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Persuasive Argumentation and

Epistemic Attitudes

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A bloody crime

Interrogation room. 1 is the main suspect. 2 is the detective. These are the only relevant pieces of information:

• c

• f

• d

• e

• b • a

a: “1 is innocent”

b: “1 was seen close to the crime scene”

c: “1 has a twin brother living in the city”

d: “1 works in a butcher’s nearby”

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Aim of the work

• Which arguments will be disclosed by agent 1 if he tries to persuade 2that he is innocent?

Thesis. It strongly depends on what 1thinks that 2thinks about the relevant information

• Previous work (e.g.(Rahwan and Larson, 2009; Sakama, 2012)) have ignored this fact

• What are the appropriate tools for capturing the

(4)

Aim of the work

• Which arguments will be disclosed by agent 1 if he tries to persuade 2that he is innocent? Thesis. It strongly depends on what 1thinks that 2thinks about the relevant information

• Previous work (e.g.(Rahwan and Larson, 2009; Sakama, 2012)) have ignored this fact

• What are the appropriate tools for capturing the

(5)

Aim of the work

• Which arguments will be disclosed by agent 1 if he tries to persuade 2that he is innocent? Thesis. It strongly depends on what 1thinks that 2thinks about the relevant information

• Previous work (e.g.(Rahwan and Larson, 2009; Sakama, 2012)) have ignored this fact

• What are the appropriate tools for capturing the

(6)

Aim of the work

• Which arguments will be disclosed by agent 1 if he tries to persuade 2that he is innocent? Thesis. It strongly depends on what 1thinks that 2thinks about the relevant information

• Previous work (e.g.(Rahwan and Larson, 2009; Sakama, 2012)) have ignored this fact

• What are the appropriate tools for capturing the epistemic component of persuasion?

(7)

Aim of the work

• Which arguments will be disclosed by agent 1 if he tries to persuade 2that he is innocent? Thesis. It strongly depends on what 1thinks that 2thinks about the relevant information

• Previous work (e.g.(Rahwan and Larson, 2009; Sakama, 2012)) have ignored this fact

• What are the appropriate tools for capturing the

(8)

Outline

AFs and Justification Status

Persuasiveness and Epistemic Persuasiveness

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Definition (Dung (1995))

An argumentation framework (AF) is a pair(A, ) where:

• A6=∅ and finite (arguments)

• ⊆A×A (attack relation)

a b is read as“a attacksb”

Including agents: a TAF is a tuple (A, ,A1,A2) where

Ai ⊆Aand i0s subgraph is defined as (Ai, i)with i= ∩(Ai ×Ai) for everyi ∈ {1,2}.

Pointed TAFs (A, ,A1,A2,a)where a∈A1∩A2 are used to

(11)

Definition (Dung (1995))

An argumentation framework (AF) is a pair(A, ) where:

• A6=∅ and finite (arguments)

• ⊆A×A (attack relation)

a b is read as “a attacksb”

Including agents: a TAF is a tuple (A, ,A1,A2) where

Ai ⊆Aand i0s subgraph is defined as (Ai, i)with i= ∩(Ai ×Ai) for everyi ∈ {1,2}.

Pointed TAFs (A, ,A1,A2,a)where a∈A1∩A2 are used to

(12)

Definition (Dung (1995))

An argumentation framework (AF) is a pair(A, ) where:

• A6=∅ and finite (arguments)

• ⊆A×A (attack relation)

a b is read as “a attacksb”

Including agents: a TAF is a tuple (A, ,A1,A2) where

Ai ⊆Aand i0s subgraph is definedas (Ai, i)with i= ∩(Ai ×Ai)for every i ∈ {1,2}.

Pointed TAFs (A, ,A1,A2,a)where a∈A1∩A2 are used to

(13)

Definition (Dung (1995))

An argumentation framework (AF) is a pair(A, ) where:

• A6=∅ and finite (arguments)

• ⊆A×A (attack relation)

a b is read as “a attacksb”

Including agents: a TAF is a tuple (A, ,A1,A2) where

Ai ⊆Aand i0s subgraph is definedas (Ai, i)with i= ∩(Ai ×Ai)for every i ∈ {1,2}.

Pointed TAFs (A, ,A1,A2,a)where a∈A1∩A2 are used to

(14)

Example of TAF

• c

• f

• d

• e

• b • a

a: “1 is innocent”

b: “1 was seen close to the crime scene”

c: “1 has a twin brother living in the city”

f: “1’s twin brother was in Venice last night”

d: “1 works in a butcher’s nearby”

(15)

Example of TAF

• c

• f

• d

• e

• b • a

1

2 2

a: “1 is innocent”

b: “1 was seen close to the crime scene”

c: “1 has a twin brother living in the city”

f: “1’s twin brother was in Venice last night”

d: “1 works in a butcher’s nearby”

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Justification Status

Definition (based on Wu et al. (2010))

Thejustification status of a for i is the outcome yielded by the function J Si :A→℘({in,out,undec}) defined as:

J Si(a) :={Li(a)| L is a complete labelling of (Ai, i)}

JS∗ is the set of possible outcomes of J S, which naturally defines an acceptance hierarchy:

strong acceptance{in} > {in,undec} > {undec} =

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Justification Status

Definition (based on Wu et al. (2010))

Thejustification status of a for i is the outcome yielded by the function J Si :A→℘({in,out,undec}) defined as:

J Si(a) :={Li(a)| L is a complete labelling of (Ai, i)}

JS∗ is the set of possible outcomes of J S, which naturally defines an acceptance hierarchy:

strong acceptance{in} > {in,undec} > {undec} =

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Examples

• c

• f

• d

• e

• b • a

1

2 2

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Definition (Persuasiveness of a set of arguments)

LetG = (A, ,A1,A2,a) and B ⊆A1 the resulting pointed TAFis GB := (A, ,A

1,AB2,a) where AB2 =A2∪B. Let

goal∈JS∗,B is said to be persuasiveiff J S2(a) = goal

w.r.t. GB

Persuasion is understood as a change in the hearer’s

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Definition (Persuasiveness of a set of arguments)

LetG = (A, ,A1,A2,a) and B ⊆A1 the resulting pointed TAFis GB := (A, ,A

1,AB2,a) where AB2 =A2∪B. Let

goal∈JS∗,B is said to be persuasiveiff J S2(a) = goal

w.r.t. GB

Persuasion is understood as a change in the hearer’s

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•c •f

•d •e

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•c •f

•d •e

•b •a 2 2 w0 •c •f •d •e

•b •a 2 w1 •c •f •d •e

•b •a 2

1,2

w2

1

2

1,2

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Epistemic argumentative models

Definition (Schwarzentruber et al. (2012))

An pointed model for (A, ,A1,A2) is (M,w) =((W,R,D),w) where:

• W 6=∅(possible worlds) with w ∈W

• R: Ag→℘(W ×W) (accessibility relations)

• D: (Ag×W)→℘(A)(awareness function) s.t. D1(w) =A1 andD2(w) =A2.

1. If wRiu, thenDi(w)⊆ Di(u) (Positive Introspection) 2. If wRiu, thenDj(u)⊆ Di(w) (General Negative

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Epistemic argumentative models

Definition (Schwarzentruber et al. (2012))

An pointed model for (A, ,A1,A2) is (M,w) =((W,R,D),w) where:

• W 6=∅(possible worlds) with w ∈W

• R: Ag→℘(W ×W) (accessibility relations)

• D: (Ag×W)→℘(A)(awareness function) s.t. D1(w) =A1 andD2(w) =A2.

1. If wRiu, thenDi(w)⊆ Di(u) (Positive Introspection) 2. If wRiu, thenDj(u)⊆ Di(w) (General Negative

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Definition (Communication Model)

Acommunication pointed model

(M,w)+b:= ((W,R,D+b),w) where

D+b: (Ag×W)(A) is defined by cases for each i Ag

and each v ∈W as follows:

Di(v)∪ {b} if b ∈ D1(w)

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Examples

1 :A 2 :{a,b,e}

w0

1 :A 2 :{a,b}

w1

1 :{a,b}

2 :{a,b,e} w2

1

2

1,2

1,2

1 :A 2 :{a,b,d,e}

w0

1 :A 2 :{a,b,d}

w1

1 :{a,b,d}

2 :{a,b,d,e} w2

1

2

1,2

1,2 (M,w0)+d

(M,w2)+d

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Epistemic Persuasiveness

Definition (Epistemic-based persuasive arguments)

Let (M,w) be a pointed model for (A, ,D1(w),D2(w),a),

let goal∈JS∗, we say that B⊆ D1(w) ispersuasive from 1’s perspectiveiff J S2(a) = goal w.r.t.

(D+B 2 (w

0), D+B 2 (w

0)) for all w0 W s.t. wR 1w0.

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Epistemic Persuasiveness

Definition (Epistemic-based persuasive arguments)

Let (M,w) be a pointed model for (A, ,D1(w),D2(w),a),

let goal∈JS∗, we say that B⊆ D1(w) ispersuasive from 1’s perspectiveiff J S2(a) = goal w.r.t.

(D+B 2 (w

0), D+B 2 (w

0)) for all w0 W s.t. wR 1w0.

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Plain Persuasion and Epistemic Persuasion

1 :A 2 :{a,b,d}

w0

1 :A 2 :{a,b,d} w1

1 :{a,b,d}

2 :{a,b,e,d} w2

1

2

1,2

1,2 Assume goal ={in}

Figure 2: (M,w0)+d

• {d} is persuasive from 1’s perspective in (M,w0) • {d} is not actually persuasive

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Plain Persuasion and Epistemic Persuasion

1 :A 2 :{a,b,d}

w0

1 :A 2 :{a,b,d} w1

1 :{a,b,d}

2 :{a,b,e,d} w2

1

2

1,2

1,2 Assume goal ={in}

Figure 2: (M,w0)+d

• {d}is persuasive from 1’s perspective in (M,w0)

• {d} is not actually persuasive

(32)

Plain Persuasion and Epistemic Persuasion

1 :A 2 :{a,b,d}

w0

1 :A 2 :{a,b,d} w1

1 :{a,b,d}

2 :{a,b,e,d} w2

1

2

1,2

1,2 Assume goal ={in}

Figure 2: (M,w0)+d

• {d}is persuasive from 1’s perspective in (M,w0) • {d}is not actually persuasive

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Plain Persuasion and Epistemic Persuasion

1 :A 2 :{a,b,d}

w0

1 :A 2 :{a,b,d} w1

1 :{a,b,d}

2 :{a,b,e,d} w2

1

2

1,2

1,2 Assume goal ={in}

Figure 2: (M,w0)+d

• {d}is persuasive from 1’s perspective in (M,w0) • {d}is not actually persuasive

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A Logic for Argument Disclosure

LetA6=∅ and finite and define L+!(A)

ϕ::=ownsi(a)| ¬ϕ|ϕ∧ϕ|iϕ|[+a]ϕ|[a!]ϕ a∈A i ∈ {1,2}

(M,w)ownsi(a) iff a ∈ Di(w)

(M,w) iϕ iff (M,w0)ϕ ∀w0 s.t. wRiw0

(M,w)[a!]ϕ iff (M,w)a! ϕ

(M,w)[+a]ϕ iff (M,w)+a ϕ

where (M,w)a! = ((W,R,Da!),w) and Da!

i (v) = Di(v)∪ {a}

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A Logic for Argument Disclosure

LetA6=∅ and finite and define L+!(A)

ϕ::=ownsi(a)| ¬ϕ|ϕ∧ϕ|iϕ|[+a]ϕ|[a!]ϕ a∈A i ∈ {1,2}

(M,w)ownsi(a) iff a ∈ Di(w)

(M,w) iϕ iff (M,w0)ϕ ∀w0 s.t. wRiw0

(M,w)[a!]ϕ iff (M,w)a! ϕ

(M,w)[+a]ϕ iff (M,w)+a ϕ

where (M,w)a! = ((W,R,Da!),w) and Da!

i (v) = Di(v)∪ {a}

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Axioms

All propositional tautologies (Taut)

`i(ϕ→ψ)→(iϕ→iψ) (K) `ownsi(a)→iownsi(a) (PI)

` ¬ownsi(a)→i¬ownsj(a) (GNI)

Rules

From ϕ→ψ and ϕ, infer ψ MP From ϕinfer iϕ NEC

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Reduction Axioms

`[+a]ϕ↔(owns1(a)→[a!]ϕ)∧(¬owns1(a)→ϕ) (Def+)

`[a!]ownsi(a)↔ > (Atoms=) `[a!]ownsi(b)↔ownsi(b) wherea6=b (Atoms6=)

`[a!]¬ϕ↔ ¬[a!]ϕ (Negation)

`[a!](ϕ∧ψ)↔([a!]ϕ∧[a!]ψ) (Conjunction)

`[a!]iϕ↔i[a!]ϕ (Box)

From ϕ↔ψ, infer δ↔δ[ϕ/ψ] SE

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Plain Persuasiveness and Epistemic Persuasiveness

Proposition

Given a pointed model (M,w) for(A, ,A1,A2,a), let B ⊆A be persuasive from the speaker’s perspective. Let

Ai :={ai ∈A|M,w owns2(ai)∧ ¬1owns2(ai)}.

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Two main pending tasks

1. Capturing persuasive sets and EB-persuasive sets in the object language:

• Including new kind of variables in the propositional fragment to talk about the attack relation (following (Doutre et al., 2014, 2017))and the belonging of each argument to certain subsets

2. Dropping the assumption of credulous agents

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Two main pending tasks

1. Capturing persuasive sets and EB-persuasive sets in the object language:

• Including new kind of variables in the propositional fragment to talk about the attack relation (following (Doutre et al., 2014, 2017))and the belonging of each argument to certain subsets

2. Dropping the assumption of credulous agents

(42)

Two main pending tasks

1. Capturing persuasive sets and EB-persuasive sets in the object language:

• Including new kind of variables in the propositional fragment to talk about the attack relation (following (Doutre et al., 2014, 2017))and the belonging of each argument to certain subsets

2. Dropping the assumption of credulous agents

(43)

Two main pending tasks

1. Capturing persuasive sets and EB-persuasive sets in the object language:

• Including new kind of variables in the propositional fragment to talk about the attack relation (following (Doutre et al., 2014, 2017))and the belonging of each argument to certain subsets

2. Dropping the assumption of credulous agents

(44)

Two main pending tasks

1. Capturing persuasive sets and EB-persuasive sets in the object language:

• Including new kind of variables in the propositional fragment to talk about the attack relation (following (Doutre et al., 2014, 2017))and the belonging of each argument to certain subsets

2. Dropping the assumption of credulous agents

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References i

References

Martin WA Caminada and Dov M Gabbay. A logical account of formal argumentation. Studia Logica, 93(2-3):109, 2009. Sylvie Doutre, Andreas Herzig, and Laurent Perrussel. A

dynamic logic framework for abstract argumentation. In

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References ii

Sylvie Doutre, Faustine Maffre, and Peter McBurney. A

dynamic logic framework for abstract argumentation: adding and removing arguments. In International Conference on Industrial, Engineering and Other Applications of Applied Intelligent Systems, pages 295–305. Springer, 2017. Phan Minh Dung. On the acceptability of arguments and its

fundamental role in nonmonotonic reasoning, logic

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References iii

Iyad Rahwan and Kate Larson. Argumentation and game theory. In Argumentation in Artificial Intelligence, pages 321–339. Springer, 2009.

Chiaki Sakama. Dishonest arguments in debate games.

COMMA, 75:177–184, 2012.

Fran¸cois Schwarzentruber, Srdjan Vesic, and Tjitze Rienstra. Building an epistemic logic for argumentation. In Logics in Artificial Intelligence, pages 359–371. Springer, 2012. Yining Wu, Martin Caminada, and Mikotaj Podlaszewski. A

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