the following way: The sender has

a private information state t ∈ T and has to choose a message m ∈ M to communicate it. The receiver

has to interpret this message by choosing an interpretation a ∈ A as response to a received message m. So

the sender is playing due to a strategy σ ∈ [T → M ], the receiver to a strategy ρ ∈ [M → A]. For each

information state t there exists exactly one appropriate interpretation a, and vice versa. So we call (ti, ai)an appropriate pair. Furthermore there is a cost function c(m) ∈ [0, 1] that assigns to each message its

degree of complexity. Successful communication means that the receiver gets the interpretation appropriate

to the information state the sender wanted to communicate. In this case, both agents get an utility value of

1, otherwise 0 for failure. To be exact this value is reduced by the complexity of the used message. Taken

together, we get a utility function for both players shown in definition (1).

U(t,m, a) =�

1− c(m) if (t, a) is an appropriate pair

0− c(m) else(1)

This utility function depicts both interlocutors with coinciding interests, namely to communicate success-

fully. The speaker’s information state should correspond to the receiver’s interpretation. Thus for a given

information state t the speaker should use that message m, which she believes the receiver would construe

with the interpretation a matching the given information state t. From the receiver’s point of view for a

received message m he should choose an interpretation a matching the information state t he believes the

sender wants to communicate by sending m. E.g. at the students’ party buffet table you choose the interpre-

tation for the word ’chips’ according to your belief about whether she’s using AE or BE and therefore about

her strategy using this word. On the other way around she is using this word according to her belief about

you’re using AE or BE, about your strategy to construe it. All in all to find the best strategy for language

use in such situations each dialogue partner needs beliefs about the interlocutor’s strategy.

With given beliefs in form of probability functions we get the following expected utility functions (EU)

for a communicative situation between a sender z1 and a receiver z2:

EU (z1,z2)S (m|t) =

�

a∈AB(z2,z1)R (a|m)× U(t,m, a)

EU (z2,z1)R (a|m) =

�

t∈TB(z1,z2)S (t|m)× U(t,m, a) (2)

Rather EU (z1,z2)S (m|t) returns the expected utility of z1, if she has information state t and uses message m

to communicate it to z2. And EU (z2,z1)R (a|m) returns the expected utility of z2, if he receives m from z1 and

interprets it as a. As described above both interlocutors’ strategies depend on beliefs about the opponents’

strategy. In this sense B(z2,z1)R (a|m) is sender z1’s belief about the receiver z2, namely the probability that

z2 is interpreting with a, if z1 sends m to him. Also B(z1,z2)S (t|m) is the receiver z2’s belief about the sender,

the probability that sender z1 wants to communicate information state t, if she sends m to him.

But where do these beliefs come from? I think that they are largely result of collected information about

a particular dialogue partner regarding her language use.6 E.g. at the students’ party buffet table after my

partner’s first expression “These chips are awesome! You should try!“ I was undecided about her strategy,

i.e. if she’s using AE or BE. Thus my belief about she wants to communicate information state chips or

french fries is .5 to .5. After her second expression “Especially in combination with fish!“ I made a beliefupdate which results in a belief that it is much more probable that she wants to communicate information

6Further other parameters could affect these beliefs like context information, information at third hand et cetera. To keepthis model simple, such parameters will be neglected.

3

state french fries instead of information state chips by using the word “chips“. In general my belief of my

interlocutor’s strategy is the result of experiences of previous communicative situations with this dialogue

partner about the same topic. A game which includes such beliefs is called fictitious play and goes back

to Brown (1951), who’s “method in question can be loosely characterized by the fact that it rests on the

traditional statistician’s philosophy of basing future decisions on the relevant past history.“ In his account

for zero-sum games a player’s belief is a mixture represented by all the interlocutor’s past plays. I want to

adopt this idea for signaling games.

As already mentioned a sender’s belief is equivalent to the probability that her interlocutor will interpret

with a if she sends m. And according to fictious play learning this probability depicts exactly the rate her

interlocutor interpreted with a in all previous situations she sent m. E.g. assume z1 and z2 had the same

kind of communicative situation many times before and that function #(z1, z2,m) returns the number of

times z1 has sent message m to z2. Likewise #(z2, z1,m �→ a) returns the number of times z2 has interpreted

message m with a, whereby z1 has sent m. Because of these observations z1 has belief B(z1,z2)R (a|m) about

z2 interpreting message m with a as shown in equation (3). In the same way an evaluation of receiver z2’s

observations about z1’s behaviour leads to belief B(z2,z1)S (t|m) as shown in equation (4).

B(z1,z2)R (a|m) =

�#(z2,z1,m�→a)

#(z1,z2,m) if #(z1, z2,m) > 01/|A| else

(3)

B(z2,z1)S (t|m) =

�#(z1,z2,t�→m)

#(z1,z2,m) if #(z1, z2, t) > 01/|T | else

(4)

It is obvious that if you’ve never had the same communicative situation before, there is no possibility to have

a belief depending on previous situations. In this case, sender and receiver are undecided. In detail, if the

sender sends a message she has never sent before, she believes that each possible receiver’s interpretation has

the same probability, namely 1/|A|. Equally if the receiver receives a message he has never received before,

he believes that each possible sender’s information state has the same probability 1/|T |.Summarized we have definitions of a sender’s and receiver’s EU about a decision in a communicative

situation with coinciding interests, namely successful communication. EU depends on beliefs about the

interlocutor’s strategy and these beliefs are result of a learning process based on observations of previous

behaviour. Thus in absence of a more sophisticated calculation agents play a myopic strategy. And this is

absolutely suitable in following the goal to conceive a model for language use based on general rules. There

is also a second important feature of this account that becomes conspicuous in section 4 and 5, namely

that because of these belief functions agents cannot directly access the global probability function Pr(t)7, a

distribution over states which is especially important for the division of pragmatic labour to mark prototypical

and rare cases. We’ll see that this independence makes the setting much more realistic. However, so far the

expected utility value depends on only one dialogue partner. To be more realistic in illustrating evolutionary

processes, an expected utility is suitable that depends on multiple communication partners. An appropriate

function is presented in the next section.

3 Communication in a group

Now the next step is to explain how an agent communicates to more than one interlocutor. A general rule

is to avoid one-on-one communication in favour of group communication to minimize communication costs.

E.g. et your students’ party while you are talking to your partner some further students come along and

a group discussion ensues. Now you want to communicate that the french fries are awesome. Using the

expression “french fries“ or “chips“ depends on your belief about the group constellation, to be exact if you

believe that there are more AE or BE speakers. Thus if you talk to eight collegians and believe that six of

them are BE speakers, you would use “chips“ on the risk of both AE speakers would misunderstand. Taken

7e.g. like in Parikh’s (1992) model

4

together now you have to choose a strategy that depends on an EU regarding a group of interlocutors. The

easiest way to do it is to compute the average value. Formally agent zi as a sender wants to communicate

to a group Z of receivers. Then her expected utility regarding this group EU (zi,Z)S (m|t) is the average value

of all group members’ z expected utilities:

EU (zi,Z)S (m|t) =

�z∈Z EU

(zi,z)S (m|t)|Z| (5)

As a rational sender zi would choose a message that maximizes her expected utility regarding this group.

The following strategy function σ(zi,Z)(t) returns all messages m ∈ M that maximize her EU , if she wants

to communicate information state t ∈ T :

σ(zi,Z)(t) = arg maxm∈MEU (zi,Z)S (m|t) (6)

To keep sender and receiver behaviour symmetrical, the receiver also chooses his strategy depending on an

EU regarding a group of senders, and exactly in the same way like a sender does. His expected utility

regarding a receiver group EU (zi,Z)R (a|m) is the average value of all group members’ z expected utilities:

EU (zi,Z)R (a|m) =

�z∈Z EU

(zi,z)R (a|m)|Z| (7)

And as a rational receiver zi would choose an interpretation that maximizes his expected utility regarding

this group. The following strategy function ρ(zi,Z)(m) returns all interpretations a ∈ A that maximize his

EU , if he wants to construe a receipted message m ∈M :

ρ(zi,Z)(m) = argmaxa∈AEU (zi,Z)R (a|m) (8)

With these strategy functions every agent communicates with a group of interlocutors by using an unique

strategy. This means for the sake of a distinct language use an agent has to abstract from the individuality

of his interlocutors by finding the best consensus for a group of dialogue partners, or regarding this model

by computing the average expected utility for this group. In the next section I will present an idealized

simulation procedure of an evolutionary process, which is constructed for a multi-agent system and applies

these strategy functions to groups of direct neighbours.

4 Simulation procedureFor my model I implemented an application to simulate an evolutionary process, which consists of consec-

utively executed evolution steps. For this application it is possible to choose the size of a toroid n × mmulti-agent lattice and the game parameters for a signaling game: the state set T , the message set M , the

set of interpretations A8, the cost function for messages c : M → R and a global probability function over

states Pr(t) 9which depicts the probability of an information state t according to the whole population.

Now I want to present the procedure of the evolutionary process. To simulate such a process, each

agent zi runs through evolution steps, whereby one evolution step simulates a communicative situation and

contains the following four sub-steps:

1. Assignment of information states to neighbours:

The set of each agent zi’s neighbours is called his neighbourhood Ni = {zj |zj is a neighbour of zi}.8T and A must contain appropriate elements9A probability function Pr(t) is needed that regulates the occurrence of information states over the complete society. It is

important to mention that in this model the agents don’t use this probability function in their computation. But because thisfunction regulates the distribution of information states in the multi-agent lattice (as we will see in sub-step 1), Pr(t) has anindirect influence on the agents strategy in a long run sense, because the occurrence of information states affects the learningprocess.

5

In this first sub-step zi acts as an informer who gets information states he has to communicate to all

zj ∈ Ni, exactly one information state to each neighbour, but not necessarily the same information state

to all neighbours. Figure 1 depicts a possible situation of agent zi. In my simulation this allocation of

information states is realized by a random process, but according to Pr(t)10.

2. Sending the best message to each neighbour:

In the second sub-step agent zi acts as a sender. After the first sub-step each of agent zi’s neighbours

is assigned with one information state. Now all neighbours assigned with the same state t constitute

a neighbour receiver group Ni(t) = {zj ∈ Ni|zi wants to communicate t to zj}. So zi will commu-

nicate t by sending a best message m ∈ σ(zi,Ni(t))(t) to each neighbour11

: ∀zj ∈ Ni : zi sends m ∈σ(zi,Ni(t))(t) to zj . Figure 2 depicts a possible best strategy for the situation in figure 1.

3. Construing each received message with the best interpretation:

In the third sub-step agent zi acts as a receiver. Because of the fact that every agent has a neigh-

bourhood of 8 agents, he is also member of every neighbour’s neighbourhood. Thus after the second

sub-step he has received 8 messages from all his neighbours (e.g. figure 3). Now all neighbours he

receives message m from constitute a neighbour sender group Ni(m) = {zj ∈ Ni|zj sends message mto zi}. So agent zi will construe each message m with a best interpretation a ∈ ρ(zi,Ni(m))(m)12

. This

means that ∀z ∈ Ni : zi interprets message m from z with ρ(zi,Ni(m))(m) (e.g. figure 4).

4. Updating his belief about the neighbours:

In the last sub-step agent zi acts like a learner. After the third sub-step every agent has had a

communicative situation with all of his neighbours and the veil has been lifted yet. Now he can

observe the information that are needed to update the belief functions for all of his neighbours (see

section 2).

zi

z1 z2

z3

z4z5z6

z7

z8

t1 t2

t2

t3t1t3

t3

t2

zi

Figure 1: Agent zi wants tocommunicate his private in-formation states t1, t2, t3 tohis neighbourhood, exactlyone state to each neighbour

zi

z1z5

z2 z3 z8

z4 z6 z7

m1

m2

m3

Figure 2: Agent zi usesan unique strategy, if hechooses one message foreach of his neighbour re-ceiver groups Ni(t)

zi

z1 z2

z3

z4z5z6

z7

z8m2 m2

m1

m3m3m2

m1

m1

Figure 3: Agent zi re-ceives messages m1, m2,m3 from his neighbour-hood, exactly one messagefrom each neighbour.

z4z5

z3 z7 z8

z1 z2 z6

a3

a1

a2

zi

Figure 4: Agent zi usesan unique strategy, if hechooses one interpretationfor each of his neighboursender groups Ni(m)

In a nutshell this is what happens in one evolution step: Every agent in a society wants to communicate

information states to different neighbour receiver groups. He does this by computing the expected utility

value for each neighbour. But because he has to find a consensus for each neighbour receiver group, he

uses this value to compute an average expected utility value. This revision leads to an unique pure sender

strategy that depicts roughly a distinct language production. Then every agent receives a message from

all of his neighbours. Like in the former case he abstracts from each neighbours expected utility value by

10This means: if e.g. Pr(t1) = .8 then 80% of all allocated information states are t1, only the exact assignment is occasional.11In most cases, their is only one best message. If there are more than one best message, the simulation picks out on of the

best messages randomly12Like in the former case: if there are more than one best interpretation, the simulation picks out on of the best interpretation

randomly

6

using the average expected utility value for each neighbour sender group and therefore he uses an uniquepure receiver strategy that depicts roughly a distinct language interpretation. After each of both stepsevery agent can observe production and interpretation behaviour of his neighbours and uses this informationto update his beliefs about them. This update step is part of a learning process and affects beliefs ofneighbours and thus the expected utility about them and hence his strategy in the next evolution step.Ergo an evolutionary process is a procedure of sequent evolution steps that depicts the progress of strategiccommunication behaviour of agents in a society-like structure.

5 Simulations & Results

In this section I want to apply this simulation procedure to signaling games. In a first trial, I set balancedgame parameters like Zollmann (2008) did for his model to compare my results with his. In a second trial,I set parameters according to the division of pragmatic labour.

5.1 Monkeys and predators

As an introductory example I examine a signaling game with balanced game parameters, which I call monkeyand predator-game. Think of a population of monkeys, which can articulate two extremely noisy screeches“Uhh“ and “Ahh“ and use them as messages mu and ma, each message with the same costs: c(mu) =c(ma) = 0.1. It is important for their survival to use these signals for two information types of danger,e.g. a predator from the sky ts and a predator from the ground tg. Both information states have theappropriate interpretations as and ag. Surviving depends on the right interpretation: Seeking refuge ona tree is good against ground predators but perilous against sky predators, but to hide in a bush leadsexactly to a converse result. Thus the monkeys should favour successful communication with a distinctstrategy. To model this society I constituted a 30× 30 multi-agent lattice and to examine the evolutionarybehaviour I started a simulation run for 90 evolution steps. In this trial A1 I chose a balanced globalprobability, viz. Pr(ts) = Pr(tg) = .5. I was interested in both strategies representing a distinct language:S1 = {tg → mu → ag, ts → ma → as} and S2 = {tg → ma → ag, ts → mu → as}.

As you can see in figure 5, while at the beginning no one is playing S1 or S2, most of the monkeys learnone of both strategies after ca. 20 evolution steps and this status keeps stable over time. But if it is importantto survive, why doesn’t any monkey learn a distinct language? The answer is shown in figure 6: There arestable local areas of S1 and S2 players, but both strategies are highly incompatible among each other, thuson the boundaries between both areas agents are torn between both strategies. They switch permanently

S1 S2 other

0 10 20 30 40 50 60 70 80 90

150

300

450

600

750

900

Figure 5: Number of players with different strategies:Course of 90 evolution steps (Trial A1)

S1 S2 other

Figure 6: Distribution of players with different strategiesin a 30× 30 lattice after 90 evolution steps (Trial A1)

7

between S1, S2 and a babbling strategy13. Hence while the core areas of S1 or S2 are stable, the boundary

areas are highly variable. As a final remark to this trial I have to point out that it is sheer coincidencethat there are more S2 than S1 players. It is possible that in a new run with the same parameters thenumber of the latter type of player is larger or both are (almost) equal. By way of comparison in Zollmann’smodel after a while all agents are playing one of both strategies, distributed in regions like in my model, butthese regions are not distributed by undecided agents, furthermore there is a possibility for the emergenceof isolated regions of babblers.

5.2 Division of pragmatic labour

With this system I examined Horn’s (1984) Division of pragmatic labour, a rule that says that informa-tion that describes the prototypical case is communicated by an unmarked message, and information thatdescribes a rare case is communicated by a marked expensive message. A prominent example is that theunmarked message “Lee stopped the car“ suggests the prototypical case that he does it with the brake pedal,whereas the marked message “Lee made the car stop“ suggests an uncommon case, maybe by pulling theemergency brake. For this phenomenon the game parameters are set in the following way: T = {tp, tr},M = {mu,mm}, A = {ap, ar}. Distinction between marked and unmarked expression is achieved by messagecosts, so c(mu) = 0.1, c(mm) = 0.2. To distinguish the prototypical and rare case the probability setting isPr(tp) = .7 and Pr(tr) = .3. The size of the agency lattice is set on 30× 30 again.

In trial B1, I started a simulation over 90 evolution steps and examined the strategies agents are playing.I wanted to find out how many agents act according to Horn strategy or Anti-Horn strategy (see figure 10for detail). The result shown in figure 7 reveals that eventually every agent plays a Horn strategy. Notice:There is no chance for the emergence of Anti-Horn players; they number zero in almost every evolution step.

In a following trial B2, I changed the probability distribution to Pr(tp) = .6 and Pr(tr) = .4. Nowyou can observe that small groups of Anti-Horn players arise amidst a superiority of Horn players. Theinterior of these groups is stable over time. Only on the boundaries between both areas players are switchingpermanently between Horn, Anti-Horn and either of them like in the Monkeys and predators-game. In figure8 you can see that after ca. 20 evolution steps there evolved a stable group of around 45 Anti-Horn playersof all 900 players in this run. Furthermore there are around 90 agents playing neither Horn nor Anti-Hornand surrounding the Anti-Horn players. Figure 9 shows the distribution of agents in the lattice after 90evolution steps.

Horn Anti-Horn other

0 10 20 30 40 50 60 70 80 90

150

300

450

600

750

900

Figure 7: Number of players with different strategies:Course of 90 evolution steps (Trial B1)

Horn Anti-Horn other

0 10 20 30 40 50 60 70 80 90

150

300

450

600

750

900

Figure 8: Number of players with different strategies:Course of 90 evolution steps (Trial B2)

13Babblers choose the same message for both states or the same interpretation for both messages or all at once.

8

The crucial period for the emergence of these stable strategies occurs early on in the simulation. Thus it is

reasonable to investigate the initial stages of trial B2 in more detail. First of all, I subdivided the result in

two courses, one for sender strategies, one for receiver strategies. Second, I evaluated all possible strategies.

There are exactly four different strategies both for sender and receiver. Beside Horn and Anti-Horn, for

one thing there is the Smolensky-Strategy. According to this Strategy the sender uses the cheap message

for both information states and the receiver interprets each message with the prototypical interpretation.

For another thing there is the Anti-Smolensky strategy, which means acting the other way around. All four

strategies for sender and receiver are represented in figure 10.

Horn Anti-Horn other

Figure 9: Distribution of players with different strategies

in a 30× 30 lattice after 90 evolution steps (Trial B2)

Strategy sender receiver

Horn tp → mu mu → aptr → mm mm → ar

Anti-Horn tp → mm mu → artr → mu mm → ap

Smolensky tp → mu mu → aptr → mu mm → ap

Anti-Smol. tp → mm mu → artr → mm mm → ar

Figure 10: Different kind of strategies for sender and

receiver: Horn, Anti-Horn, Smolensky & Anti-Smolensky

The results for the first 20 evolution steps are depicted in figure 11 for the sender and 12 for the receiver.

As you can see in the first evolution step, all agents play as senders according to the Smolensky strategy,

because they have no belief about their neighbours and thus the best thing they can do is to minimize their

costs by sending the cheap message mu. And because all agents send mu to their neighbours means all

agents receive mu from all neighbours in the first step, they can only choose one interpretation and thus

they play the Smolensky or the Anti-Smolensky strategy by chance. With this in mind, no agent is playing

the Horn or Anti-Horn strategy in the first step.

Horn Anti-Horn Smol. Anti-Smol.

0 5 10 15 20

150

300

450

600

750

900

Figure 11: Number of players regarding their sender

strategy: Course of 20 evolution steps (Trial B2)

Horn Anti-Horn Smol. Anti-Smol.

0 5 10 15 20

150

300

450

600

750

900

Figure 12: Number of players regarding their receiver

strategy: Course of 20 evolution (Trial B2)

9

In the second evolution step the number of Horn and Anti-Horn players increases very sharply both for

sender and receiver, while the number of Smolensky players declines sharply. It is an interesting fact that

the number of Anti-Horn players reaches its peak very early and then decreases to a stable number. The

reason for the strong upward movement is the probability .4 of tr, that means the system has a tendency for

40% of Anti Horn players. But Horn players exert social pressure by assimilating isolated Anti-Horn players,

thus the number of Anti-Horn players increase finally to around 10%.

With trial B3 as the last examination regarding this phenomenon I wanted to explore the influence of the

complexity of the marked message on the agents’ strategic behaviour. Trial B1 showed that a probability

distribution of Pr(tp) = .7 and Pr(tr) = .3 leads to an emergence of Horn players only. Then while

maintaining these values I successively raised the cost value of message mm to examine the emergence course

of Horn players. As you can see in figure 13, the higher the complexity of the marked message, the longer

the emergence process of Horn players takes. And there is another interesting result. If the c(mm) is higher

than .61, there is no emergence of Horn players. Figure 14 shows the number of senders regarding the Horn

and Smolensky strategy for c(mu) = .1 and c(mm) = .61. As you can see in the first evolution step, every

agent is playing the Smolensky strategy, but in the second step, there is an upward jump of Horn players to

around 100. This is an initial step for the emergence of Horn players. But by changing the costs of c(mm) to

.61001, i.e. raising them marginally, there is no jump of Horn players and all agents keep to the Smolensky

strategy. In fact, multiple trials confirm that the difference between both message costs has a crucial upper

limit; it should not exceed .51. This value seems to be a cut-off point for the emergence of Horn players.

c(mm): 0.2 0.5 0.6

0 10 20 30 40 50 60 70 80 90

150

300

450

600

750

900

Figure 13: Number of Horn players for different cost val-

ues of message mm: Course of 90 evolution steps (Trial

B3)

Horn Smolensky

0 10 20 30 40 50 60 70 80 90

150

300

450

600

750

900

Figure 14: Number of players with sender strategy Horn

or Smolensky: Course of 90 evolution steps with c(mm) =.61 (Trial B3)

5.3 Conclusion

With this paper I presented a model of a multi-agent system to simulate evolutionary processes in language

change. This model involves a classical signaling game between sender and receiver, both acting as rational

agents. An entity of this model is a n×mmulti-agent lattice in a toroid structure of spatial arrangement; each

agent can only communicate with his eight direct neighbours. These settings are really close to Zollmann’s

(2008) model. The crucial difference is the way an agent chooses his strategy. In Zollmann’s model agents

are easily imitating the best neighbour strategy in the next round. In my model a decisions for choosing a

strategy is influenced (i) by beliefs about the particular interlocutors and (ii) by the need to find a consensus

for a group of different neighbours. Beliefs about interlocutors are the result of a learning-by-observation

process. This learning process is really simple and agents play a myopic strategy. A consensus between

different neighbours is easily reaches by computing the average EU . All in all the cognitive capabilities

10

required in my model reflect more sophistication than those in Zollmann’s model, but nevertheless they arenot overly demanding. This makes my model appropriate for the examination of simple communication thatshould depict the evolution of general rules for conventional language use. To simulate such an evolutionaryprocess, agents of mymodel are communicating in consecutively performed evolution steps, in each one everyagent sends messages, interprets received messages and updates his beliefs about his neighbours. For thesake of evaluation I used this model to simulate evolutionary processes.

While the examination of the Monkeys and predators-game was an introductory example to motivate theadequateness of signaling games for the research of language change processes in society and to comparemy results with these of Zollmann, my main focus concerns pragmatical phenomena like the Division of

pragmatic labour, examined in the second trial. The obvious most important result of trial B1 is the reallyfast emergence of Horn players. Consider that the use of Horn strategy is a consequence of combiningmessage ordered by costs with states ordered by probability. But contrary to the message costs no agent’scomputation of EU is directly influenced by the global probability Pr(t), but it has only indirect influence ofhis learning process. I think under these circumstances such a rapid evolving process of using Horn strategyas observable in my examination is really impressive. An important result of Trial B2 is that the emergenceof regions of Anti-Horn players depends on the value of stereotypicality of the prototypical state tp. Thehigher the global probability Pr(tp), the lower the probability of the emergence of regions of Anti-Hornplayers. But those regions are stable over time, if they obtain a size of around 5 Anti-Horn players. To findthe exact number and constellation as a lower limit for those stable regions is a task for further research. Butbecause of being highly incompatible with the Horn strategy, all areas of Anti-Horn players are surroundedby a variable non-stable wall of switching players. In Trial B3 a really interesting result is the existenceof a cut-off point for message cost settings. Exceeding this point prevents the emergence of Horn playersin favour of Smolensky players. Further research should include an examination of the correlation betweengame parameters that influence the existence of the aforementioned cut-off point.

References[1] Brown, G.W. (1951) ’Iterative Solutions of Games by Fictitious Play’, Activity Analysis of Production

and Allocation, T.C. Koopmans (Ed.), New York: Wiley.

[2] Horn, L. (1984), ’Towards a new taxonomy of pragmatic inference: Q-based and R-based implicature’,In: Schiffrin (ed.), Meaning, Form, and use in Context: Linguistic Applications, GURT84, 11-42,Washington; Georgetown University Press.

[3] Jäger (2006), ’Evolutionary stability of games with costly signaling’, In: M. Aloni, P. Dekker & FlorisRoelofsen (eds.), Proceedings of the 16th Amsterdam Colloquium, ILLC, University of Amsterdam,121-126.

[4] Jäger (2007), ’Convex meanings and evolutionary stability’, In: Angelo Cangelosi, Andrew D. M. Smithand Kenny Smith (eds.), The Evolution of Language. Proceedings of the 6th International Conference

(EVOLANG6), Rome, pp 139-144.

[5] Lewis, D. (1969), Convention, Cambridge: Harvard University Press.

[6] Parikh, P. (1992), ’A Game-Theoretic Account of Implicature’, In: Proceedings 4th Conference on

Theoretical Aspects of Reasoning about Knowledge (TARK), Morgan Kaufmann, Los Altos, CA.

[7] van Rooy, R. (2004), ’Signaling games select Horn strategies’, Linguistics and Philosophy 27, 493-527.

[8] Zipf, G. (1949), Human behaviour and the principle of least effort, Cambridge: Addison-wesley.

[9] Zollman, K. (2005), ’Talking to neighbors: The evolution of regional meaning’,Philosophy of Science

72, 69–85.

11

Metalanguage dynamics.

Tillmann ProssInstitute of Philosophy, University of Stuttgart

Stuttgart, Germanytillmann.pross@philo.uni-stuttgart.de

Abstract

This paper outlines the basics of Grounded Dis-

course Representation Theory, a conservative ex-

tension of Discourse Representation Theory to

the specific needs of goal-directed joint interac-

tions between humans and robots. Grounded

Discourse Representation Theory defines a gen-

eral framework for the processing of verbal and

non-verbal interaction in dialogue that combines

multi-agent based control mechanisms with the

apparatus of formal natural language semantics.

1 Introduction and Motivation

Central to the formalism of Grounded Discourse

Representation Theory (GDRT, [Pross, 2010])

which is outlined in this paper is an agent-based

theory of dynamic interpretation of semantic

representations (in the sense of Discourse Rep-

resentation Structures (DRS) in Discourse Rep-

resentation Theory (DRT), [Kamp et al., 2010])

that makes it possible to capture the dynam-

ics of interaction in dialogue as reciprocal influ-

ence between the object language of DRSs and

the metalanguage of set-theoretic model theory

against which DRSs are evaluated. In this sense,

GDRT counters the objection that the dynam-

ics of DRT-like interpretation processes “resides

solely in the incremental build-up of the repre-

sentations, and not in the interpretation of the

representations themselves.” [Groenendijk and

Stokhof, 1999, p. 10]. The approach to the dy-

namic interpretation of representations proposed

by GDRT takes a step behind the scenes of the

processing of information and interaction in dia-

logue in that it specifies how an agent deals with

the dynamics of information and interaction in

dialogue. This dynamics manifests itself on the

one hand in the form of the semantic representa-

tions that are constructed, maintained and used

during a dialogue but also - and this is the focus

of GDRT - in the model-theoretic semantics for

these representations. The central idea that this

discussion amounts is that the modelling of the

dynamics of information and interactions in their

application to dialogue must be itself dynamic.

The context of human-robot interaction from

which GDRT draws its basic motivation places

special demands on a formalism that is to pro-

vide a robot with the necessary means and in-

formation structures. Consider the following ex-

ample (1), uttered by Fred, a human, to robot

Clara. Fred and Clara are situated at a table.

On the table is a cube, a slat and a screw1.

(1) Give me the cube.

With respect to the meaning of (1), traditional

formal semantics would seek to derive the truth

conditions of (1) with the help of which inter-

preters (such as Clara) of (1) could in turn eval-

uate (1) against a set-theoretic model theory pro-

vided to them. A suitable model theory for the

analysis of this sentence should consider that (1)

1GDRT has been developed as a part of a project onjoint action of humans and robots, where the task do-main consisted of a Baufix construction kit, consisting ofwooden cubes, screws, nuts, slats with holes and otherparts.

1

is a request for the addressee to choose from

her future possibilities for action the course of

action which renders true that the speaker has

been given a cube. A formal structure that cap-

tures future possibilities models time as a tree

branching towards the future and in which links

connecting successive models (representing mo-

mentary situations in various possible future de-

velopments of the present) represent basic causal

transitions (=actions) from the first situation to

the second situation. GDRT integrates this con-

ception of time and action with the multi-agent

interpretation of CTL* put forward in [Singh,

1994]2. This allows us to think of (1) being true

iff there exists a sequence of actions in Clara’s

model of reality starting right after the utter-

ance time n and leading to a timepoint t1 > nwhere Fred has the cube

3.

But the semantic interpretation of (1) is only

half the story. Uttering (1) additionally puts a

pragmatic - normative - pressure on Clara to ex-

ecute the actions leading to t1 by adding this

sequence of actions to her intentions. An agent

interpreting (1) must translate (1) into future

actions whose realization constitutes the proper

reaction to (1) by selecting that part of future

action in which those actions are performed and

in which their final result is therefore true as

well, thereby verifying (1). The problem here is

that the successful interpretation of (1) is to be

determined with respect to possible future de-

velopments of reality that we - as designers of

robot Clara’s processing formalism - cannot fore-

see in advance and which we consequently cannot

capture in terms of models that can be specified

before the utterance time. Instead, the dynam-ics induced by the interpretation of the informa-

tion conveyed with (1) pertains to the extraction

of instructions how the model of reality against

2The combination of DRT and CTL* has, besides[Singh and Asher, 1993], not received the attention itprobably deserves.

3The account of pragmatic utterance meaning inGDRT shares its basic assumptions with plan-based the-ories of speech acts [Cohen and Perrault, 1986], whereGDRT employs the account of [Singh, 1998].

which (1) is to be evaluated must be shaped by

future action in order that the truth conditions

expressed by (1) are rendered true.

Let me explicate the point with another example,

where Fred announces her future plans to Clara.

(2) I am going to build a bike.

Put a simple way, (2) is true iff the information

that Fred is going to build a bike is contained

in the model against which (2) is evaluated by

Clara. However, if we - as designers - would

have captured this information before the utter-

ance time and provided it to Clara, the utter-

ance would have a very different status, as the

information conveyed by (2) would be already

available to Clara. Instead, the interpretation

of (2) requires Clara to adapt her modelling of

the future in such a way that it captures the

choice and commitment of Fred to those future

courses of action that bring about a bike. That

is, the interpretation of (2) as informative utter-

ance requires Clara to alter both the universe of

her model of reality (adding an individual ’bike’)

and the interpretation function (adding a (set of)

ordered pairs �a, b� to the extension of ’build’,

where a is an agent and b a bike) of the model

against which she interprets (2).

What we see here - and saw in the interpre-

tation of (1) - is another dimension of inter-

pretation dynamics than the dynamics which is

and can be captured by Standard DRT. It is a

dynamics that lies ’outside’ of the scope of se-

mantic representations themselves - the object-

language - but pertains to the metalanguage of

model theory. The way in which GDRT ac-

counts for the metalanguage dynamics that is

induced by such interpretation strategies for ut-

terances that concern the future is closely par-

allel to the notion of context change potential

in Dynamics Semantics. GDRT considers mean-

ing, one might say, as model change potential.

Two types of model-changing actions have to

be distinguished: agent-internal actions that di-

rectly affect an agent’s models (as planning) and

agent-external actions that indirectly affect an

2

agent’s models via the feedback of the results ofperceived action (e.g. the building of a bike orpassing over an object). Then the crucial ques-tion is how to capture the reciprocal dependencyof internal and external action and the inter-pretation of utterances against dynamic models.The answer of GDRT has three components. Atfirst, we - as designers - need to specify how ut-terances relate to actions. To this end, GDRTintroduces temporal anchors to the object lan-guage of DRT and defines their formal seman-tics in terms of the metalanguage of branchingtime structures. Temporal anchors render pos-sible to combine semantic - truth-conditional -and pragmatic - success-based approaches of nat-ural language meaning. Second, we must elab-orate how the anchor-based interpretation of anutterance specifies courses of action that consti-tute sequences of future actions as appropriatereaction to the utterance. To this end, GDRTemploys a formal theory of planning couched inthe metalanguage. Third, we need to explicatehow an agent like Clara can employ these mech-anisms to interpret utterances. For that pur-pose, GDRT employs a BDI-interpreter4 to con-trol the execution of intentions on the basis ofbeliefs, goals and plans, where the interpretationof DRSs is integrated into the overall frameworkof BDI-based control.

2 Overview of GDRT

In the following, for the analysis of example (1)in the framework of GDRT, I introduce onlythose fragments of GDRT necessary for the anal-ysis of (1), a full account of GDRT can be foundin [Pross, 2010]. Given an agent x at time ti,I call the agent’s configuration of her seman-tic representations (DRS) and models (Exter-nal Presentation Structure, EPS) at ti the ’cog-nitive structure’ CS of x at ti, CS(x)(ti) =�DRS(x, ti), EPS(x, ti)�. Figure 1 pictures the

4I do not introduce the functioning of a BDI-interpreter here but refer the reader to e.g. the simpleversion given in [Singh et al., 1999]

cognitive structure of Clara right after she hasconstructed a DRS K1 for (1), where her EPSpresents initial information delivered by her ob-ject recognition about the objects on the tablein front of her. As Clara is assumed to be co-operative, she has invoked an attempt to inter-pret DRS K1, presented as an internal actionint-a?K1 in the EPS.The main DRS K1 in figure 1 represents (1) withthe help of two related events e1 and e2. e1

represents the intentional nature of the actionin question, namely that the plan correspond-ing to giving the cube is or should be amongstthe active intentions of Clara. The relation be-tween the propositional attitude INT and thecontent of the attitude DO is represented by thenested use of temporal anchors for e1 (denotingchoice and commitment wrt. e1) and e2 (denot-ing the plan for giving a cube)5. Temporal an-choring in GDRT is built upon the idea that onthe one hand, humans structure their temporalperception along “goal relationships and causalstructures” [Zacks and Tversky, 2001] and on theother hand, human beings structure their futureexperiences in terms of the causal relations thatthey perceive between them and their plan-goalstructures or intentions. Formally, this dual roleof temporal entities is captured by the introduc-tion of temporal anchors (extending the theoryof anchors put forward by [Asher, 1986, Kamp,1990]), two-place relations between discourse ref-erents for eventualities (which I call floaters) andrepresentations of causal, plan-goal and inten-tional structures (which I call sources). The tem-poral structure imposed by the additional con-straints on e1 and e2 represents the present tenseof (1), where e2 starts after e1 and e1 starts af-ter n, Clara’s present now defined by the utter-ance time t0. DRS K3 represents the goal ofthe plan for giving a cube. The goal of the plandenoted by e2 is represented with the state s1.Besides the anchors for e1, e2, s1 and n, also the

5Intentions are analyzed as events as they denote dy-namic control over the execution of plans by the BDI-interpreter.

3

K1 : �

n, e1, i, q

�e1, iINTK2 :

e2

�e2, i DOK3 :

s1

�s1, handhuman(q)�hold(s1)e2 <beg s1

�

give(e1)e1 <beg e2

n <beg e2

�

n ⊆ e1, be(e1), �n, t0�cube(q), �q,

−→? �

, t0�

t0, a, b, ccube(a) on− table(a)slat(b) on− table(b)screw(c) on− table(c)

int-a:?K1

t1

1

Figure 1: The cognitive state of Clara at t1, CS(i)(t1): Representation K1 of utterance (1) “Give methe cube.” (left) making up DRS(i)(t1) and presentation of Clara’s EPS at t1 (right), EPS(i)(t1).Note that from an epistemic point of view, Clara’s EPS presents a complete picture of the model-theoretic information available to her but further steps of interpretation will extend Clara’s EPS.

discourse referent for ’the cube’ has an anchor.The initial representation of (1) in GDRT hasan unresolved anchor for ’the cube’, i.e. the ref-erential source of the floater x is unspecified inthat it lacks a specification of the object of ref-erence, represented as a variable anchor source?. The fact that ’the cube’ has been identifiedby means of definite description is captured bythe arrow over the variable source ?. This ar-row constrains interpretations of K1 by requiringan unique anchor source for q. The interpreta-tion process for a DRS such as K1 spelled outin the following seeks to resolve both temporaland non-temporal anchors to metalanguage en-tities (objects for things, temporal structures foreventualities) of the model theory EPS againstwhich the representation is evaluated. In thefollowing, I call discourse referents for eventu-alities time-individuals and other discourse ref-erents thing-individuals. Unresolved anchors arecalled variable anchors.

Clara’s interpretation attempt of K1 proceedsalong the following general procedure. Supposethat �K, tj� is a DRS belonging to DRS(x, ti) -normally tj would be the instant ’now’ and thatis what I will assume here - and that x attemptsto interpret K at ti. As a first step x must findanchor sources for all those thing anchors of theanchor set Anch of K where anchor sources arevariable. Each such variable anchor source is of

one of two sorts: (1) direct6 or (2) anaphoric, inaddition, each anchor may be constrained witha definiteness constraint. If s is a variable di-rect anchor source, then a (non-variable) an-chor source that can replace it must be an ob-ject from EPS(x, ti); when the variable anchorsource is anaphoric, then a non-variable anchorsource replacing it must be a thing-individualfrom DRS(x, ti). If for any variable thing-anchorno suitable sources can be found, then the inter-pretation of K stops (but can be continued, seesection 5 below). Suppose that it is possible tofind a suitable non-variable anchor source to re-place each of the variable thing anchor sourcesoccurring in anchors of K. Then there will bea nonempty set G of functions g each of whichis defined on the set of ’floaters’ of anchors inK and maps each floater onto a suitable non-variable anchor source. In the next step of in-terpretation, each function g in G can be usedto identify the time-individuals from �K, tj� bychecking whether their branching-time semanticscan be embedded into EPS(x, ti) at tj . If thisstep succeeds, the respective g is stored in a setF ⊆ G to check in a final step whether K asa whole has at least one successful anchoring

6I exclude the distinction between external and inter-nal anchors that is drawn in the full version of GDRT,as this would involve additional elaborations on symbolgrounding and object recognition.

4

h ⊆ F that enables identification of all condi-tions c1, . . . , cn ∈ K with respect to EPS(x, ti)at tj7. If there exists such a successful anchor-ing for K, I say that K has a successful ’plain’interpretation. ’Plain’ because no manipulationsof EPS(x, ti) were necessary.If the plain interpretation of K as describedin the last section fails, i.e. no successful an-choring of K could be established, ’reactive’ in-terpretation comes into play - this what is re-quired in the case of interpreting (1). It could bethe (pragmatic) meaning of K that EPS(x, ti)has to be changed by the interpreter of K toEPS(x, tk) with i < k in order to render possi-ble a successful anchoring of K in EPS(x, tk)8.The appropriate reaction is guided in partic-ular by the time-individuals contained in K.These time-individuals specify a course of ac-tion (via their branching-time semantics) whichis to be executed in order to bring about theconditions that render a successful plain inter-pretation of K possible. That is, in responseto a failed plain interpretation of K with re-spect to EPS(x, ti) at tj , the interpreter ofK should perform some actions which resultin EPS(x, ti) being transformed into a modelstructure EPS(x, tk) which allows for a success-ful plain interpretation of K at tk. Technically,this is achieved with the formulation of a seman-tic (object-language) and a pragmatic (metalan-guage) identification of time-individuals, specify-ing the conditions that identify time-individualsin plain interpretation mode (corresponding toclassical truth-conditional semantics) and in ad-dition the actions which are to be undertakenin order to make a given time-individual ’true’via an execution of reactive interpretation (cor-

7Complex conditions are not discussed here, but are

analyzed in [Pross, 2010] in accordance to the dynamic

semantics of DRT.8The decision in which cases reactive interpretation

is allowed is not discussed here. However, there are some

basic cases in which reactive interpretation is not allowed,

e.g. if the reaction concerns the manipulation of agent’s

own history. That is, a question such as “Did you build

a bike?” should not receive an reactive interpretation.

responding to the unfolding of the pragmatic im-pact of the utterance)9.

3 An application of GDRT

The procedure outlined in the last section is ap-plied to (1) in figure 2, where formal details areprovided in the next section. For the analysisof (1), the required information for the resolu-tion of the variable anchor source for ’the cube’is given with the following simplified semantic-pragmatic concept, where the [SEM] part speci-fies the DRS representation and the [PRG]-partthe identification conditions in metalanguage as-sociated with the discourse reference marker xfor ’cube’, where I call ’cube’ the handle of x.

Sem-Prag-Concept 1 cube

SEMxcube(x), �x, a�

PRGacube(a)

A rudimentary semantic-pragmatic concept for atime-individual adapted to the use of ’give’ in (1)can be stated as follows, where the [SEM] partspecifies the semantic contribution of ’give’ andthe [PRG] part a metalanguage specification ofthe pragmatic profile of ’give’, a plan10.

Sem-Prag-Concept 2 give

9The two options of interpretation (reactive/plain)

come close to the distinction between declarative and im-

perative semantics [Gabbay, 1987] or from another point

of view, conceptual and procedural meaning [Sperber and

Wilson, 1993]10

Serious attempts of modelling ’give’ must of course

be more fine-grained and specify invocation, context and

feedback conditions for [PRG] and a more detailed con-

nection between [SEM] and [PRG]. I also exclude the con-

tribution of tense and aspect which is spelled out in detail

in [Pross, 2010] as well as the integration into a syntactic

framework such as the lambda-calculus

5

SEM

x, egive(e)

�e, xDO

y, s�s, handhuman(y)��y, a�

�

PRG t0 - ext-a:grasp(a) - t1 - ext-a:present(a) - t2

4 Formal Definitions

This section sketches the basic formal ideas un-

derlying the account of semantics and pragmat-

ics in GDRT as discussed with examples (1) and

(2). In order to capture the dynamic nature of

the EPS structure and consequently of the model

theory against which DRSs are evaluated, we

first need a specification of EPSs and the branch-

ing structure of EPSs.

Definition 1 EPS vocabulary

• A set TR of EPS reference markers forthings: {a1, . . . , an, . . .}

• For each n > 0 a set Reln of n-place predi-cate constants for handles {C1, . . . , Cm, . . .}

• A set Times of EPS times {t0, . . . , tn, . . .}11

Definition 2 Syntax of EPSs and EPS condi-tions

1. If U ⊆ TR�

Times, Con a (possibly empty)set of conditions then �U, Con� is an EPS

2. If R1 ∈ Reln and a1, . . . , an, . . . ∈ TR thenR1(a1, . . . an) is an EPS-condition

3. A time-indexed EPS is a tuple �t, �U, Con��.

The branching structure of time-indexed EPSs

- the EPS structure - can be formally described

in terms of a modal model structure (cf. [Singh,

1994], [Emerson, 1990]).

11The numerical subscripts are used only to clarify thedesign of the EPS structure.

Definition 3 EPS StructureAn EPS structure is a tuple E = {T, I, Actions}of an agent x at time t, where

• T = �<,TimesA� is a time structure of anagent x at time t, where TimesA ⊆ Timesand T is a labeled directed graph with nodeset TimesA, arc set Actions and node labelsgiven by I. In addition, we require the graphof T to be a tree.

• I associates times t ∈ TimesA with EPSs,i.e. I is a function from TimesA to EPSsaccording to definition 212.

• Actions is a function from pairs �t, t�� of ad-jacent members of TimesA to the set of in-ternal and external actions available to anagent.

For the use of EPS structures as models for the

interpretation of the language of DRSs, it is use-

ful to convert the ’raw form’ of the EPS structure

into the logically more manageable form of sets

and assignment functions. Here we make use of

the function I from EPS times to EPSs (defi-

nition 3.) That is, with a given EPS structure

E = {T, I, Actions} of an agent x at time t we

are provided with the following sets:

Definition 4 EPS sets of an agent x stored inher EPS structure at t

• The set of EPS times TimesA=

{t0, .., tn, . . .} (occurring in Dom(I)13)

• The set of EPS things Things= {a, b, c, . . .}(occurring in the universes of EPSs inRan(I))

12That is, the interpretation I of an EPS-timet ∈ TimesA is a function from time indices tto EPSs as defined by a set of time-indexed EPSs�t1, �U1, Con1��, . . . �tn, �Un, Conn��, ...

13I write Dom(F ) for the domain and Ran(F ) for therange of a function F .

6

K1 : �

n, e1, i, q

�e1, iINTK2 :

e2

�e2, i DOK3 :

s1

�s1, handhuman(q)�hold(s1)e2 <beg s1

�

give(e1)e1 <beg e2

n <beg e2

�

n ⊆ e1 be(e1) �n, t0�cube(q) �q,

−→? �

, t0�

K1 : �

n, e1, i, q

�e1, iINTK2 :

e2

�e2, i DOK3 :

s1

�s1, handhuman(q)�hold(s1)e2 <beg s1

�

give(e1)e1 <beg e2

n <beg e2

�

n ⊆ e1 be(e1) �n, t3�cube(q) �q, a�

, t3�

t0, a, b, ccube(a) on− table(a)slat(b) on− table(b)screw(c) on− table(c)

int-a:?K1

t1

int-a:resolve(q)

t2

int-a:set�q, a�

t3

int-a:i-add(i, K2)

t4

ext-a:grasp(a)

t5handrobot(a)

ext-a:release(a)

t6handhuman(a)

1

Figure 2: Processing of example (1) “Give me the cube” by Clara in the framework of GDRT.

The figure shows her cognitive structure CS(i)(t5). Earlier stages of processing are recorded in

CS(i)(t5), representing a discourse history. Clara’s interpretation attempt int-a:?(K1) at t1 invokes

a plan that pushes K1 to the list of DRSs to be interpreted. Next, it is checked whether K1 contains

variable anchor sources. The variable anchor source ? in K1 at t1 triggers a plan int-a:resolve(q) for

the resolution of this source. The anchor source for q is resolved at t3 by a successful unification

of the semantic-pragmatic concept 1 under consideration of the definiteness constraint on q with

EPS(i)(t1). Once the variable anchor source for q has been resolved to a, K1 is passed over to

the main interpretation process. As a plain embedding of K1 fails at t3 - up to t3 there is no

temporal structure in which the [PRG] of ’give’ could be embedded, reactive interpretation of K1

is executed. This results in an extension of Clara’s EPS at t3 with the pragmatic part [PRG] of the

semantic-pragmatic concept for ’give’. As the EPS-path t4 − t5 − t6 is added to Clara’s intentions

by the command i-add, Clara’s BDI-interpreter executes this intention. Finally, with t5, K1 can

be embedded into EPS(i)(t6) and Clara realizes a successful interpretation of (1), where Fred has

the cube in his hands. Note that the DRS with unresolved anchors at t1 differs from the DRS with

resolved anchors at t3 in its now-anchor. The information in the EPS presents only updates to the

EPS, i.e. newer information replaces old information if there exist incompatibilities between an

existing and a new EPS condition concerning an already registered thing (e.g. a move to a different

location). Similarly, new information (e.g. a thing appearing for the first time in the agent’s area

of vision) is presented in the EPS.

7

• The set of EPS properties Properties={p1, . . . , pn, . . .} (occurring in EPSs in

Ran(I))

• The set of EPS atomic actions

Actions= {a1, . . . , an, . . .} (occurring

in Ran(Actions))

Next, we define a set of functions that assignssets of (tuples of) agents and/or things and timesto subsets of I as specified in definition 4.

Definition 5 EPS assignment functions

• A function T that assigns EPS structures to

an agent τ at t, i.e. the time structure of an

agent at t: T(τ)(t)

• A function S that assigns Scenarios to an

agent τ at t: S(τ)(t).

A scenario is an EPS structure {T, I, Actions}such that < is a linear ordering. Let R ={T, I, Actions} be an Model structure. S ={T’, I �, Actions�} is a scenario of R iff S is a

scenario, I � ⊆ I and T’ is a substructure of

T. If S is a scenario of R, then there will be

t, t� ∈ Dom(I) so that T’ is the segment (t, t�)of T. t is called the starting point of S in R.

Of particular interest are those scenarios S of Rin which t� is a leaf of T. When t is the start-

ing point of S then we write ’S(t)’. S(t) ⊆ Tdenotes the set of all scenarios of T at t. The

notation [S; t, t�] denotes an inclusive interval on

a scenario S from t to t� with t, t� ∈ S and t ≤ t�.

• A function P that assigns Plans to an agent

τ at t: P (τ)(t).

A plan P of some EPS structure R has a starting

point t. Its time structure is a subtree of T with

t as root. When t is the starting point of P , then

we write ’P(t)’. P (t) ⊆ T denotes the set of

plans at t. [P ; t, t1] denotes a plan starting at twith its goal located at t1.

• A function P that assigns Properties to

(tuples of) EPS-things �a1, . . . , an� at t:P(a1, . . . , an)(t)

The information structures of the BDI-

interpreter of an agent x provides us with

sets of plans (the knowledge base) and inten-

tions (the current configuration of the intention

stack), sow we can define

• A function Attitudes that assigns attitudes

of a certain type φ (DO or INT) to an agent

x at t:Attitudes(φ)(x)(t)

The syntactic definition of DRSs in GDRT fol-lows the standards of DRT [Kamp et al., 2010],so I skip this step and directly move on to themain point, the use of EPS structures as modelsfor DRS interpretation. In defining the modelsfor DRS interpretation we have to consider animportant point that distinguishes the modelsfor DRS interpretation in GDRT from the mod-els in Standard DRT. As the models an agentcan employ for DRS interpretation are derivedfrom the agent’s current EPS, those models onlypresent the agent’s current information aboutthe state of affairs. The ’indexed’ nature of themodels for DRS interpretation is captured byrecording the agent from whose EPS the modelwas derived and the time at which this was done.Mindful of this consideration, a model M forthe semantic definition of the language of DRSscould be defined as follows.

Definition 6 A model M at a time t of an

agent x is a tuple

M(x)(t) = �P, S,T,P,PRG,Things,Attitudes�

M differs from traditional models (e.g. thosefor the interpretation of Standard DRSs) in thatthere is no ’interpretation function’ included,i.e. a predefined function that maps predicatesand individual constants to their model-theoreticcounterparts. Instead, the concept of an ’inter-pretation function’ is replaced by two compo-nents. First, this is the PRG function, whichcontains the pragmatic profiles associated withthe handles of discourse reference markers in therespective semantic-pragmatic concepts. The

8

identification conditions provided by the [PRG]part of an individual guide the transformationof a DRS with variable thing anchor sourcesto a DRS with a set of possible thing individ-ual anchor source resolutions. The other com-ponent is the identification procedure for time-individuals, which determines at runtime the ’ex-tension’ of time-individuals in that it identifiessets of thing individual anchor sources amongthe set of possible thing anchor resolutions thatsatisfy the identification conditions [PRG] of thetime-individuals with respect to ↑ M . In addi-tion, we have to consider the fact that adding aDRS to DRS(x)(t) may result in a revised setof referents, conditions and anchors. So a suc-cessful anchoring must be defined with respectto the existing anchors of DRS(x)(t). That is,suppose a sequence of DRSs in DRS(x)(t) con-sisting of a set of reference markers U ⊆ Ref ,a set of conditions Con = {C1, . . . , Cn} and aset of anchors G = {A1, . . . , Am} is given. Anupdate to DRS(x)(t) with K will result in thesets U

updateK , Con

updateK , G

update, where Gupdate is

a successful anchoring of K with respect to M iffG

update extends G in a way that it identifies thefloaters in U

updateK with EPS entities given the

pragmatic identification conditions [PRG] asso-ciated with the floaters. Formally, a successfulanchoring is defined as follows:

Definition 7 Successful anchoring14 of a DRSK

Given sets of possible anchorings G, H, a modelM and a DRS K

• ��G, H�� �M K iff G ⊂D H and for all γ ∈ConK : H �M K, where A ⊂D B reads as:’the domain of A is a subset of the domainof B’

• G �M K reads as: G successfully anchorsK in M and

• ��G, H�� �M K reads as: H extends G to asuccessful anchoring of K in M .

14Successful anchoring mirrors the notion of a verifyingembedding in DRT.

Definition 8 Successful interpretation of aDRS K

• A DRS K has a successful interpretationin a model M iff there exists a successfulanchoring G for K in M that extends theempty anchoring ξ.

• I write �M K iff there exists a successfulanchoring G such that ��ξ, G�� �M K.

• When G �M γ, where γ is a DRS-condition,I say that G identifies γ in M .

In addition, interpretations can be determinedwith respect to a time, a scenario, a plan anda model, which will be written as �M,S,P,t K. Inthe EPS (e.g. as part of a plan), an interpre-tation attempt of a DRS K can be triggered bythe command int-a :?K. The EPS constituentswhich were identified as a successful interpreta-tion of a DRS K with respect to a model anda time are denoted by [K]M,t. If the respectiveEPS constituents have not been identified yet,[K]M,t triggers an interpretation attempt of K,int-a :?K.

Thing-individuals are identified as follows.

Definition 9 Identification of thing-individuals.

• �x, a� �M,t handle(x) iff PRGhandle(x) ∈P(a)(t)

Constraints imposed by definite descriptions arecaptured by the following clause.

• �x,−−−−→source� �M,t handle(x) iff there is ex-

actly one source with which PRGhandle(x)can be identified in M at t.

Time-individuals in present tense are resolvedvia the following clauses, where the reactive in-terpretation of a time-individual is formulatedin terms of metalanguage actions, i.e. the addi-tion of beliefs (b-add), goals (g-add) or intentions(i-add).

9

Definition 10 Identification of time-individuals in present tense.

• �s, R(x1, . . . , xn)� �M,t handle(s)plain: iff ∃G = {�x1, a1�, . . . , �xn, an�} sth.PRGhandle(a1, . . . , an) ∈ P(a1, . . . , an)(t);reactive: b-add(x, t, PRGhandle(a1, . . . , an))

• �e, xDOK� �M,S,P,t handle(e)plain: iff ∃[S; t, n] ∈ S(x)(t) and∃[P ;n, t1] ∈ T(x)(n) sth. (S ∪ P ) ∈PRGhandle(e) and �M,t1 K and [K]M,t ∈Attitude(Do, x, t);reactive: g-add(x, PRGhandle(e))

• �e, xINTK� �M,S,P,t handle(e)plain: iff ∃[S; t, n] ∈ S(x)(t)and ∃[P ;n, t1] ∈ T(x)(n) sth.(S ∪ P ) ∈ PRGhandle(e) and[K]M,t ∈ Attitude(Int, x, t);reactive: i-add(x, PRGhandle(e))

5 The logic behind GDRT

With this picture of dynamic partial models

in mind, consider another utterance of Fred to

Clara.

(3) Show me all cubes.

In the intended application scenario of GDRT,

an agent x will always find herself in a specific

situation in which she is supposed to evaluate

her semantic representations by default15

. We

- as designers - do not want that the agent in-

terprets (3) as involving quantification over an

infinite set of possibly existing cubes but as per-

taining to the possible anchors for cubes pro-

vided by the current model of reality. However,

the staged design of interpretation in GDRT

supports the implementation of ’switches’ be-

tween this situation-bounded interpretation and

non-situation-bounded interpretation. At each

15In its semantic conception of evaluation in specific sit-uations, GDRT is reminiscent of the information limita-tion proposed by situation semantics [Barwise and Perry,1983]

level of interpretation via the resolution of an-

chors, the agent can adopt different logical atti-

tudes towards the interpretation of DRSs. That

is, depending on the situation, different log-

ics (classical or non-classical) can be employed

by an agent for the semantic interpretation of

DRSs. GDRT models can be incomplete in sev-

eral ways. First, the extensions (the referents)

of DRS thing-individuals may be unknown to

an agent and thus missing in the agent’s mod-

eling of reality. Second, the extensions of DRS

time-individuals may be unknown to an agent

and thus are not contained in the agent’s models.

Third, the models against which an agent eval-

uates DRSs involving quantification over thing-

individuals have finite domains. At first sight,

these limitations seem to be in conflict with two

fundamental assumption of bivalent formal se-

mantics. First, that models are complete in

that they include all information that is rele-

vant with respect to evaluation; with respect to

complete models, a sentence evaluates to either

true or false. Second, that models provide in-finite quantification domains. The account of

this problem in GDRT makes use of the possi-

bility to intervene into the interpretation process

of semantic representations. That is, in princi-

ple GDRT allows an agent to take different logi-

cal attitudes towards the interpretation of DRSs

depending on the context of interpretation such

as supervaluation semantics [van Fraassen, 1966]

or Kleene Logic [Kleene, 1952] to deal with in-

complete information due to unresolved anchors.

However, the practically motivated default ac-

count of incomplete information implemented in

GDRT forces an agent to put on hold the inter-

pretation of DRSs until the given models have

been extended with the information necessary for

the resolution of anchors of a DRS. Extensions

of models may result from asking for more infor-

mation (concerning e.g. the reference of thing-

individuals) or bringing about certain state of

affairs (concerning e.g. the reference of time-

individuals). With respect to the finite quan-

tification domains GDRT models provide, it is

10

possible to drop the restriction of GDRT quan-tifiers to the closed world of the robot’s finitemodel structures by switching from finite to in-finite quantification domains with the definitionof an ’inflated’ model ↑ M obtained from a fi-nite model M by adding a finite or infinite set ofobjects Unk (for Unknown) to the set Things,resulting in the set ↑Things replacing Thingsin M16. Finally, given the limitation of the in-tended application scope of GDRT and its pri-marily practical motivation, issues of e.g. decid-ability may not be as relevant from that specificpoint of view as they are from the more generalmetaphysical point of view.

6 Comparison and Conclusion

While there exist numerous semantic approachesto dialogue processing, the in-depth discussion ofthe semantics-pragmatics interface and GDRT’sintegration of DRT-based formal semantics andpragmatic planning with temporal anchors isnew to the literature. Consequently, it is dif-ficult to compare GDRT with other approachesto dialogue processing. Prominent pragmatic ac-counts based on plan recognition are limited topropositional logic (to name some:[Cohen andPerrault, 1986, Grosz and Sidner, 1986, Pollack,1990, Singh, 1994] and do not explicitly spell outthe connection between propositional planningand complex semantic representations of natu-ral language whereas GDRT integrates planninginto the formal semantics of complex DRSs.On the other hand, approaches to discourse pro-cessing that are built on top of DRT such asSDRT [Asher and Lascarides, 2003] tackle theproblems presented in this paper on a differentlevel of analysis than GDRT does. As SDRTadopts the formal semantics underlying DRT,the considerations on metalanguage dynamicsput forward with GDRT apply to SDRT too.However, SDRT and GDRT can be considerednatural companions in DRT-based dialogue pro-

16One could even think of the option of finite models

with extensions spelled out in [Bonevac and Kamp, 1987]

cessing: GDRT does not spell out how sequencesof DRSs constructed and interpreted during adialogue are (rhetorically) interconnected so itis at this point where the mechanisms of SDRTcan be connected to GDRT. In turn, it wouldbe interesting to see how GDRT can flexibilizeSDRT’s logical system of axioms and inferenceby providing the possibility to ground rhetori-cal structures of and pragmatic inferences fromDRSs in an action-theory based account of prag-matics. As GDRT is ’backwards’-compatible toDRT, axiom-based reasoning about agents anddiscourse (e.g. [Asher and Lascarides, 2003]) isput back into the game if an agent x ’freezes’her cognitive structure CS(x)(t) and uses theanchors in CS(x)(t) to reconstruct a static uni-verse and interpretation function that can be em-ployed as a classical model theory.This paper introduced the basics of GDRT,where the systematic use of anchors allows tocombine normative (pragmatic) and descriptive(semantic) approaches to discourse processing. Icall GDRT normative in the sense that its cen-tral goal is to derive appropriate future options of(re-action) that serve the realization of discoursegoals. Theories such as DRT are descriptive inthe they describe the processes which are sup-posed to take place in the minds of the discourseparticipants when they try to make sense of agiven discourse. The combination of descriptivesemantic and normative pragmatic meaning viathe concept of dynamic interpretation proposedin this paper probably constitutes the main tech-nical innovation of GDRT with respect to DRT.

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12

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2 This does not attempt to explain the following classic: “How can you spot an extroverted mathematician?” “He looks at your shoes when you talk.”

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3 CSD is a stronger form of the stochastic dominance mentioned earlier that gives us more tractable results in the tradition of Maskin and Riley (1998). This takes the form, in some examples, of shifted uniform distributions and stretched distributions.

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4 The handy mnemonic here is that the shape of the B is symmetric, and the shape of the b is not.

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Split antecedents and non-singular pronouns in dynamic semantics!

Jos Tellings

jos.tellings@cantab.net

Abstract. Literature on the problem of split antecedents has focused on singular pronouns.However, the construction involved is also possible in combination with non-singular pronouns, inparticular plural pronouns and disjunctive pronouns (‘he or she’, ‘him or her’). Besides a theoreticalinterest, there is an empirical argument for studying them: the experiment I have conductedshows that non-singular pronouns are used more frequently than singular pronouns. I proposean extension of DPL (Groenendijk and Stokhof 1991) to include variable complexes. These arecomplexes consisting of two variables that can each bind a di!erent quantifier. I will show that thissolution works both for singular and disjunctive pronouns, while I argue that for plural pronounsno additional theory is required.

1 Introduction

The problem of split antecedents (henceforth PSA) occurs when a pronoun refers back to a disjunctiveclause (or split antecedent). Some typical examples are given below (I will refer to such sentences witha split antecedent as SA-sentences):

(1) If I see John or Bill, I will wave to him.

(2) %If I see John or Bill, I will wave to them.

(3) If I see John or Mary, I will wave to him or her.

(4) A professor or an assistant professor will attend the meeting of the university board. He willreport to the faculty. (from Groenendijk and Stokhof 1991:88)

(5) If Mary catches a fish or John traps a rabbit, Bill will cook it. (from Stone 1992:377)

(6) If Mary hasn’t seen John lately, or Ann misses Bill, she calls him. (from Stone 1992:378)

The essence of the problem of split antecedents is that the actual referent of the pronoun may dependon the situation. A frequent interpretation of the pronoun in SA-sentences is as referring to the uniqueperson that turns out to satisfy the predicate the split antecedent occurs in (e.g. the unique person Iwill see, the unique person that will attend the meeting, and so forth). I will refer to this interpretationas the singular interpretation.

PSA has been studied to some extent in several semantic theories. The earliest studies I know ofare in static Montague semantics (Rooth and Partee 1982, Partee and Rooth 1983). Then the problemsparkled interest in dynamic semantics (Groenendijk and Stokhof 1991:88-9, Stone 1992, Kamp andReyle 1993:205-6), and has more recently received new interest (Elbourne 2005, section 2.7.2, Wang2005, section 7.4, Schlenker 2010). However, this literature has almost exclusively looked at PSA incombination with singular pronouns ((1)), while other pronouns are possible as well ((2) and (3)).Plural pronouns can be used, as in (2), and another possibility is a pronominal construction I referto as a disjunctive pronoun. Disjunctive pronouns are used when there is a mixed-gender antecedent,and they take the form ‘he or she’, ‘him or her’ and so forth (as in (3)). Because of the lack of dataof this type, I conducted an experiment in which I tested how English native speakers use these two

!This paper is a condensed and slightly expanded version of a part of my Master’s thesis at the University of Cambridge(Tellings 2010), which is available from the author. I am most grateful to my supervisor, Kasia Jaszczolt. I also thankNicholas Asher for discussion on PSA. I thank the Department of Linguistics at the University of Cambridge for partiallysupporting my experiment.

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types of pronouns. The main conclusion was that both plural and disjunctive pronouns are used moreoften than singular pronouns, which gives a second, empirical argument for studying them.

I will first describe my experiment and discuss its results. Subsequently, I will propose a dynamicsolution for PSA with singular and disjunctive pronouns. This works by adding a new type of variableto DPL, a variable complex. I will argue that standard plural dynamic theories will be adequate todeal with plural pronouns.

Some data cited herein have been extracted from the British National Corpus Online service1,managed by Oxford University Computing Services on behalf of the BNC Consortium. All rights inthe texts cited are reserved. Examples from the BNC are cited with a text identifier and sentencenumber of the form ‘ABC.123’, as is prescribed on the BNC website.

2 Data survey

An experiment in the form of a questionnaire was conducted to test English native speakers’ use ofdisjunctive and plural pronouns in SA-sentences. I will first describe the goals of my experiment, andsubsequently the results.

2.1 Goals and content

The questionnaire served two goals. Firstly, I wanted to test how the use of plural and singularpronouns depends on (lexical) context and world knowledge. It is relatively easy to construct sentencesin which one reading is clearly more natural than others. The main question is: do speakers preferplural pronouns in sentences with a prevalent plural reading, and do they prefer singular pronouns insentences in which a singular reading is most natural?

A second purpose of the questionnaire was to gain some insight into the way disjunctive pronounsare used, i.e. how mixed-gender antecedents are dealt with. It is important to observe that both pluraland disjunctive pronouns have a di!erent use, viz. to avoid gender bias. The sentences in (7) bothcontain simple antecedents that are gender-neutral (‘student’, ‘candidate’), and both a disjunctive anda plural pronoun can be used to refer to them:

(7) a. If a student likes semantics, he or she should come to this workshop.b. When the candidate has settled ask if they would like to smoke. (BNC, BNA:1548)

It is possible that plural pronouns can have the same role in SA-sentences and can be used withmixed-gender antecedents. A problem in the questionnaire design is that it is di"cult to detect thisuse: it is not clear if a participant uses a plural pronoun because of the plural reading of the sentence,or as a gender-neutral pronoun. To overcome this di"culty, I introduced pairs of nearly identicalsentences, which for expository reasons I will refer to as gender pairs. A gender pair consists of twosentences that are identical, except that in one sentence there is a same-gender antecedent (e.g. ‘Johnor Bill’, ‘Mary or Linda’), and in the other sentence a mixed-gender antecedent (e.g. ‘Fred or Sue’)2.The advantage of this method is that, within one gender pair, the prevalent reading is maintained:the factor of same-gender / mixed-gender antecedent has been isolated, for which a gender pair isa minimal pair. We can thus compare the use of disjunctive and singular pronouns, independent ofwhich reading is preferred.

As an example of the kind of information this set-up may give, suppose that in one gender pair,a participant chooses a singular pronoun in the same-gender sentence, but a plural pronoun in thecorresponding mixed-gender sentence. Then there is a strong indication of the use of a gender-neutralplural pronoun. The participant should give the same answer in both sentences of a gender pair, exceptthat he or she has to solve the ‘problem’ of a mixed-gender antecedent. If the participant switchesfrom a singular to a plural pronoun in the mixed-gender sentence, this must have been caused by the

1Available at http://www.natcorp.ox.ac.uk.2To avoid further complexity, the male-female order is maintained throughout the questionnaire. So, for example,

‘John or Bill’ would correspond with ‘Fred or Mary’ and ‘Linda or Sue’ would correspond with ‘Michael or Dorothy’. Ihave been using four di!erent names in each gender pair to bring some variation in the questionnaire.

2

presence of a mixed-gender antecedent. Therefore, the plural pronoun was used as a gender-neutralpronoun (and not because the sentence had a plural reading).

2.2 Procedure

Participants The questionnaire was completed by twenty students (graduate and undergraduate,in the age range of 19 to 25) from Trinity College, Cambridge who were native speakers of English andwere not reading linguistics. The questionnaire was sent out by e-mail and could be filled-in on theparticipants’ own computer. The participants were paid after returning the completed questionnaire.

Materials Each item in the questionnaire contains a ‘mini-story’ consisting of a small number ofsentences or a part of a sentence. Each mini-story ends with two possible continuation sentences: thefirst with a singular/disjunctive pronoun (depending on the type of the antecedent), the second with aplural pronoun. This order is constant throughout the questionnaire. The participant is instructed toconsider which of these sentences (if any) are adequate as a continuation of the story in question, byanswering a multiple-choice question (see Figure 1). The participant can favour precisely one sentence(answer A or B), both sentences (C) or neither sentence (D). The formulation and order of the fouroptions is, again, identical for each item in the questionnaire. Finally, there is room for the participantto leave some remarks or comments at each mini-story.

– 1 –

The department’s 5000 study grant will go to Anne or Mary.

1. She has applied well before the deadline.

2. They have applied well before the deadline.

A Sentence 1 is more adequate.

B Sentence 2 is more adequate.

C Sentence 1 and 2 are equally adequate.

D Neither sentence is adequate.

Answer:Remarks:

– 2 –

. . .

Figure 1: Final form of the questionnaire

The questionnaire consisted of thirty mini-stories in the following distribution:

seven mini-stories in which a singular reading was most natural;

seven mini-stories in which a plural reading was most natural;

seven mini-stories in which both a singular and a plural reading were natural;

nine filler items.

I shall refer to the groups of seven as ‘categories’, labeled ‘singular’, ‘plural’ and ‘ambiguous’, respec-tively. Each category was built up as follows: three gender pairs (as described above) and one storywith a non-animate antecedent (taking the pronoun ‘it’ rather than ‘he’ or ‘she’). The nine filler itemswere mini-stories that looked similar (they also had continuation sentences with singular and pluralpronouns), but had nothing to do with disjunction. The thirty items were randomly ordered, with theconstraint that two items from one gender pair were never adjacent. In order to avoid any possible

3

bias caused by the order of the items, four such random orders were generated. The resulting fourversions of the questionnaire were distributed evenly under the participants (five each).

The twenty-one relevant mini-stories (i.e. excluding the filler items) are collected below, bearinglabels (Q1) to (Q12). For reasons of space and easy reference, they are printed in this text in anabbreviated form. The real format used in the questionnaire is as in Figure 1. The example fromFigure 1 is printed in abbreviated form as (Q7) below. The underlined part corresponds with thetwo continuation sentences: the first sentence with the singular/disjunctive pronoun and the secondone with the plural pronoun. The same mini-story, but with di!erent names (as described above),constitutes the other half of a gender pair. A sentence like (Q7) is thus a template that correspondsto one gender pair, i.e. to two items in the questionnaire. In two cases, the nature of the mini-storyrequired some other adaptations (besides the names) to the same-gender story to serve as a mixed-gender variant. In those cases, the two stories of the gender pair are listed separately as a. and b. (asin (Q6) and (Q9)). Stories (Q4), (Q8) and (Q12) are the ones containing a non-animate antecedentand are not part of a gender pair, so they correspond to just one item in the questionnaire.

Singular category

(Q1) I think someone in our street will win the lottery jackpot this week. If Michelle from nr. 12 orLaura from nr. 18 would win, she/they would go on holiday to South Africa.

(Q2) Professor Richard Smith from London or Dr. Paul Brown from New York will be elected asMaster of the college. He/They will take residence in the Master’s Lodge.

(Q3) The general will send out Frank or Harry to Afghanistan. He/They will join the troops in theKandahar region.

(Q4) After the congress, we will know if New York or Tokyo will host the Olympic Games. It/Theywill also host the Paralympic Games.

Plural category

(Q5) I am organizing a conference. It will be opened by Michael or George. He is an expert / Theyare experts in the field.

(Q6) a. If Suzy or her best friend Helen would win first prize in the lottery this month, she/theywould go shopping in Milan.

b. If James or his wife would win first prize in the lottery this month, he or she/they wouldgo on holiday to Spain.

(Q7) The department’s 5000 study grant will go to Anne or Mary. She has / They have appliedwell before the deadline.

(Q8) Glasgow or Manchester will be chosen this year’s Capital of Culture. It has / They haveinvested loads of money in theatres and music education.

Ambiguous category

(Q9) a. If I see Mark or his best mate David in town today, I wave to him/ them.b. If I see Anthony or his wife in town today, I wave to him or her/ them

(Q10) I expect to see Linda or Elizabeth at the party tonight. I will invite her/them to dinner.

(Q11) I have decided to employ Brian or Steven as a teacher. I will show him/them the school nextweek.

(Q12) If the United States or the United Kingdom would be attacked by terrorists, it/they will fighteven harder against terror.

2.3 Results and discussion

I will now proceed to discuss of the results of the experiment. I will do this for each of the two goalsseparately, because it is useful to have di!erent presentations of the results for each case. I shall use theabbreviations s.g. and m.g. for ‘same-gender antecedent’ and ‘mixed-gender antecedent’, respectively.Because the answer options were identical for each question, they can be interpreted as follows:

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A = singular pronoun (s.g.) / disjunctive pronoun (m.g.) is more adequateB = plural pronoun is more adequateC = both pronouns are equally adequateD = neither pronoun is adequate

I shall be using them as such in the discussion below.

2.3.1 First goal

Results The basic results are collected in Table 1: it gives the responses per category and per type.There was one non-response, in question (Q4).

Category Type A B C D n

Singular

s.g. 21 23 5 11 60

m.g. 30 13 13 4 60

(Q4) 3 6 2 8 19

Sum: 54 42 20 23 139

% 39% 30% 14% 17%

Plural

s.g. 2 48 7 3 60

m.g. 2 51 6 1 60

(Q8) 1 18 1 0 20

Sum: 5 117 14 4 140

% 4% 84% 10% 3%

Ambiguous

s.g. 8 29 8 15 60

m.g. 13 26 14 7 60

(Q12) 6 14 0 0 20

Sum: 27 69 22 22 140

% 19% 49% 16% 16%

Grand total

s.g. 31 100 20 29 180

m.g. 45 90 33 12 180

(Q4,8,12) 10 38 3 8 59

Sum: 86 228 56 49 419

% 21% 54% 13% 12%

Table 1: Responses per category and per type

Discussion If the choice between a singular or plural pronoun reflects the availability of a plural orsingular reading, we would expect a high percentage of A responses in the singular category, and ahigh percentage of B responses in the plural category. For the ambiguous category, we would expectC responses for participants who recognize both readings, or A or B for participants who recognizeone reading.

Table 1 shows that this hypothesis holds very well for the plural category: 94% of the responsesjudged a plural pronoun adequate (B+C), and 84% judged a plural pronoun more adequate than asingular / disjunctive pronoun (B).

In the singular category, however, these numbers are 53% (A+C) and 39% (A). In fact, if oneonly looks at the s.g. sentences in the singular category, there are more responses favouring a pluralpronoun ( 23 ) than a singular pronoun ( 21 ). The ambiguous category lies between the other twocategories, as expected, but also shows a general favour for plural pronouns.

We can conclude, then, that there is an overall preference for plural pronouns: in 67% of allresponses, the plural pronoun was judged adequate (B+C), while in 54% of all responses it wasconsidered more adequate than a singular / disjunctive pronoun (B). On the other hand, only 34% ofall responses favoured a singular / disjunctive pronoun (A+C), and 21% judged a singular/disjunctive

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pronoun more adequate than a plural pronoun. Also, we observe there are much more D responses inthe singular and ambiguous categories, which may suggest that speakers do not like singular readingsof SA-sentences.

I conclude that the factor of context and world knowledge that determines the plausibility of asingular or plural reading is an important factor, but it is not the only one. For the plural category it isobvious: whenever a plural reading is readily available, speakers consistently choose a plural pronoun.On the other hand, there is a relatively high number of plural pronouns in the singular category, whichleads to the conclusion that plural pronouns can sometimes receive a singular interpretation. I willreturn to this issue in section 3.4. Finally, I note that there are other, more pragmatic factors thatinfluence the choice of a plural or a singular pronoun, for example rhetorical structure. I refer thereader to Tellings (2010) for details.

2.3.2 Second goal

Results In order to say something about the di!erences between mixed-gender and same-genderantecedents, I have presented the results somewhat di!erently (Table 2). For the sake of clarity, letme give an example of how to read this table. I shall denote the two answers a participant gave forthe two items of a gender pair as an answer pair, that is an ordered pair, in which the first element isthe answer for the s.g. story, and the second element is the answer for the m.g. story. So, for example,for a particular participant and a particular gender pair, the answer pair ‘(B,A)’ denotes that thisparticipant answered B in the s.g. story, and A in the corresponding m.g. story of the gender pairunder consideration. In other words, the participant preferred a plural pronoun in the s.g. story, anda disjunctive pronoun in the corresponding m.g. story.

Table 2 collects the number of each possible answer pair, split by category (Table 2(a-c)) and intotal (Table 2(d)). Let me use Table 2(a) as an example to further explain how it should be read. Asmentioned above, each category contains three gender pairs. This results in sixty answer pairs for thetwenty participants. The rows in Table 2(a) correspond with the answers given for the s.g. stories,and the columns correspond with the answers given for the m.g. stories. Each entry in the table isthe number of such answer pairs (in the singular category). So, the entry ‘16’ is the number of (A,A)answer pairs. That is, in sixteen cases, participants chose a singular pronoun in the s.g. story, anda disjunctive pronoun in the corresponding m.g. story. Similarly, the entry ‘6’ corresponds with thenumber of (B,A) answer pairs. The row and column labeled ‘Sum’ give the total number of s.g. andm.g. answers, respectively. The first row of Table 2(a) thus gives the following information:

There were 21 A responses on the s.g. sentences from the singular category. From these 21, 16had A in the corresponding m.g. sentence of the gender pair, 1 had B, 3 had C and 1 had D.

Discussion Firstly, Table 2 gives information about the total number of responses on m.g. sentencescompared to s.g. sentences (in the cumulative columns and rows). The main conclusion is that thegeneral preference of plural pronouns is less strong in mixed-gender sentences: Table 2(d) shows thatthere were 31 singular pronouns as opposed to 45 disjunctive pronouns in gender pairs. Also therewere fewer plural pronouns in mixed-gender sentences: 100 versus 90. Also, mixed-gender sentencesshow more C’s and fewer D’s.

Secondly, Table 2 gives interesting results on the response pattern in gender pairs. Expectedanswer patterns include those in which the participant gave the same answer for both sentences in agender pair. These correspond with the answer pairs that lie on the diagonal of the tables in Table 2.Otherwise, as explained earlier, the answer pair (A,B) may be expected, which indicates the use of agender-neutral plural.

We find that in less than 60% (107/180) of the responses, the same answer was given in bothsentences of a gender pair. The (A,B) answer pair is quite rare with only 5 occurrences. Two commonanswer pairs are (B,C) ( 18 ) and (D,A) ( 15 ). The answer pair (B,C) may be interpreted as ‘theparticipant judged a plural pronoun adequate in both sentences, but in the m.g. sentence he or she alsojudged a disjunctive pronoun adequate’. The (D,A) answer pair may be interpreted as ‘the participant

6

m.g.

A B C D Sum

s.g.

A 16 1 3 1 21

B 6 8 7 2 23

C 0 2 3 0 5

D 8 2 0 1 11

Sum 30 13 13 4 60

(a) Singular category

m.g.

s.g.

A B C D Sum

A 0 2 0 0 2

B 2 43 3 0 48

C 0 4 3 0 7

D 0 2 0 1 3

Sum 2 51 6 1 60

(b) Plural category

m.g.

s.g.

A B C D Sum

A 4 2 1 1 8

B 1 19 8 1 29

C 1 3 4 0 8

D 7 2 1 5 15

Sum 13 26 14 7 60

(c) Ambiguous category

m.g.

s.g.

A B C D Sum

A 20 5 4 2 31

B 9 70 18 3 100

C 1 9 10 0 20

D 15 6 1 7 29

Sum 45 90 33 12 180

(d) Grand total

Table 2: Answer pairs per category and in total

did not like the s.g. sentence, but preferred a disjunctive pronoun in the m.g. sentence’. Interestingly,both pairs burn down to the same pattern: (B,C) and (D,A) are precisely those answer pairs thatshow that a participant prefers disjunctive pronouns over singular pronouns.

The main conclusions are that there is a slight preference of disjunctive pronouns over singularpronouns (45 disjunctive pronouns as opposed to 31 singular pronouns). Furthermore, it seems thatspeakers generally like sentences with a disjunctive pronoun better than sentences with a singularpronoun, because there were more D-responses in the s.g. sentences (29 D’s in the s.g. sentences asopposed to 12 D’s in the m.g. sentences).

Participants’ remarks In total, 13 participants gave 67 remarks of some kind in the space provided(excluding remarks on filler sentences). Most remarks state an ambiguity, or say that the choicedepends on the situation or the context. The remarks most interesting for our purposes concerneddisjunctive pronouns. Two participants remarked that they connect a disjunctive pronoun with amore formal register (e.g. generated by the situation of (Q2)). Another remark stated that disjunctivepronouns sound ‘artificial’, and one participant would avoid disjunctive pronouns altogether.

3 Variable complexes

In Groenendijk and Stokhof’s (1991) framework of dynamic predicate logic (DPL), ordinary disjunction(!) is an internally and externally static connective. To deal with SA-sentences a new connective,program disjunction ("), is introduced, which is externally dynamic. I will show that this leads totechnical problems, and I propose an alternative solution that does not introduce a new connective,but instead adds a new type of variable to the syntax.

3.1 Groenendijk and Stokhof’s program disjunction

Program disjunction, denoted ", is imported from another logic into DPL, with the intended semantics!! " "" = !!" " !"". Groenendijk and Stokhof use example (4), repeated below:

(4) A professor or an assistant professor will attend the meeting of the university board. He willreport to the faculty.

7

I will raise two objections against the use of program disjunction for SA-sentences as intended byGroenendijk and Stokhof. My first objection regards the representation of a sentence like (4), claimedto look like this:

(8) [!x.P (x) " !x.Q(x)] #H(x)

Here, P , Q and H stand for the predicates ‘professor’, ‘assistant professor’ and ‘report to the fac-ulty’, respectively. Crucially, the variable x in the second conjunct is supposed to be bound by bothquantifiers in the first conjunct. This seems not possible, because it is in contradiction with claimsGroenendijk and Stokhof themselves have made earlier: “[o]nly existential quantifiers can have activeoccurrences, and for any formula, only one occurrence of a quantifier in that formula can be active”(p. 59, my emphasis). In (8) however, there are two active quantifiers for the variable x.

An additional problem I see with (8) comes from compositionality, one of the main reasons forintroducing DPL. The first part of (4) may be constructed as !x.P (x) " !y.Q(y). I do not see whythere is no ‘free choice’ of dummy variables in DPL. Of course, now there is a problem, because therecannot be a single variable binding both !x and !y.

My second objection against program disjunction concerns the dynamic qualification it has beengiven by Groenendijk and Stokhof: unlike all other connectives, it is said to be internally static, butexternally dynamic. How can a connective pass on variable bindings to disjuncts yet to come, but notfrom its left disjunct to its right disjunct? Assuming for the moment that Groenendijk and Stokhofare right, the definitions of binding pairs (bp), active quantifier occurrences (aq) and free variables(fv) for " should reflect this possibility. I come to the following:

(9) Program disjunctionbp(! " ") = bp(!) " bp(")aq(! " ") = aq(!) " aq(")fv(! " ") = fv(!) " fv(")

This seems obvious, but it is quite subtle. The first and third line reflect that " is internally static:there are no additional binding pairs between the disjuncts. These lines are the same for the corre-sponding definitions for $. The second line expresses external dynamicity. However, this cruciallydi!ers from the other externally dynamic connective, conjunction:

(10) aq(! # ") = aq(") " {!x % aq(!) | !x /% aq(")} (p. 58).

The second clause in (10) is just there to disallow double binding of the sort that is supposed to bepossible for program disjunction.

It is easy to give an example in which there is an anaphoric relation between two disjuncts, thusshowing that (9) cannot be right, and program disjunction should be internally dynamic as well:

(11) Mary or a colleague she works with on the project, will attend the meeting of the universityboard. She will report to the faculty.

In this example there is some binding between a variable x in the second disjunct and a quantifier inthe first disjunct. Because of the close similarity between (4) and (11), it would be undesirable to saythat they contain di!erent connectives.

3.2 An alternative approach: variable complexes

In order to avoid the problematic translation in (8), using program disjunction, I propose a di!erentsolution. Instead of introducing a new type of disjunction, I add some structure to the pronoun. Thishas the advantage of not having to add a new connective to the language. The additional structureon the pronoun is justified because we are dealing with an unusual kind of anaphoric binding: a singlepronoun to a disjunctive clause. For example in DRT, pronouns referring to plural antecedents arealso di!erent from ordinary pronouns.

To circumvent the problematic binding in (8), I propose that pronouns to disjunctive antecedentsare in fact variable complexes, consisting of two variables. The basic representation would then be

8

[!x.P (x) " !y.Q(y)] #H(x ! y). Here, x ! y is a variable complex formed from x and y, and it canbind to the quantifiers !x and !y. I use the notation ! to distinguish it from ordinary disjunction.Let me first give translations of two earlier sentences using !:

(4!) A professor or an assistant professor will attend the meeting of the university board. He willreport to the faculty.[!x.(professor(x) # attend(x)) " !y.(assistant-professor(y) # attend(y))] # report(x ! y)

(5!) If Mary catches a fish, or John traps a rabbit, Bill will cook it.!xy.(Mary(x) # John(y) # [!w.(fish(w) # catch(x,w)) " !z.(rabbit(z) # trap(y, z))]) $!u.(Bill(u) # cook(u,w ! z))

A variable complex x! y is a variable in the sense that it can occur inside a predicate just like simplevariables, but it is more than a simple variable in the sense that it can bind to two quantifiers. Letme make this more precise.

The set of variables is to be divided into simple variables and variable complexes. Variable com-plexes are formed out of simple variables by the following syntactic rule:

(12) When x, y are simple variables, x ! y is a variable complex.

The division between simple variables and variable complexes ensures that there is no ‘nesting’ ofvariable complexes (e.g. (x ! y) ! z). Recall that ! is not a connective, so it makes no sense to talkabout the dynamicity of !. In fact, we should now revise the term ‘internally static’ to ‘internally staticw.r.t. simple variables’ and ‘internally static w.r.t. variable complexes’. In these terms, disjunctionshould be internally dynamic w.r.t. simple variables and externally dynamic w.r.t. variable complexes.The following rule shows how variable complexes are bound:

(13)(!x.P (x) " !y.Q(y)) (x ! y)

Finally, we must specify the meaning of a predicate in which a variable complex occurs. For mostcases this is straightforward, e.g. for a two-place predicate P , the meaning of P (x ! y, z), where z isa simple variable, is !v.((v = x " v = y) # P (v, z)).

More complex cases, e.g. those which involve more than one variable complex, require a morecareful definition. Consider (6), repeated below:

(6) If Mary hasn’t seen John lately, or Ann misses Bill, she calls him.

This involves a predicate call(x!y,w!z). We do not want to include the possibility that, for example,call(x, z) holds (Mary calls Bill). Therefore the order between the various variable complexes shouldbe maintained. The general rule for an n-place predicate with k variable complexes can be stated asfollows:

(14) Let P (t1, . . . , tn) be an n-place predicate such that tj is a variable complex if and only if j % K,with K = {i1, . . . , ik} & {1, . . . , n}. Write the variable complexes ti1 , . . . , tik as x1!y1, . . . , xk!yk, respectively. Then P (t1, . . . , tn) means

!v1 . . . vk.

!!

k"

i=1

(vi = xi) "k"

i=1

(vi = yi)

#

# P (u1, . . . , un)

#

, where u! =

$

t! if ! /% K;

v! if ! % K.

A similar rule is required when one wants ‘non-elliptical’ translations of SA-sentences. Observe that(4!) is an ‘elliptical’ translation in that attend occurs twice. Alternatively, one could choose a moredirect translation as follows3:

(15) [!x.professor(x) " !y.assistant-professor(y)] # attend(x ! y) # report(x ! y)

Again, an ordering rule should ensure that ‘attend(x)# report(y)’ does not occur. Without a direct

3I am indebted to an anonymous reviewer for this point.

9

advantage for non-elliptical translations, I would suggest to stick with elliptical translations, thusavoiding to add another ordering rule.

3.3 Variable complexes and disjunctive pronouns

The main argument to use the same formal translation for singular and disjunctive pronouns is theobservation that they are interpreted in the same way. Singular and disjunctive pronouns both have asingular interpretation as their default reading. Also observe that the existence of disjunctive pronounsis a result of English morphology, and that the (informal) logical form of the following sentences isexactly the same:

(16) a. If a student likes semantics, he or she should come to this workshop. (=(7a))b. If a man likes semantics, he should come to this workshop.

With these observations in mind, it is not di!cult to shift the use of x ! y for singular pronouns topronouns with a singular interpretation (i.e. including disjunctive pronouns). In fact, it seems verynatural to me to represent ‘he or she’ by x! y: x and y correspond to the internal pronouns ‘he’ and‘she’, and the use of ! instead of ! reflects our intuition that the disjunction is not descriptive (i.e.that there should be no ordinary ! in the LF).

My experimental results show a slight preference of disjunctive pronouns over singular pronounsin gender pairs (recall the relatively high number of (D,A) and (B,C) answer pairs, as well as theabsolute di"erence 45/31 in Table 2(d)). This may suggest that there is a (formal) di"erence betweensingular and disjunctive pronouns, and therefore count as an argument against the suggestions I justmade. However, I believe that the observed di"erence can be explained in another way. I see twofactors that may cause a preference for disjunctive pronouns. Firstly, as some participants of myexperiment have reported, disjunctive pronouns are sometimes associated with a more formal register.It is plausible that the di"erence in surface form and the stylistic di"erence cause speakers to preferdisjunctive pronouns in certain cases. Secondly, and more importantly, recall that the vast majority ofdisjunctive pronouns is used in combination with simple antecedents (as in (7a)). It is therefore veryplausible that the common use of (7a) has as a consequence that speakers use disjunctive pronounsmore easily in SA-sentences.

The idea of variable complexes gives rise to conjecturing a further correspondence between singularand disjunctive pronouns. For one might say that there is also a syntactic correspondence, in the sensethat singular pronouns are syntactically of the form ‘he or he’, ‘she or she’ and so forth. For economyreasons we do pronounce these forms as ‘he’ and ‘she’, respectively4. This idea has the advantage thatx! y has a correlate in syntax. This solves for example the problem that in the formal language, ‘sheor he’ is equally well possible as ‘he or she’ (x ! y and y ! x). Once we can explain why the syntaxdoes not (or rarely) output ‘she or he’ (e.g. for lexical or sociolinguistic reasons), by the syntacticcorrelate, it follows that ‘she or he’ does not appear in the formal semantics either.

3.4 Plural pronouns

When a plural pronoun is used with a split antecedent, it can take either a singular or a pluralinterpretation. The latter case is the most common: with this use the plural pronoun just refers to thesummation of the disjuncts. The case of singular interpretation should be divided into two subcases,corresponding to the same gender / mixed gender di"erence. I will now discuss these three casesseparately.

Plural interpretation In the majority of cases a plural pronoun is used in its usual manner, i.e.it refers to x" y, the summation of the two disjuncts x and y. My experiment shows that especiallywhen a plural reading is natural, speakers frequently use plural pronouns in this way.

4The reader should keep the direction of the argument in mind. I am not claiming that singular pronouns shouldhave this form and should thus be translated as x ! y. Rather, from technical problems with !, I proposed variablecomplexes for singular pronouns, and later applied them to disjunctive pronouns as well. Now ‘reasoning backwards’, Isuggest that singular and disjunctive pronouns have the same underlying syntactic form.

10

Various dynamic theories for plurals have been proposed (see e.g. van der Berg 1996, Ogata 2002).For example, Ogata (2002:275) uses a DRT-style approach with plural variables X which denote setsof individual variables. Any such theory should be able to deal with this case, as long as the disjunctsare available for summation.

Singular interpretation The subcase of a singular interpretation with a mixed-gender antecedentis the easier one. This is an instance of a plural being used as a gender-neutral pronoun, comparableto (7b). The results of my experiment did not show a frequent use of a social plural, but they did notrefute it either. Answer pairs (A,B) and (A,C) which correspond best with the use of a social pluralwere rare (5 and 4 occurrences in total, respectively). Answer pairs (B,B), however, which occurredmore frequently, may include cases in which a speaker chose a plural pronoun because of the m.g.antecedent, but because of the plural reading in the s.g. sentence.

When plural pronouns are used in this way, I do not think they cause major technical problems.Gender-neutral plurals are just an additional strategy for referring to antecedents which are not spec-ified for gender, and the plural pronoun can be translated by the usual variable complex, used forsingular interpretations. Any explanation for how gender-neutral plurals are used for singular, simpleantecedents would also apply to the similar use of plural pronouns with split antecedents.

The final case, combining a singular interpretation and a single-gender antecedent, is the most di!-cult one. In a simple example like (2), repeated below, the plural pronoun seems to have a singularinterpretation, i.e. it has the same meaning as when a singular pronoun would be used at that place(note that we cannot replace ‘them’ by ‘John and Bill’):

(2) %If I see John or Bill, I will wave to them.

While Wang (2005:473) claims that not all speakers like (2), my experiment showed that speakers douse plural pronouns in this way: in the singular category, for s.g. sentences, 23 plural pronouns werepreferred as opposed to 21 singular pronouns.

On the other hand, it is clear that a plural pronoun with a singular interpretation is not alwayspossible. So, this use of plural pronouns is still restricted in some way. I have no full explanation forhow speakers use them, but I guess that it has something to with the fact that in uttering (2), I amasserting something about two individuals: for both John and Bill I assert that I will wave to themwhen I see them. This inferred ‘conjunction’ may elicit a plural pronoun.

For a formal translation, one may simply translate the plural pronoun by a variable complex andobtain the singular interpretation in that way. There is no obvious syntactic relation between x ! yand ‘them’, but this also depends on a syntactic account on the nature of these plural pronouns.

Summarizing, I claim that we do not need additional theory to deal with plural pronouns. Theyeither have a singular interpretation, in which case variable complexes can be used, or they havea plural interpretation, in which summation can be used. The latter is readily available in varioustheories.

4 Conclusion

The results of my experiment show that plural pronouns are used much more frequently than singularpronouns in SA-constructions. In a lesser degree, also disjunctive pronouns are preferred to singularpronouns in the same sentences. Also, sentences with a singular reading were more often judged inap-propriate than sentences with a plural reading or with a mixed-gender antecedent (more D responses).These findings are remarkable, given the literature’s strong inclination to singular pronouns.

I discussed technical problems with Groenendijk and Stokhof’s (1991) program disjunction ap-proach to PSA. As an alternative, I proposed to use variable complexes, which are formed by com-bining two simple variables. A variable complex can bind to two di"erent quantifiers. This solves thetechnical problems, but variable complexes can also be used for disjunctive pronouns. This has thefurther advantage that there is a syntactic correlate between variable complexes and pronouns. This

11

syntactic correlate may be extended to singular pronouns, in saying that they have the underlyingform ‘he or he’ etc.

For plural pronouns, I distinguished three di!erent uses, for none of which additional formaltheory is necessary. The major case is the use of plural pronouns with a plural interpretation, so thatsummation of the disjuncts can be used. The more di"cult cases are those in which a plural pronountakes a singular interpretation. My experiment gives evidence that this usage actually occurs, but itis unclear in what way it is restricted: in which cases do speakers use plural pronouns in this way andhow can we explain this distribution? I leave this problem for further research.

Future work I have investigated other aspects of PSA in Tellings (2010), in particular the morepragmatic side of the problem, concerning di!erent readings SA-sentences can have and their influenceon anaphora. Nevertheless, I think more work can be done on this problem. Both theoretical andexperimental work is welcome in the area of PSA, especially on disjunctive and plural pronouns.Experiments may focus on other factors that influence the availability of anaphora to split antecedents,or they may investigate how plural pronouns with a singular interpretation are used. Theoreticalquestions may concern n-ary disjunction, or more di"cult cases like VP-disjunction.

References

van der Berg, M.H. (1996). Some Aspects of the Internal Structure of Discourse. The Dynamics ofNominal Anaphora. Ph.D. thesis, University of Amsterdam. URL http://www.mhvdberg.com/

publications/dissertation.pdf.

Elbourne, P.D. (2005). Situations and Individuals. Cambridge, MA: MIT Press.

Groenendijk, J. and Stokhof, M. (1991). Dynamic predicate logic. Linguistics and Philosophy 14(1),39–100. doi: 10.1007/BF00628304.

Kamp, H. and Reyle, U. (1993). From Discourse to Logic. Dordrecht: Kluwer.

Ogata, N. (2002). Dynamic Semantics of Plurals DPL!

Q. Electronic Notes in Theoretical ComputerScience 67, 263–283. doi: 10.1016/S1571-0661(04)80553-X.

Partee, B.H. and Rooth, M. (1983). Generalized conjunction and type ambiguity. In R. Bauerle,C. Schwarze and A. von Stechow (eds.) Meaning, use and interpretation of language, 361–383.Berlin: Walter de Gruyter.

Rooth, M. and Partee, B.H. (1982). Conjunction, Type Ambiguity, and Wide Scope “Or”. InD. Flickinger, M. Macken and N. Wiegand (eds.) Proceedings of the 1982 West Coast Conferenceon Formal Linguistics. Stanford University.

Schlenker, P. (2010). Singular Pronouns with Split Antecedents. URL https://files.nyu.edu/

pds4/public/Disjunctive_Antecedents.pdf, unpublished ms.

Stone, M.D. (1992). Or and Anaphora. In C. Barker and D. Dowty (eds.) SALT II: Proceedingsfrom the Conference on Semantics and Linguistic Theory, 367–385. Columbus, OH. URL http:

//eric.ed.gov/ERICWebPortal/contentdelivery/servlet/ERICServlet?accno=ED352828.

Tellings, J.L. (2010). Disjunction and anaphora in dynamic semantics. Master’s thesis, University ofCambridge. Available from the author.

Wang, L.I.C. (2005). Dynamics of plurality in quantification and anaphora. Ph.D. thesis, the Universityof Texas at Austin. URL www.lib.utexas.edu/etd/d/2005/wangl66171/wangl66171.pdf.

12

Dynamics of Implicit and Explicit Beliefs

Fernando R. Velazquez-Quesada

Abstract

The dynamic turn in epistemic logic is based on the idea that notionsof information should be studied together with the actions that modifythem. Dynamic epistemic logics have explored how knowledge andbeliefs change as consequence of, among others, acts of observationand revision. Nevertheless, the omniscient nature of the representedagents has kept finer reasoning acts, like inference, outside the picture.

Following proposals for representing non-omniscient agents, re-cent works have explored how implicit and explicit knowledge change asa consequence of acts of inference, consideration and even forgetting.The present work proposes a next step towards a common frameworkfor representing finer notions of information and their dynamics. Wepropose a model for representing not only implicit and explicit beliefs,but also how these notions change as consequence of acts of revision,inference and retraction.

1 Introduction

Epistemic logic (16) and its possible worlds semantics is a powerful andcompact framework for representing an agent’s information. Their dynamicversions (10) have emerged to analyze not only information in its knowledgeand belief versions, but also the actions that modify them. Nevertheless,agents represented in this framework suffer from the so called logical om-niscient problem: their information is closed under logical consequence,therefore hiding finer reasoning acts, like inference.

Based on the awareness approach of (14), recent works have exploreddynamics for non-omniscient agents. In particular, they have studied howacts of observation, inference, consideration and forgetting affect the notionsof implicit and explicit knowledge (3; 15; 4; 9). The present work follows theprevious ones, now focussing on the notion of beliefs. Combining worksfrom the existing literature, we propose a framework for representing anagent’s beliefs in its implicit and explicit versions. Then, we show how torepresent not only the already studied act of revision (also called upgrade),but also the acts of inference (on beliefs) and retraction.

1

2 Modelling implicit and explicit beliefs

This section recalls a framework for implicit and explicit information and a

framework for beliefs. We will combine them to get our static model.

Epistemic logic. The frameworks of this section are based on that of epis-temic logic (EL). The EL language extends the propositional one with formu-

las of the form�ϕ: “the agent is informed aboutϕ”. Semantically, EL-formulas

are evaluated in Kripke models: tuples M = �W,R,V�where W is a non-empty

set of possible worlds, V is an atomic valuation function and R is an ac-

cessibility relation, indicating which worlds the agent considers possible.

Boolean connectives are interpreted as usual, and �ϕ is true at a world wiff ϕ is true in all the worlds the agent considers possible from w.

Non-omniscient agents. The formula � (ϕ→ ψ) → (�ϕ → �ψ) is valid

in Kripke models: the agent is informed about all logical consequences of

her information. One of the most influential solutions to this omniscienceproblem is awareness logic (14), which extends the EL language with formulas

of the form Aϕ (“the agent is aware ofϕ”), and Kripke models with a function

A that assigns a set of formulas to the agent in each possible world (Aϕ is

true at w iff ϕ ∈ A(w)). They key point is the difference between implicit and

explicit information. The agent is implicitly informed about ϕ iff ϕ is true

in all the worlds she considers possible (�ϕ), but to be explicitly informed

about ϕ the agent also needs to be aware of it (�ϕ ∧Aϕ).

With a different interpretation of the A-sets, the models of (11; 12) go

one step further, adding relations between worlds representing changes in

information. Then, (17; 18) extend them by assigning to the agent not only

formulas but also rules. Now we have an agent that is non-omniscient, but

also able to perform inferences.

Representing beliefs. Plausibility models (5; 2; 1), discussed in (19; 6)

under a different name, are Kripke models in which the accessibility relation,

≤, is asked to be a well-preorder: a reflexive, transitive relation such that, for

every non-empty U ⊆W, U has ≤-maximal elements.1 These models allow

us to represent beliefs following this intuition: we believe what is true in

the situations we consider most likely to be the case.

2.1 Combining the models

Our static framework combines the mentioned ideas.

Definition 2.1 (Language L). Given a set of atomic propositions P, formulas

ϕ and rules ρ of the plausibility-access language L are given, respectively, by

ϕ ::= p | Aϕ | Rρ | ¬ϕ | ϕ ∨ ψ | [∼]ϕ | [≤]ϕρ ::= ({ϕ1, . . . ,ϕnρ},ψ)

1From an observation of (1), a well-preorder is a connected conversely well-founded preorder.

2

where p ∈ P. Formulas of the form Aϕ are read as “the agent acknowledges theformulaϕ as true” and those of the form Rρ are read as “the agent acknowledgesrule ρ as truth-preserving”. For the modalities, [∼]ϕ is read as “ϕ is true in allthe worlds the agent cannot distinguish from the current one” and [≤]ϕ is readas “ϕ is true in all the worlds the agent considers more plausible than the currentone”. Other boolean connectives as well as the diamond modalities �∼� and�≤� are defined as usual. We denote by L f the set of formulas of L, and byLr its set of rules.

We emphasize that a rule ρ is a tuple consisting of a finite set of formulas,its premises pm(ρ), and a single one, its conclusion cn(ρ). A rule could havebeen defined in a simpler way, as an implication tr(ρ) (the rule’s translation)whose antecedent is the (finite) conjunction of ρ’s premises and whoseconsequent is ρ’s conclusion, or in a more general form, with its premisesgiven by an ordered sequence or a multi-set, allowing us to deal with sub-structural logics where order and multiplicity matters. Our choice fits betterwith the notion of truth-preserving inference, to be explored in Section 3.2.Sometimes we will write a rule ρ as pm(ρ)⇒ cn(ρ).

Definition 2.2 (Plausibility-access model). Let P be a set of atomic proposi-tions. A plausibility-access model is a tuple M = �W,≤,V,A,R�where �W,≤,V�is a plausibility model over P and

• A : W → ℘(L f ) is the access set function, assigning to the agent a set offormulas of L in each possible world,

• R : W → ℘(Lr) is the rule set function, assigning to the agent a set ofrules of L in each possible world.

Functions A and R can be seen simply as valuation functions with a partic-ular range, assigning a set of formulas and a set of rules to the agent at eachpossible world. Moreover, the relation ∼, defined as the union of ≤ and itsconverse (∼ :=≤ ∪ ≥), represents the agent’s indistinguishability relation.

Here it is important to emphasize our interpretation of the A-sets. Dif-ferent from (14) and (4), we do not interpret A(w) as “the formulas the agentis aware of at world w”, but rather as “the formulas the agent has acknowledgedas true at world w”, closer to the ideas in (11; 12; 17; 18; 15). While the firstinterpretation is “a matter of attention, and does not imply any attitude pro orcon” (4), the second one goes beyond simple consideration, and indicates apositive attitude towards the formulas.

Definition 2.3 (Semantic interpretation). Let M = �W,≤,V,A,R� be a plau-sibility-access model, and take a world w ∈ W. Atomic propositions andboolean operators are interpreted as usual. For the remaining cases,

3

(M,w) � Aϕ iff ϕ ∈ A(w)

(M,w) � Rρ iff ρ ∈ R(w)

(M,w) � [≤]ϕ iff for all u ∈W, w ≤ u implies (M,u) � ϕ(M,w) � [∼]ϕ iff for all u ∈W, w ∼ u implies (M,u) � ϕ

In order to talk about what happen at the most plausible worlds, the

formula [≤]ϕ is too strong. It holds at a world w when ϕ is true in all theworlds that are more plausible than w, and that includes not only the most

plausible ones, but also those laying between them and w. As observed

in (7; 20; 1), the sequence of modalities �≤� [≤] does the work: �≤� [≤]ϕ is

true at w iff ϕ holds in the most plausible worlds from ϕ. Our definitions

of implicit and explicit beliefs, combining the mentioned ideas with those

from (15) and (4), are shown in Table 1.

The agent implicitly believes formula ϕ BImϕ := �≤� [≤]ϕ

The agent explicitly believes formula ϕ BExϕ := �≤� [≤]

�ϕ ∧Aϕ

�

The agent implicitly believes rule ρ BImρ := �≤� [≤] tr(ρ)

The agent explicitly believes rule ρ BExρ := �≤� [≤]

�tr(ρ) ∧ Rρ

�

Table 1: Implicit and explicit beliefs about formulas and rules.

Observe the definitions. The agent believes a formula ϕ implicitly iff ϕis true in the most plausible worlds, but in order to believe it explicitly, the

agent should also acknowledge ϕ as true in these worlds. For the case of

rules, the agent believes a rule ρ implicitly iff its translation is true in the

most plausible worlds (that is, iff ρ preserves truth), but in order to believe

ρ explicitly she should also acknowledge that the rule is truth-preserving.

Note also the result of the interpretation of the A-sets. While an agent

in (14) and (4) is non-omniscient due to lack of attention (she does not

need to be aware of the possibility of ϕ), our agent has full attention, but

still she is non-omniscient because she fails to recognize some formulas as

true (she does not need to be aware that ϕ is the case). This may seem a

small difference, but in fact the interpretation of the A-sets determines the

reasonable operations over them, as we briefly discuss at the beginning of

Section 3.2.

3 Dynamics of implicit and explicit beliefs

We have a framework for representing implicit and explicit beliefs. We now

look at their dynamics by introducing actions that modify them.

4

3.1 Explicit upgrade

The χ-upgrade operation (5; 2; 1) modifies the plausibility relation ≤ to putthe χ-worlds at the top, therefore changing (revising) the agent’s beliefs. Inour setting this operation comes in two flavors, depending on whether italso adds χ to the A-sets (explicit upgrade) or not (implicit upgrade). Here isthe definition of the first one.

Definition 3.1 (Explicit upgrade). Let M = �W,≤,V,A,R� be a plausibili-ty-access model and χ a formula in L. The model Mχ⇑+ = �W,≤�,V,A�,R�differs from M in the plausibility relation and in the access set function:

≤� := (≤;χ?) ∪ (¬χ?;≤) ∪ (¬χ?;∼;χ?)A�(w) := A(w) ∪ {χ} for every w ∈W

The new plausibility relation is given in a dynamic logic style. It statesthat, after an upgrade with χ, “all χ-worlds become more plausible than all¬χ-worlds, and within the two zones, the old ordering remains” (2). Of course,there are other definitions for a new plausibility relation that put χ-worldsat the top while preserving the required properties of the relation (2; 1; 13).The presented one, radical upgrade, only shows one of many possibilities.

In order to express the effect of an explicit upgrade over the agent’sbeliefs, we extend the language with formulas of the form �χ ⇑+�ϕ: “itis possible to perform an explicit χ-upgrade after which ϕ holds”. There is noprecondition for this action (the agent can perform an explicit upgradewhenever she wants), so the semantic interpretation is as follows.

Definition 3.2. Let M = �W,≤,V,A,R� be a plausibility-access model, andtake a world w ∈W:

(M,w) � �χ⇑+�ϕ iff (Mχ⇑+ ,w) � ϕ

Note how the explicit upgrade operation is a total function: it can alwaysbe executed (i.e., there is no precondition) and always yields one and onlyone model. Then, the semantic interpretation of the ‘box’ explicit upgrademodality, defined as [χ⇑+]ϕ := ¬�χ⇑+�¬ϕ, collapses to

(M,w) � [χ⇑+]ϕ iff (Mχ⇑+ ,w) � ϕ

Note how an explicitχ-upgrade puts at the top of the plausibility relationthose worlds where χ holds in M, but such worlds do not need to makeχ true in Mχ⇑+ . In other words, an explicit χ-upgrade does not necessarilymake the agent believe χ explicitly. This is because, besides beliefs aboutfacts, our agent also has high-order beliefs (i.e., beliefs about beliefs andso on) which can change after an upgrade. Nevertheless, if χ is purelypropositional, the worlds satisfying it at M also satisfy it at Mχ⇑+ ; then, afteran explicit χ-upgrade, the agent will believe χ implicitly. Moreover, sinceχ is added to the A-sets, the agent will believe χ explicitly.

5

3.2 Inference

Non-omniscient beliefs can be modified in other ways. In particular, beliefs

can be extended by making explicit what is already implicit, that is, the

agent can use inference. But it is a stretch to assume that an inference

allows the agent to acknowledge as true any formula at any time. It is more

reasonable to assume that the agent can do it when she has a justification;

here is where rules play a role.

Our inference operation, defined below, extends the agent’s explicit be-

liefs according to the following intuitive idea:

if the agent believes explicitly a rule and all its premises, then she can performan inference with that rule after which she will explicitly believe its conclusion.

Definition 3.3 (Inference on beliefs). Let M = �W,≤,V,A,R� be a plausibili-

ty-access model and σ a rule in L. The model M�→Bσ= �W,≤,V,A�,R� differs

from M just in the access set function, given for every w ∈W as

A�(w) :=

�A(w) ∪ {cn(σ)} if pm(σ) ⊆ A(w) and σ ∈ R(w)

A(w) otherwise

The operation adds σ’s conclusion to those worlds where the agent

already has σ and its premises. Since it does not modify the plausibility

relation, the operation preserves models in the intended class.

Just as before, we extend the language, this time with formulas of the

form ��→Bσ�ϕ: “it is possible to perform a belief inference with σ after which

ϕ holds”. This time, there is a precondition: in order to perform a belief

inference, the agent needs to believe explicitly both the rule and its premises.

Definition 3.4. Let M = �W,≤,V,A,R� be a plausibility-access model, and

take a world w ∈W:

(M,w) � ��→Bσ�ϕ iff (M,w) � BExσ ∧ BExpm(σ) and (M�→B

σ,w) � ϕ

Just like explicit upgrade, the inference operation is functional. Different

from it, inference is not total: there is a precondition that should be satisfied

for the operation to take place. The semantic interpretation of the ‘box’

modality for the inference operation, [�→Bσ]ϕ := ¬��→B

σ�¬ϕ, collapses to

(M,w) � [�→Bσ]ϕ iff (M,w) � BExσ ∧ BExpm(σ) implies (M�→B

σ,w) � ϕ

It is not difficult to see that we have the following validity:

�BExσ ∧ BExpm(σ)

�→ [�→B

σ] BExcn(σ)

In words, “if the agent believes explicitly a rule and its premises, then any appli-cation of the rule will make her believe explicitly the rule’s conclusion”.

6

Here is a good idea to make a brief comparison with other modal frame-works representing inference. The works of Duc (11; 12) and Jago (17; 18)also use models that assign a set of formulas to the agent in each possibleworld, with Jago also assigning rules. The difference with respect to ourproposal is the definition of explicit information (knowledge, in their case)and the representation of inference. They define explicit knowledge, in ourterminology, as what the agent has in her A-set (Aϕ). Then, they representinference with a relation for each rule, asking for each relation to be faithfulto the rule’s intuition: for every rule ρ, a world u is Rρ-reachable from aworld w iffw contains the rule’s premises (in Jago’s case, also the rule) and uextends w with ρ’s conclusion. Another important detail is that these workslimit the agent’s implicit and explicit knowledge to propositional formulas.

A first dynamic epistemic attempt is (21), where inference is defined as anoperation over models, still using the same definition for explicit knowledgeand still limited to propositional information. Then, in (15), the authorsdefine explicit knowledge by looking at the A-sets in all the indistinguishableworlds, also allowing the agent to have high-order implicit and explicitknowledge. Our work follows them, defining explicit beliefs by looking atthe A-sets in the most plausible worlds, and defining inference (on beliefs)as a model operation.

3.3 Retraction

There are situations in which the agent simply retracts some explicit belief,that is, she decides that she will not believe in it anymore. There maybe many reasons for this, from a simple sudden lack of confidence on thesource of the belief to a more interesting inference-dual scenario, suggestedby an anonymous referee, in which an agent believes a rule and its premises,but instead of accepting the rule’s conclusion she decides to drop some ofthe premises. In our framework, the difference among these possibilities isjust the precondition for the operation, but the effect is achieved in all thecases by simply removing the chosen formula from the A-sets.

Definition 3.5 (Retraction). Let M = �W,≤,V,A,R� be a plausibility-accessmodel and χ a formula in L. The model M−χ = �W,≤,V,A�,R� differs fromM just in the access set function, given for every w ∈W as

A�(w) := A(w) \ {χ}Again, the operation preserves models in the intended class.

Technically, this operation is similar to the forgetting operation of (8)and the dropping operation of (4). We just emphasize that, given our inter-pretation of A-sets, removing a formula from the sets does not mean that theagent becomes unaware of ϕ; it means that the agent stop acknowledgingthe formula as true.

7

These actions are represented in the language by formulas of the form�−χ�ϕ, read as “it is possible to retract χ and after it ϕ holds”. Just like anupgrade, the agent can retract a formula in any situation.

Definition 3.6. Let M = �W,≤,V,A,R� be a plausibility-access model, andtake a world w ∈W:

(M,w) � �−χ�ϕ iff (M−χ,w) � ϕ

Just like the explicit upgrade case, the semantic interpretation of the ‘box’modality for the retraction operation, [−χ]ϕ := ¬�−χ�¬ϕ, collapses to

(M,w) � [−χ]ϕ iff (M−χ,w) � ϕ

The following validity shows that our definition behaves as we intend:

�−χ�¬BExχ

In words, “after retracting χ, the agent does not believe it explicitly anymore”.

3.4 A brief example with the language

Suppose you believe explicitly that Chilly Willy is a bird (b) and it isnot a penguin (¬p): BEx(b ∧ ¬p). You also believe explicitly the rule σ,which says that if Chilly Willy is a bird and not a penguin, then it flies (f):BEx�(b ∧ ¬p)⇒ f

�. Then, you can make an inference step to make explicit

the implicit belief “Chilly Willy flies”:�BEx�(b ∧ ¬p)⇒ f

�∧ BEx(b ∧ ¬p)

�→ [�→B

(b∧¬p)⇒ f ] BEx f

But now suppose that a reliable (but still fallible) source informs you thatChilly Willy is a penguin. First, you can retract your belief “Chilly Willyflies”, but you should also retract your “Chilly Willy is not a penguin” belief.Then, you can explicitly upgrade your beliefs, making you believe explicitlythat Chilly Willy is a penguin.

BEx f ∧ �− f ��¬BEx f ∧ �−¬p� �¬BEx¬p ∧ �p⇑+�BExp

��

4 Conclusions and further work

We have provided a framework for representing implicit and explicit beliefs.We have also defined three actions that modify them: upgrade, inference andretraction. We have not discussed axiom system, but we can get one for thestatic part by merging those of the frameworks in which it is based. For thedynamic part, we can follow a standard dynamic epistemic logic techniqueand provide reduction axioms.

8

Some questions arise. Among them, (1) we would like a more sys-tematic discussion about the effects of these operations: how are currentimplicit/explicit beliefs affected? What does the agent need to believe im-plicitly/explicitly something after each one of them? (2) A multi-agenttreatment is in order.

Other interesting questions have been posed by an anonymous referee.(3) Our agent is not only non-omniscient; in general she also does not havea complete set of reasoning tools in the sense that what she can eventu-ally derive does not need to cover everything that follows logically fromher explicit beliefs. The former could be defined by a ‘Kleene-start’ thatcomputes the ‘derivable’ closure of the agent’s explicit beliefs, and it wouldbe interesting to look at their properties. (4) The acts of explicit upgrade,inference and retraction deal with formulas. What about rules? In particu-lar, the agent can retract a rule, but she can also infer new ones from thoseshe already has. In (21), the author proposes to use structural rules, likemonotonicity or cut, in order to extend the rules the agent knows. Whichare the adequate structural operations for the case of beliefs?

And there are more interesting questions. First, by combining thisframework with previous ones for dynamics (observation and inferences)of implicit and explicit knowledge, we get an agent with a richer set ofreasoning abilities. For example, what do we get from an inference with abelieved rule and known premises? And from an inference with a knownrule but only believed premises? Second, the Chilly-Willy example of be-fore is actually a classical one about non-monotonic reasoning. What is theprecise relation between dynamics of implicit and explicit beliefs and non-monotonic reasoning? In particular, what is the relation between inferenceon beliefs and default reasoning?

Acknowledgements The author thanks three anonymous referees for theircomments and suggestions.

References

[1] A. Baltag and S. Smets. A qualitative theory of dynamic interactivebelief revision. In G. Bonanno, W. van der Hoek, and M. Wooldridge,editors, Logic and the Foundations of Game and Decision Theory (LOFT7),volume 3 of Texts in Logic and Games, pages 13–60. Amsterdam Univer-sity Press, 2008.

[2] J. van Benthem. Dynamic logic for belief revision. Journal of Applied

Non-Classical Logics, 17(2):129–155, 2007.

[3] J. van Benthem. Merging observation and access in dynamic logic.Journal of Logic Studies, 1(1):1–17, 2008.

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[4] J. van Benthem and F. R. Velazquez-Quesada. Inference, promotion,and the dynamics of awareness. Technical Report PP-2009-43, Insti-tute for Logic, Language and Computation (ILLC), Universiteit vanAmsterdam (UvA), 2009.

[5] O. Board. Dynamic interactive epistemology. Games and Economic

Behavior, 49(1):49–80, October 2004.

[6] C. Boutilier. Conditional logics of normality: A modal approach. Ar-

tificial Intelligence, 68(1):87–154, 1994.

[7] C. Boutilier. Toward a logic for qualitative decision theory. In J. Doyle,E. Sandewall, and P. Torasso, editors, KR 94, pages 75–86, Bonn, Ger-many, May 1994. Morgan Kaufmann.

[8] H. van Ditmarsch and T. French. Awareness and forgetting of factsand agents. In Proceedings of the 2009 IEEE/WIC/ACM International Joint

Conference on Web Intelligence and Intelligent Agent Technologies (WI-IAT

2009), Milan, 2009.

[9] H. van Ditmarsch, A. Herzig, J. Lang, and P. Marquis. Introspectiveforgetting. Synthese (Knowledge, Rationality and Action), 169(2):405–423,July 2009.

[10] H. van Ditmarsch, W. van der Hoek, and B. Kooi. Dynamic Epistemic

Logic, volume 337 of Synthese Library Series. Springer, 2007.

[11] H. N. Duc. Reasoning about rational, but not logically omniscient,agents. Journal of Logic and Computation, 7(5):633–648, 1997.

[12] H. N. Duc. Resource-Bounded Reasoning about Knowledge. PhD thesis,Institut fur Informatik, Universitat Leipzig, Leipzig, Germany, 2001.

[13] J. van Eijck and Y. Wang. Propositional dynamic logic as a logic ofbelief revision. In W. Hodges and R. J. G. B. de Queiroz, editors,WoLLIC, volume 5110 of Lecture Notes in Computer Science, pages 136–148. Springer, 2008.

[14] R. Fagin and J. Y. Halpern. Belief, awareness, and limited reasoning.Artificial Intelligence, 34(1):39–76, 1988.

[15] D. Grossi and F. R. Velazquez-Quesada. Twelve Angry Men: A study onthe fine-grain of announcements. In X. He, J. F. Horty, and E. Pacuit,editors, LORI, volume 5834 of Lecture Notes in Computer Science, pages147–160. Springer, 2009.

[16] J. Hintikka. Knowledge and Belief: An Introduction to the Logic of the Two

Notions. Cornell University Press, Ithaca, N.Y., 1962.

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[17] M. Jago. Rule-based and resource-bounded: A new look at epistemic

logic. In T. Ågotnes and N. Alechina, editors, Proceedings of the Work-shop on Logics for Resource-Bounded Agents, organized as part of the 18thEuropean Summer School on Logic, Language and Information (ESSLLI),pages 63–77, Malaga, Spain, August 2006.

[18] M. Jago. Epistemic logic for rule-based agents. Journal of Logic, Languageand Information, 18(1):131–158, 2009.

[19] P. Lamarre. S4 as the conditional logic of nonmonotonicity. In J. F. Allen,

R. Fikes, and E. Sandewall, editors, KR 91, pages 357–367, Cambridge,

MA, USA, April 1991. Morgan Kaufmann.

[20] R. Stalnaker. On logics of knowledge and belief. Philosophical Studies,

128(1):169–199, March 2006.

[21] F. R. Velazquez-Quesada. Inference and update. Synthese (Knowledge,Rationality and Action), 169(2):283–300, July 2009.

11

Wunder and Stone

An Empirical Model of Strategic Dialoguein Group Decisions with Uncertainty

Michael WunderRutgers University

mwunder@cs.rutgers.edu

Matthew StoneRutgers University

matthew.stone@rutgers.edu

Abstract. We explore the dialogue implications of a strategic voting game witha communication component and the resulting dialogue from lab experiments. Thedata reveal the role of communication in group decision making with uncertaintyand personal biases. The experiment creates a conflict in human players betweenselfish, biased behavior and beneficial social outcomes. Unstructured chats show-case this conflict as well as issues of trust, deception, and desire to form agreement.

1 Introduction

Communication has a long history within game theory. From the

earliest examples of signaling games [Lewis1969] to their most re-

cent acceptance as an essential component of interacting computer

agents [Shoham and Leyton-Brown2009], the exchange of information al-

ters both strategies and incentives. Examples of cheap talk [Aumann1990,

Aumann and Hart2003], common in matrix-type games, can influence the

outcome under certain circumstances. Natural dialogue shows a human re-

sponse to problem-solving situations.

Decision theory has also influenced linguistics. Phenomena such as the

maxim of quantity [Grice1975] are rooted in the tradeoff between level of

detail and efficiency in cooperative language. [Koit and Oim2000] presents

a model similar to Balance-Propose-Dispose [DiEugenio et al.2000] whereby

actors take communicative action when it is expected to increase pleasant

feelings or decrease unpleasant ones.

Previous work has investigated information sharing among networks of

individuals with partial information [Choi et al.2005] or a personal bias to-

1

An Empirical Model of Strategic Dialogue in Group Decisions with Uncertainty

wards sending misinformation [Wang et al.2009]. These experiments use the

given task to explore how actors reason about the game.

In this work we present strategic heuristic models by way of a task for a

small group to make a simple binary decision with both of the above forces

at work. Given noisy private signals about a risky policy, players must de-

cide whether to take the risky route. Players may be biased toward one

outcome. Before the vote, there is an opportunity to chat with others using

natural language. The data provide insight into the various mechanisms of

persuasion and trust people bring to such games. Our insights into sub-

jects’ information and decision making offer a complementary perspective

to previous studies of collaborative dialogue such as [DiEugenio et al.2000].

There has been some investigation into the theory linking economic

principle to indirect speech [Pinker et al.2008], although it lacks empiri-

cal grounding. Building on previous findings in linguistics and behavioral

economics, we propose an empirically based model motivated by limited

decision-theoretic reasoning which a bounded rational agent could use to

play these kinds of games like people do.

2 Experiment: Voting Game with Chats

In a series of behavioral economics experiments, five players assigned to

a group were told to decide between two collective actions with different

payouts. Two different scenarios were set up.

• Game 1: One Risky Policy, Vote Yes or No. Policy succeeds with

probability p. If a group votes yes, it gets average score p. If a majority

votes no all get 0.5.

• Game 2: Two Opposing Policies: A or B. If a group votes A, it gets

average score p and otherwise 1− p.

All players i receive a noisy signal si that nobody else sees, drawn from

a distribution centered on p. They converse with other players through

anonymous chat boxes, and then are asked to vote on the policy. In addition

to si, players are given a bias, a personal payoff adjustment ppai that shifts

the amount a player receives when the policy is passed.

While a true optimized strategy in this complex task might consider a

multi-party multi-round game, practically speaking each actor must employ

heuristics to approximate best action. In the parlance of behavioral eco-

nomics, this game is not structured to test how advanced reasoning can

become, and so players quickly reach diminishing returns for additional ex-

ertion. Because our experiment has obvious cross purposes, it allows for rich

communication and negotiation, and diverse preferences can lead to several

sensible heuristics in response.

2

Wunder and Stone

Figure 1.1: A model of the sequence of iterated reasoning and negotiation

steps. The arrows represent flow of action, as inputs cause new moves.

3 Dialogue Moves and Negotiation

There are several key actions that are needed for a complete dialogue process

in the course of a single round, and there is a definite sequence of moves that

people consistently follow (see Figure 1.1). First, each participant has the

option to share his private signal. As a result of social pressure, a number is

reported almost every time. Since a player controls only this piece of infor-

mation, there is an opportunity to falsify what she says to push the group’s

decision towards an outcome that is personally beneficial to her or to her

subgroup. Second, the subjects must decide one of the two options at their

disposal, and frequently discuss the merits as well as intention of the deci-

sion. Figure 1.1, drawing from similar diagrams in [DiEugenio et al.2000],

details the flow of action as a single round runs its course.

Define the value Vi(X) as the expected value if player i votes for policy

X and δij is the sensitivity of i’s payoff value to j’s reported signal. Consider

that player i uses the following heuristic to decide upon si:

si = argmaxsi(g�

j

δji(si − si) + λδii(si − si)) (1.1)

where g represents sympathy with the effect of a falsely reported signal upon

the group, and λ is a parameter that measures the willingness to exaggerate

(or to consider exaggerating) in order to benefit the score of i. Unless i is

the swing vote, the δ sensitivies of Vi to si might be negligible to always

make lying the best strategy, or to make it worth the effort.

In about 90% of the chat situations, players wait until neighboring play-

ers have shared the relevant information or misinformation to begin a new

phase to consider the other major decision, the vote itself. Before the actual

vote, typically there is discussion about the merits of each choice, and some

agreement may be reached. In some cases, people will aggregate signal data

as evidence to use towards this decision. An example is to note the average

signal or whether a majority of signals supports one view.

Our general formula evaluates a policy X by combining and weighting

the available information for n players as follows. The first term is the

stochastic component of the score, divided between known private signal si

3

An Empirical Model of Strategic Dialogue in Group Decisions with Uncertainty

and reported public signals 1n

�nj=1 sj , while the second term is certain bias:

Vi(X) = ((1− α)si(X) + α1n

n�

j=1

sj(X))(1− r) + βppai(X) + u (1.2)

where r measures risk aversion, α is the tendency to accept the averagereported signal as approximating the true underlying percentage over one’sown signal, β weights the personal bias, and u is an unknown factor (assumedto be zero). The voting decision can then be formed as Vote X if Vi(X) >Vi(−X), and vote −X otherwise. If X is Y , Vi(N) = 0.5. Note that α, ameasure of trust, depends on a player’s estimation of the group’s values forg (altruism) and λ (ability to lie). r is usually not significant, but there areexceptions.

While the signal, aggregate reported signal, and bias data are known,the three parameters (r, α,β) are subjective values. In the voting behavior,we observe a distribution that covers the range from zero to one. In a sense,the second phase consists of a period of discovering and negotiating theseparameter values, although not directly. Players do occasionally mentionsuch factors as risk and bias, but most discussion simply revolves aroundthe conclusion. The decision shifts with the weightings of each individual asthe result of further reasoning. While we can say that agreement is a goal,it is not the only goal in this miniature democracy–commitments are looseat best. A voter may find that the parameter settings of others create anopposing conflict. It appears to be that only a small fraction of players areon the fence once provided with the evidence from others.

Two competing factors that affect the voting decision are reliance onone’s known personal signal (weighted by 1 − α) and the available publicknowledge, weighted by α. By isolating the information ultimately availableto each player we can identify the goal of most players (see Table 1.1). Inmost cases, the interests of the individual and that of the group, measuredby the average signal, line up with the chosen option. Where there areconflicting indicators, players will choose one condition over the other. Self-interested voters disrupt the dialogue more often. When the voter ends upchoosing in the group interest, she waits for 92% of the neighbor’s signalson average before announcing a vote. In cases that a player indicates a voteand votes selfishly, he has only received 86% of the information.

For the purpose of modeling communicative strategy, it is important toknow how the voting decision is aligned with the phrases used to indicate andpropose courses of action. We have identified five major phrase types thatare used when someone would like to indicate voting preference, which areDeclarative, Suggestive, Question, Imperative, and Everyone says. Differentforms can have very different connotations even with the same root words.

Using the relative frequency of such phrases, it is possible to answerquestions about how people negotiate based on the expected outcomes. The

4

Wunder and Stone

Table 1.1: Percentage of voter type in experimentsType of voter A/B Y/N

Self-interest vote (s + ppa) 0.12 0.11Vote based on group interest 0.14 0.19Both factors align with vote 0.54 0.60

Neither factor aligns with vote 0.20 0.10

types of utterances do indicate how strong the evidence is, and in turn howlikely they are to follow through with the vote expressed in the message. Inaddition, we have found that people use different negotiation tactics basedon their interests, such as personal versus group. First we filter out therelevant indicating messages from the chat logs. Next we check each ofthese vote indicators for certain giveaway keywords, such as “I think” forSuggestive phrases, and messages starting with verbs for the Imperatives.

4 Experimental Results

We have posed a number of questions of the corpus from this experiment.

• How are conversations organized?

Typically, there are two main phases to each conversation. Players firstexchange signals, then they discuss the merits of each choice. Finally theyannounce decisions by either coming to an agreement or not. Selfish votersare less patient than group voters. Mostly these tasks take much less timethan is available and so players fill the remaining time with side talk. Thereasons discussed include the average signal, biases, and riskiness.

• Do people negotiate differently based on their interests?

Group voters say “we” in 20 out of 117 (0.17) mentions of voting indicationfor A/B and 55 out of 264 (0.21) in the Y/N task. In contrast, self-interestvoters use “we” 7 out of 76 (0.09) indications in the A/B game and 27 outof 177 (0.15) in Y/N. One likely explanation for these figures is that voterswho identify more with the group tend to verbally reveal that propensity.

• Do the phrases people use impact or predict their scores?

Scores for players stating “we” average approximately 5 points lower thanplayers who state “I” in both tasks. In the Y/N case, this difference putsit outside the 95% confidence bound. Again with the Y/N task, playersmentioning what “everyone” is doing do about 9 points worse (although forA/B that gap narrows to 3 points or so).

• Who lies and how much does it pay off?

We have found that liars do take advantage of gullible partners occasionally.Somewhere between 10% and 20% of the signals passed to others are ad-justed in some way. There is also some evidence that too much exaggerationcan backfire on the liar as well as the group.

5

BIBLIOGRAPHY

5 Conclusion

In keeping with previous negotiation models [DiEugenio et al.2000] we cansee our dialogues as playing out of a formal process involving a set of movesand arguments. Our data opens up the possibility to characterize how thespecific moves that people choose reflect their interests and expectationsabout reaching agreement, as well as their interests and strategies for successin the underlying domain task.

6 Acknowledgements

The authors are grateful for their support via HSD-0624191. We would alsolike to thank R. Lau, M. Littman, B. Sopher, D. Anderson, and J. Birchbyfor their involvement in this project.

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