Inferences and Dialogues in Ludics

DialogueArgumentation theory

InferencesSummary

Back to Logics: Ludics

Inferences and Dialogues in Ludics

Christophe Fouqueré & Myriam Quatrini

Laboratoire d’Informatique de Paris-Nord & Institut de Mathématiques de LuminyA.N.R. LOCI

Natural Language and Computer Science(NLCS’13)

June 28, 2013

Christophe Fouqueré & Myriam Quatrini Inferences and Dialogues in Ludics 1 / 45


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Objectives of this work

Define a fundamental framework:that combines representations of dialogues and ofreasoning

that contains only one kind of objects=> composition of objects logically sound

that does not refer to formulas when there is no necessity(e.g. arguments in dialogues)=> stay in a monotonous setting



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Logics, Argumentation, Reasoning

In Logics, two notions of inference:Cut-eliminationModus Ponens

and a relation between computation/proofs and formulas:Typing

They may be used to model dialogues and reasoning:

Computation Logics Dialogue/Reasoningdynamics of computation Cut-elimination dialogue processcomputation of a result Modus Ponens reasoning processtyping formula semantics of utterances



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Logics and Contradiction

Remark that:A Modus Ponens between a formula and its negationinduces a contradiction.A formula may be denoted by its set of proofs.A term may be used as an argument for a function even if itis incorrect as the level of types (A⊕ B vs A).

Motto– Use proofs to model dialogues and argumentation structures– Use formulas to define/characterize utterances, . . .



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A rough RepresentationDialog actDialogue designs

Interaction in dialogues

A rough attempt to focus on interaction in natural languagedialogues:

associate actions to interventions

actions are polarized: positive for the locutor whoproduces them, negative for the locutor who receives themactions/interventions of each locutor are organized in atree-like structure: a designthe dialogue is the trace of the interaction between twodesigns



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Example (A. Dumas, Le Comte de Montecristo)

Dialogue between Edmond and Faria:

F I1 : What was your life at this time?E I2 : I was ready to become captain of the Pharaon;

I was about to marry a beautiful young girl.F I3 : Was anyone interested in you not becoming

the captain of the Pharaon?E I4 : [. . . ] Only one man [. . . ],F I5 : Who was he?E I6 : Danglars.F I7 : Well, tell me about that young girl . . .



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Example (A. Dumas, Le Comte de Montecristo)We associate to each intervention an action: κ1, . . . , κ7 and ajustification relation between them.

κ1

κ2

κ3

κ4

κ5

κ6

κ7

Each intervention refers to a previous one.Action κ1 is initial: F initiates the dialogue.Action κ7 is justified by action κ2.

I1 : What was your life at this time?I2 : I was ready to become captain of the Pharaon;

I was about to marry a beautiful young girl.I3 : Was anyone interested in you not becoming

the captain of the Pharaon?I4 : [. . . ] Only one man [. . . ],I5 : Who was he?I6 : Danglars.I7 : Well, tell me about that young girl . . .



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Example (A. Dumas, Le Comte de Montecristo)The interventions polarised and organised according to bothjustification and chronological relations:

κ7

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

κ2 κ2

κ1 κ1Point of view of Faria Point of view of Edmond



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Example (A. Dumas, Le Comte de Montecristo)The dialogue as a step by step interaction between twodesigns:

κ7

κ6

κ5

κ6

κ4

κ5

κ3

κ4

κ7κ3

κ2κ2

κ1 κ1Faria’s viewpoint Edmond’s viewpoint

F I1 : What was your life at this time?E I2 : I was ready to become captain

of the Pharaon;I was about to marry a beautiful

young girl.F I3 : Was anyone interested in you

not becoming the captainof the Pharaon?

E I4 : [. . . ] Only one man [. . . ],F I5 : Who was he ?E I6 : Danglars.F I7 : Well tell me about

that young girl. . .



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What is a dialog act?

Following Landragin, we define a dialog act as

“the minimal unit of communication in a dialogic context”.



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What is a dialog act?

A dialog act is a communicational fact whose role is to allow thedialogue to progress and to determine the dialogue shape:

Definition (Dialog Act)A dialog act κ is:

either a proper dialog act, i.e. a tuple (+/−,L, I,e)(see next slide)or a particular positive dialog act, called daimon and noted(†,e), that shows the end of an interaction that went well.



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What is a dialog act

When the dialog act is a proper one κ = (+/−,L, I,e):

L is the focus of κ : the address on which the act islocalized with respect to the dialogic interaction oneconsiders,I = {L1, . . . } is the ramification of κ : the openingscreated by the dialog act on which new dialog acts may beproduced,e is the expression of the dialog act, i.e. the language (orcommunication) fact by which the dialog act is reflected:

a (sequence of) proposition(s), word, prosodic element, nonverbal sign.



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Interpreting a Dialogue / Dialogue in progress

A dialogue is represented as an interaction between two dialogdesigns, i.e. designs with dialog acts.

Dialog designs are built incrementally from theinterventions, i.e. the turns of speech of the two persons.To each intervention corresponds a set of dialog acts thatcompletes the current design of the speaker.Such a set may be a unique dialog act, an alternatesequence of dialog acts or a partial design.The design completed by such a set should still be adesign.Finally, an intervention must begin and end with a positivedialog act: hence, if not ending with a daimon, theintervention of a speaker allows the addressee to speak.



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Formalization of controversiesJuridic controversy

The controversy: a particular case of dialogue

Among the dialogues, controversies own some specificfeatures:

they are composed by means of only a few speech acts:assertions, argumentations, denegations, concessions . . .

a notion of “gain” arises.It is out of purpose in most cases of dialogues.But it is essential when the dialogue is a controversy.



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How give an account of these specific dialogues ?

Lots of argumentation studies make use of games theoryBUT there are two important distinctions between theargumentative frame and the model of games.

the only relevant rule in the argumentative frame is the onerelative to keeping the dialogue convergent.in games, each player makes an elementary move. Indialogues, one intervention may contain several speechacts.



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A case study

We consider an example developed by Prakken. It is a juridiccontroversy between a plaintiff and a defendant.

Plaintiff claimed that defendant owes him 500 euro, because both signed acontract ; himself delivered but defendant did not pay.Defendant will give two counter arguments: 1. he does not recognize hissignature, 2. he was insane during the signature of the contract.

I1 Plaintiff: I claim that defendant owes me 500 euro.I2 Defendant: I dispute plaintiff’s claim.. . . . . .J4 Judge (deciding the dispute):

I am conviced by plaintiff’s evidence that defendant’s signatureunder the contract is authentic. Yet I cannot grant plaintiff’s claimsince the fact that defendant looked normal during thenegociations is insufficient to conclude that defendant’sinsanity could not be known to plaintiff: he might have knownif he had checked the court’s register.Therefore I deny plaintiff his claim.



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The dispute in Ludics

We represent the dispute between plaintiff and defendant by

an interaction between two designs,

as if this controversy were a dialogue between only these twolocutors.

Inside this representation, the interventions of the judge aregiven into account by means of some forced dialog acts foreither one locutor or the other one.



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The three first interventions: I1 . . . I3

I1I2

I3

....L31`

....L32`

....L33`

....L34`

`L2

L1``L0

....`(L31

,)L32,L33

,L34

L2``L1

L0`Plaintiff Defendant

I1 : I claim that defendant owes me 500 euro.I2: I dispute plaintiff’s claim.I3 : Defendant owes me 500 euro by r1 since we conclude a valid sales contract,

I delivered but defendant did not pay.

And next intervention:

I4: I concede that plaintiff delivered and I did not pay,but I dispute that we have valid contract.

Look at speech acts modelling . . .Christophe Fouqueré & Myriam Quatrini Inferences and Dialogues in Ludics 18 / 45


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Towards a model of the speech acts of controversy

To claim a thesis This corresponds to posing aproposition on which the interlocutor may continue thedialogue by negating it or by conceding it or by askingmore explanations.

In Ludics: (+,L, I,e).To argue, to negate, to ask, to request somejustification . . . .

They are represented as a claim, by a positive properdialog act.



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The speech acts of controversy

To concede. This consists in accepting one assertion thatthe interlocutor claimed. The dialogue will not continue onthis element.

sequence (+,L, {L′},e0)(−,L′, ∅,e1)

(+,L, {L′},e0) expresses what is conceded, its focus is theone created by the affirmation that one concedes and itsramification is a singleton.

(−,L′, ∅,e1) makes the conceded affirmation disappear: itis focused on L′, i.e. the immediate previous act created, ithas an empty ramification.



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Winning a controverse

During a controversy, one of the locutors becomes loser when:

It is his turnNo addresses are avalaibleHe is obliged to play the daïmon.

The winner has the last word.



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The fourth intervention : I4

I4: I concede that plaintiff delivered and I did not pay, but Idispute that we have valid contract.

We give an account of this one not by a unique dialog act butby a sequence of dialog acts:

speech act dialog act expression

concession (+, L33 , {L41}, e1) I concede that(−, L41 , ∅, e2) plaintiff delivered

concession (+, L34 , {L42}, e3) (I concede that)(−, L42 , ∅, e3) I did not pay

negation (+, L32 , {L43}, e4) but I dispute that we havea valid contract



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The dispute

until the intervention I4: I concede that plaintiff delivered and Idid not pay, but I dispute that we have valid contract.

I4

L31`

....`L43

L32`

`L42∅

L33`

`L41∅

L34`

`L2

L1``L0

....L43`

`L32

L42`L32

∅

`L32,L34

L41`L32

,L34∅

`L31,L32

,L33,L34

L2``L1




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The dispute continues

I5

I6

I7....L51`

....L7``L6

L52``L43....

....`L7,L51

L6`L51

`L51 ,L52

L43`

....Plaintiff Defendant

I5 : We have a valid contract by r2

since this document is a contract signed by us.I6 : I dispute that this is my signature.I7 : Why?



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The first judge’s intervention: J1

J1 : By r3 the party who invokes a signature under a document which is notan avidavit has the burden to prove that it is authentic when this is disputedso plaintiff must prove this is defendant’s signature.

It occurs inside the dispute as an intervention of thedefendant.It is a sequence of dialog acts:it starts with a positive one (arguing), opens threeelements.The two first elements are immediately closed: it is notpossible to discuss neither a law (r3) nor a fact (thedocument is not an avidavit).It gives the turn to plaintiff.



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The first judge’s intervention: J1

J1: By r3 the party who invokes a signature under a document which is not anavidavit has the burden to prove that it is authentic when this is disputed soplaintiff must prove this is defendant’s signature.

J1

....L51`

....`J10

,J11,J12

L7``L6

L52``L43....

`J′10

∅

J10`

`J′11

∅

J11`

....J12`L51

`L7,L51

L6`L51

`L51 ,L52

L43`

....Plaintiff Defendant



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The Judge’s sentence

Before

L16`....L51`π1

L′91`

....`J10

,J11,J12

π2

`L0

`L16,L′91ω....


It is the defendant’s turn. Two addresses are available:

L′91(his signature is not valid)

L16 (his illness)

After

J4`†

L16`....L51`π1

`J41∅

L′91`

....`J10

,J11,J12

π2

`L0

`L16∅

J41`L16

`L16,L′91ω....


I am convinced by plaintiff’s evidence thatdefendant’s signature under the contract is authentic.Yet I cannot grant plaintiff’s claimsince the fact that defendant looked normal during thenegotiations is insufficient to conclude that defendant’sinsanity could not be known to plaintiff: he might have knownif he had checked the court’s register.Therefore I deny plaintiff his claim.



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Dialogue: surface and context modellingJustifications as designs in the cognitive base

Back to intervention I3 of the plaintiff

I3 : Defendant owes me 500 euro by r1 since we conclude avalid sales contract, I delivered but defendant did not pay.

This expresses a contradiction (pay/not pay) according toplaintiff’s viewpoint.How can we make this explicit without yielding a logicalabsurdity?

The judge overcomes this contradiction when giving thesentence.How can we modify plaintiff’s knowledge?



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Inferential rôle of interventions

We consider a framework close to ISU(Informative states update, Asher, Ginzburg, . . . )

Dialogue surface: an interaction between designs madeof dialogue acts.Dialogue context: one cognitive base for each locutorcontaining a state of their knowledge, their entitlements,. . . , and updated after each intervention. Such a dialoguecontext is a set of designs.

An intervention is built from elements of the cognitive base.



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Back to intervention I3 of the plaintiff

I3 : Defendant owes me 500 euro by r1 since we conclude avalid sales contract, I delivered but defendant did not pay.

In the modelling of the dialogue:Dr1

...L31 `

Dval−contract

...L32 `

Ddel

...L33 `

Dnot−paid

...L34 `

` L2

Plaintiff’s base includes designs Dr1 , Dval−contract , Ddel andDnot−paid , that are delocalized to build his argumentation.



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Plaintiff’s designs

Designs modelling facts he put forward and that areconceded, Ddel et Dnot .paid , together with a design D¬payinterpreting the fact: ‘to pay’ is the logical negation of ‘notto pay’:

Ddel =` LDel∅Dnot−paid =` LnotPaid

∅D¬pay =

` †

LPay `∅

The design Dval−contract (the validity of a contract relies onthe validity of the signature):

Dval−contract =

` Lsign∅

Lnon−sign `` Lval−contract



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Interaction in plaintiff’s base

Finally, there is a design corresponding to the legal rule r1:

Dr1 =

` LPay∅

LDel ` LPay∅

` Lnot−Del ,LPay

Lsign ` Lnot−Del ,LPay∅

` Lnot−sign,Lnot−Del ,LPay

Lval−contract ` Lnot−Del ,LPay

` Lnot−val−contract ,Lnot−Del ,LPay

LCont−and−Del ` LPay

and

` Lsign∅

Lnot−sign `` Lval−contract

Lnot−val−contract `` LDel

∅

Lnot−Del `` LCont−and−Del

gives ` LPay∅



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Interaction in plaintiff’s base

The result of the interaction between

Dr1 =

` LPay∅

LDel ` LPay∅

` Lnot−Del ,LPay

Lsign ` Lnot−Del ,LPay∅

` Lnot−sign,Lnot−Del ,LPay

Lval−contract ` Lnot−Del ,LPay

` Lnot−val−contract ,Lnot−Del ,LPay

LCont−and−Del ` LPay and

` Lsign∅

Lnot−sign `` Lval−contract

Lnot−val−contract `` LDel

∅

Lnot−Del `` LCont−and−Del

gives ` LPay∅



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A “contradiction” in plaintiff’s base

Designs Dpay and D¬pay are in plaintiff’s base. Their interactionyields a contradiction:

∅` LPay

†

`∅

LPay ` gives†

`

This design is not a formula corresponding to a logicalabsurdity!

The aim of the judge consists in overcoming this contradiction.



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Example ctd (simplified)

The judge overcomes the contradiction as he decides that legalrule r1 cannot be applied.

We interpret the judge’s argument as: the legal rule r1needs two premisses (and not one as plaintiff supposes).

In other words, the validity of a contract requires twoconditions (hence two sub-addresses): validity of thesignature and non-insanity of those signing.



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Example ctd (simplified)

Formally, plaintiff’s base is changed (by interactions)either to ‘delete’ the (false) design corresponding to legalrule r1 plaintiff has.or to ‘refine’ design Dval−contract .

‘Deletion’ and ‘refinement’ are in fact done by inferring (hencewith interactions) new designs.



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Specificities of our approach

With respect to the usual approaches (concerning the juridiccontroversies):

We refine speech acts as sequences of (more elementary)dialog acts.Instead of considering a game between three players, thejudge directly intervenes inside the interaction between twoprotagonists.Reasoning is modelled at a more primitive level than withformulas.



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Argumentation and Logics

The notions of formal proof and of formula are simultaneouslytoo demanding and too simplistic to account forargumentation:

during a controversy, one can agree on the validity of astatement, without this statement to be necessarily fullydetermined as a formal proposition,one may concede the correctness of a position withoutnecessarily having explored all its premises,a locutor can win a debate, simply because her addresseesurrendered.



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Argumentation and Ludics

Why Ludics?

Designs are the basic objects: Formulas and proofs arenot given a prioriLudics gives a uniform framework for dealing with differentlevels, and for combining elements from these levels.Interaction is the main ingredient for modelling reasoningand dialogue.



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Ludics in a nutshell

In Ludics:objects are abstract proof trees: the designsinteraction between objects may be done to

test the validity of an object,infer a result (similar to Modus Ponens)

interaction between two dual designs is the core oforthogonalityclosed sets of objects help to recover formulas, hencerecover logics



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Key points:No more formulas but addresses.No more logical rules but actions:

A⊥ ` B⊥ `` (A⊗ B)⊕ (A⊗ C)

corresponds to one positive action:(+, L, {L1, L2})

` A⊥,B⊥ ` A⊥,C⊥

(A⊗ B)⊕ (A⊗ C) `corresponds to two negative actions:(−, L, {L1, L2}) and (−, L, {L3, L4})



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No more proofs but designs:φ ` φ

` φ⊥, φ

φ⊗ φ⊥ `

ψ ` ψ

` ψ⊥, ψ

ψ ⊗ ψ⊥ `

` ((φ⊥ ⊕ φ)⊗ (ψ⊥ ⊕ ψ))⊕ ρ

becomes (three notations)

(+, L21, L3)

(−, L1, {L21, L22}

(+, L21, L3)

(−, L1, {L21, L22}

(+, L0, {L1,K1}) or

L3 ` L22

` L21, L22

L1 `

K3 ` K22

` K21,K22

K1 `` L0 or

κ5

κ4

κ3

κ2

κ1



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A design of base (L0) ` L1, . . . ,Ln is a set of alternatesequences of actions, satisfying a few constraints:

In each sequence, actions are either initial, i.e. in the base,or justified by an action that precedes in the sequence.

The set is closed by prefix, total, positive, coherent.



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Interaction between designs

No more cut-elimination but interaction

An interaction occurs between designs based on dualaddresses:

L7 ` L6

` L6, L22

L5 ` L22

` L4, L22

L3 ` L22

` L21, L22

L1 `` L0

L6 `` L5

L4 `` L3

L21 ` L22 `` L1

L0 ` or

κ7

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

κ2 κ2

κ1 κ1



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Interaction begins on the design that has a positive base.Each time a positive action κ is visited, the travel continues on the dual (neg-ative) action κ in the other design (if it exists), then goes to the unique positiveaction after κ in this design (bottom-up reading).

κ7

κ8†

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

...

κ2 κ2

κ1 κ1

?



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. . . and so on.

κ7

κ8†

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

...

κ2 κ2

κ1 κ1

?



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. . . if the negative action does not exist, the interaction fails, diverges.

κ7

κ8

†

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

...

κ2 κ2

κ1 κ1

?



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. . . if the daimon is reached, the interaction succeeds, converges.

κ7κ8

†

κ6 κ6

κ5 κ5

κ4 κ4

κ3 κ3 κ7

...

κ2 κ2

κ1 κ1

?



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Results in Ludics

Two designs are orthogonal when their interactionsucceeds.Separation: A design is fully defined by its counter-designs.A formula is retrieved as a set of designs closed bybi-orthogonality: such sets are called behaviors.Completeness: A good design belongs to theinterpretation of a formula iff it is the interpretation of aproof.A formula is valid when the behavior that interprets itcontains such a good design.


Inferences and Dialogues in Ludics

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