ASPIC-END: A Structured Argumentation Framework to Model - - PowerPoint PPT Presentation

ASPIC-END: A Structured Argumentation Framework to Model Explanations of the Liar Paradox

PhDs in Logic IX, Ruhr-Universit¨ at Bochum J´ er´ emie Dauphin & Marcos Cramer

University of Luxembourg

May 2, 2017

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Introduction

Reasoning about conflicting information? Formalism we will focus on: Formal Argumentation Human reasoning: a lot of world knowledge Logical paradoxes: mostly abstract knowledge but also defeasible conflicting arguments

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Outline

1

Introduction The Liar Paradox Abstract Argumentation Frameworks Explanatory Argumentation Frameworks

2

ASPIC-END and a paracomplete solution Natural deduction arguments and explananda Attacks and explanations Argument selection

3

Rationality Postulates

4

Conclusion

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The Liar Paradox

This sentence is a lie.

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The Liar Paradox

This sentence is a lie. True or false?

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The Liar Paradox

This sentence is a lie. True or false? Many arguments in the philosophical literature

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Abstract Argumentation Frameworks

Abstract Argumentation Framework[1]

An abstract argumentation framework (AF) is a pair A, →, where A is a set of arguments and → ⊆ A × A is an attack relation. Example: [1] introduced by Dung in 1995

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Explanatory Argumentation Frameworks

Explanatory Argumentation Frameworks[2]

An explanatory argumentation framework (EAF) is a tuple A, X, →, , where A is a set of arguments, X is a set of explananda, → ⊆ A × A is an attack relation and ⊆ A × (A ∪ X) is an explanation relation from arguments to either explananda or other arguments. [2] introduced by ˇ Seˇ selja and Straßer in 2013

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Example

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Outline

1

Introduction The Liar Paradox Abstract Argumentation Frameworks Explanatory Argumentation Frameworks

2

ASPIC-END and a paracomplete solution Natural deduction arguments and explananda Attacks and explanations Argument selection

3

Rationality Postulates

4

Conclusion

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ASPIC-END

Argumentation Theories

An argumentation theory is a triple AT = (L, R, n), where: L is a logical language closed under negation (¬) and the unary

• perator Assumable.

R = Ris ∪ Rd is a set of intuitively strict (Ris) and defeasible (Rd) rules of the form ϕ1, . . . , ϕn ϕ and ϕ1, . . . , ϕn ⇒ ϕ respectively (where ϕi, ϕ ∈ L), and Ris ∩ Rd = ∅. n is a partial function such that n: R → L Goal: build an EAF

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Example: a paracomplete solution

Define L to be the sentence ”L is false”. If L is true, i.e. ”L is false” is true, then L is false, which is absurd. So L is not true, i.e. L is false. So ”L is false” is true, i.e. L is true. So we have the absurdity that L is both true and false from no assumption. One possible solution: L is neither true nor false. When concluding that L is false because L is not true, we are making the assumption that any sentence is either true or false. Even though applicable in many situ- ations, this principle is not applicable to problematically self-referential sentences like L.

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Example rules from the text

Some of the rules we can model from the text are: t1 : Ltrue “Lfalse”true t2 : “Lfalse”true Lfalse t5 : Ltrue, Lfalse ⊥ t6 : ¬Ltrue Lfalse p1 : ¬LEitherTrueOrFalse t9 : ¬LEitherTrueOrFalse ¬t6

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Arguments

An argument A on the basis of an argumentation theory (L, R, n) has one

• f the following forms:

Application of an intuitively strict rule

A1, . . . An ψ where A1, . . . An are arguments such that there exists an intuitively strict rule Conc(A1),. . . ,Conc(An) ψ in Ris with: Conc(A) = ψ, As(A) = As(A1) ∪ · · · ∪ As(An).

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Arguments

An argument A on the basis of an argumentation theory (L, R, n) has one

• f the following forms:

Application of an intuitively strict rule

A1, . . . An ψ where A1, . . . An are arguments such that there exists an intuitively strict rule Conc(A1),. . . ,Conc(An) ψ in Ris with: Conc(A) = ψ, As(A) = As(A1) ∪ · · · ∪ As(An).

Application of a defeasible rule

A1, . . . An ⇒ ψ where A1, . . . An are arguments such that there exists a defeasible rule Conc(A1),. . . ,Conc(An) ⇒ ψ in Rd and As(A1) ∪ · · · ∪ As(An) = ∅ with: Conc(A) = ψ, As(A) = ∅.

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Arguments

An argument A on the basis of an argumentation theory (L, R, n) has one

• f the following forms:

Assumption introduction

Assume(ϕ) where ϕ ∈ L with: Conc(A) = ϕ As(A) = {ϕ}

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Arguments

An argument A on the basis of an argumentation theory (L, R, n) has one

• f the following forms:

Assumption introduction

Assume(ϕ) where ϕ ∈ L with: Conc(A) = ϕ As(A) = {ϕ}

Proof by contradiction

ProofByContrad(¬ϕ, A′) where A′ is an argument such that ϕ ∈ As(A′) and Conc(A′) = ⊥ with: Conc(A) = ¬ϕ, As(A) = As(A′) \{ϕ}.

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Explananda

Explananda

There is an explanandum E such that Source(E) = A if and only if there exists an argument A such that: Conc(A) = ⊥, As(A) = ∅ and Rules(A) ⊆ Ris.

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Example arguments from the text

A1 : Assume(Ltrue), Conc = Ltrue, As = {Ltrue} A5 : ProofByContrad(¬Ltrue, A4), Conc = ¬Ltrue, As = ∅ A6 : ¬Ltrue Lfalse, TopRule = t6 A9 : Lfalse, Ltrue ⊥, As = ∅ E : Source = A9 B1 : ¬LEitherTrueOrFalse B2 : ¬LEitherTrueOrFalse ¬t6 C2 : LProblSelfRef ¬LEitherTrueOrFalse D1 : ¬LEitherTrueOrFalse ¬Ltrue D2 : ¬LEitherTrueOrFalse ¬Lfalse

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Attacks

Attacks

A attacks B if and only if A rebuts, undercuts or assumption-attacks B, where: A rebuts argument B (on B′) iff Conc(A) = −ϕ for some B′ ∈ Sub(B) of the form B′′

1 , . . . , B′′ n ⇒ ϕ and As(A) = ∅.

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Attacks

Attacks

A attacks B if and only if A rebuts, undercuts or assumption-attacks B, where: A rebuts argument B (on B′) iff Conc(A) = −ϕ for some B′ ∈ Sub(B) of the form B′′

1 , . . . , B′′ n ⇒ ϕ and As(A) = ∅.

A undercuts argument B (on B′) iff Conc(A) = −n(r) for some B′ ∈ Sub(B) such that TopRule(B′) = r, As(A) ⊆ (As(B) ∪ As(B′)) and there is no ϕ ∈ As(B′) such that −ϕ = Conc(A′) for some A′ ∈ Sub(A).

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Attacks

Attacks

A attacks B if and only if A rebuts, undercuts or assumption-attacks B, where: A rebuts argument B (on B′) iff Conc(A) = −ϕ for some B′ ∈ Sub(B) of the form B′′

1 , . . . , B′′ n ⇒ ϕ and As(A) = ∅.

A undercuts argument B (on B′) iff Conc(A) = −n(r) for some B′ ∈ Sub(B) such that TopRule(B′) = r, As(A) ⊆ (As(B) ∪ As(B′)) and there is no ϕ ∈ As(B′) such that −ϕ = Conc(A′) for some A′ ∈ Sub(A). A assumption-attacks B (on B′) iff for some B′ ∈ Sub(B) such that B′ = Assume(ϕ), Conc(A) = ¬Assumable(ϕ) and As(A) = ∅.

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Explanatory relation

Explanations

An argument A explains an explanandum E if and only if A attacks Source(E).

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Explanatory relation

Explanations

An argument A explains an explanandum E if and only if A attacks Source(E). An argument B explains another argument A (on A′) if and only if Conc(B) = TopRule(A′) for some A′ ∈ Sub(A) such that As(B) ⊆ As(A′) and Sub(B) \ B = ∅.

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Preference relation

Weakest-link preference

Let A and B be two arguments. We have that A w B if and only if Rules(A) = ∅ and: There exists ra ∈ Rules(A), such that for all rb ∈ Rules(B), we have ra ≤ rb

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Preference relation

Weakest-link preference

Let A and B be two arguments. We have that A w B if and only if Rules(A) = ∅ and: There exists ra ∈ Rules(A), such that for all rb ∈ Rules(B), we have ra ≤ rb

Defeat

An argument A defeats an argument B with respect to a preference relation if and only if: A undercuts or assumption-attacks B on B′, or A rebuts B on B′ and B′ A

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EAFs from rules

EAFs

Let AT = (L, R, n) be an argumentation theory and ≤ a preference relation defined over R. An explanatory argumentation framework (EAF) defined by AT and ≤, is a tuple A, X, →, , where: A is the set of all arguments that can be constructed from R; X is the set of all explananda that can be constructed from the arguments in A; is the preference relation over arguments obtained from lifting ≤ according to the weakest-link principle; (X, Y ) ∈→ if and only if X defeats Y with respect to , where X, Y ∈ A; (X, E) ∈ if and only if X successfully explains E with respect to , where X ∈ A and E ∈ X; (X, Y ) ∈ if and only if X explains Y , where X, Y ∈ A.

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Example arguments from the text

A1 : Assume(Ltrue), Conc = Ltrue, As = {Ltrue} A5 : ProofByContrad(¬Ltrue, A4), Conc = ¬Ltrue, As = ∅ A6 : ¬Ltrue Lfalse, TopRule = t6 A9 : Lfalse, Ltrue ⊥, As = ∅ E : Source = A9 B1 : ¬LEitherTrueOrFalse B2 : ¬LEitherTrueOrFalse ¬t6 C2 : LProblSelfRef ¬LEitherTrueOrFalse D1 : ¬LEitherTrueOrFalse ¬Ltrue D2 : ¬LEitherTrueOrFalse ¬Lfalse

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Resulting EAF

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Argument selection

Satisfactory sets

We say that S is satisfactory iff S is admissible and there is no S′ ⊃ S such that S′ >p S and S′ is admissible.

Insightful sets

We say that S is insightful iff S is satisfactory and there is no S′ ⊃ S such that S′ >d S and S′ is satisfactory.

Argumentative core extensions

We say that S is an argumentative core extension (AC-extension) of ∆ iff S is satisfactory and there is no S′ ⊃ S such that S′ is satisfactory.

Explanatory core extensions

We say that S is an explanatory core extension (EC-extension) of ∆ iff S is insightful and there is no S′ ⊂ S such that S′ is insightful.

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Postulates of closure

Closure under sub-arguments

Let Σ = (L, R, n, ≤) be an argumentation theory, ∆ = A, X, →, be the EAF defined by Σ and S be an AC-extension of ∆. Then, for all A ∈ S and A′ ∈ A, if A′ ∈ Sub(A), then A′ ∈ S.

Closure under accepted intuitively strict rules

Let Σ = (L, R, n, ≤) be an argumentation theory, ∆ = A, X, →, be the EAF defined by Σ and S be an AC-extension of ∆. Then, Conc(S) = ClRisa(S)(Concs(S)).

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Postulates of consistency

Non-triviality

Let Σ = (L, R, n, ≤) be a well-defined argumentation theory, ∆ = A, X, →, be the EAF defined by Σ, and S be an AC or EC-extension of ∆. Then, ⊥ / ∈ Concs(S).

Consistency

Let Σ = (L, R, n, ≤) be a consistency-inducing argumentation theory, ∆ = A, X, →, be the EAF defined by Σ and S be an AC or EC-extension of ∆. Then, there does not exist ϕ ∈ Concs(S) such that ¬ϕ ∈ Concs(S).

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Outline

1

Introduction The Liar Paradox Abstract Argumentation Frameworks Explanatory Argumentation Frameworks

2

ASPIC-END and a paracomplete solution Natural deduction arguments and explananda Attacks and explanations Argument selection

3

Rationality Postulates

4

Conclusion

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Conclusions

Extended ASPIC+ with explanations Explananda arise from triviality under no assumption Added Proof-By-Contradiction arguments Relaxed the notion of strict rules Shown postulates of rationality

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Future work

Empirical study on cognitive aspects of Formal Argumentation Aggregate multiple solutions to the paradox Apply ASPIC-END to scientific debates in other areas Incorporate more rules of natural deduction (reasoning by cases, ...)

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Questions

Thank you!

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