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 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 1 / 28
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 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 2 / 28
Outline Introduction 1 The Liar Paradox Abstract Argumentation Frameworks Explanatory Argumentation Frameworks ASPIC-END and a paracomplete solution 2 Natural deduction arguments and explananda Attacks and explanations Argument selection Rationality Postulates 3 Conclusion 4 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 3 / 28
The Liar Paradox This sentence is a lie. J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 4 / 28
The Liar Paradox This sentence is a lie. True or false? J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 4 / 28
The Liar Paradox This sentence is a lie. True or false? Many arguments in the philosophical literature J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 4 / 28
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 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 5 / 28
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 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 6 / 28
Example J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 7 / 28
Outline Introduction 1 The Liar Paradox Abstract Argumentation Frameworks Explanatory Argumentation Frameworks ASPIC-END and a paracomplete solution 2 Natural deduction arguments and explananda Attacks and explanations Argument selection Rationality Postulates 3 Conclusion 4 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 8 / 28
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 operator Assumable . R = R is ∪ R d is a set of intuitively strict ( R is ) and defeasible ( R d ) rules of the form ϕ 1 , . . . , ϕ n � ϕ and ϕ 1 , . . . , ϕ n ⇒ ϕ respectively (where ϕ i , ϕ ∈ L ), and R is ∩ R d = ∅ . n is a partial function such that n : R �→ L Goal: build an EAF J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 9 / 28
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. J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 10 / 28
Example rules from the text Some of the rules we can model from the text are: t 1 : Ltrue � “ Lfalse ” true t 2 : “ Lfalse ” true � Lfalse t 5 : Ltrue , Lfalse � ⊥ t 6 : ¬ Ltrue � Lfalse p 1 : � ¬ LEitherTrueOrFalse t 9 : ¬ LEitherTrueOrFalse � ¬ t 6 J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 11 / 28
Arguments An argument A on the basis of an argumentation theory ( L , R , n ) has one of the following forms: Application of an intuitively strict rule A 1 , . . . A n � ψ where A 1 , . . . A n are arguments such that there exists an intuitively strict rule Conc( A 1 ), . . . ,Conc( A n ) � ψ in R is with: Conc( A ) = ψ , As( A ) = As( A 1 ) ∪ · · · ∪ As( A n ). J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 12 / 28
Arguments An argument A on the basis of an argumentation theory ( L , R , n ) has one of the following forms: Application of an intuitively strict rule A 1 , . . . A n � ψ where A 1 , . . . A n are arguments such that there exists an intuitively strict rule Conc( A 1 ), . . . ,Conc( A n ) � ψ in R is with: Conc( A ) = ψ , As( A ) = As( A 1 ) ∪ · · · ∪ As( A n ). Application of a defeasible rule A 1 , . . . A n ⇒ ψ where A 1 , . . . A n are arguments such that there exists a defeasible rule Conc( A 1 ), . . . ,Conc( A n ) ⇒ ψ in R d and As( A 1 ) ∪ · · · ∪ As( A n ) = ∅ with: Conc( A ) = ψ , As( A ) = ∅ . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 12 / 28
Arguments An argument A on the basis of an argumentation theory ( L , R , n ) has one of the following forms: Assumption introduction Assume( ϕ ) where ϕ ∈ L with: Conc( A ) = ϕ As( A ) = { ϕ } J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 13 / 28
Arguments An argument A on the basis of an argumentation theory ( L , R , n ) has one of 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 } . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 13 / 28
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 ) ⊆ R is . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 14 / 28
Example arguments from the text A 1 : Assume( Ltrue ), Conc = Ltrue , As = { Ltrue } A 5 : ProofByContrad( ¬ Ltrue , A 4 ), Conc = ¬ Ltrue , As = ∅ A 6 : ¬ Ltrue � Lfalse , TopRule = t 6 A 9 : Lfalse , Ltrue � ⊥ , As = ∅ E : Source = A 9 B 1 : � ¬ LEitherTrueOrFalse B 2 : ¬ LEitherTrueOrFalse � ¬ t 6 C 2 : LProblSelfRef � ¬ LEitherTrueOrFalse D 1 : ¬ LEitherTrueOrFalse � ¬ Ltrue D 2 : ¬ LEitherTrueOrFalse � ¬ Lfalse J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 15 / 28
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 ) = ∅ . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 16 / 28
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 ). J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 16 / 28
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 ) = ∅ . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 16 / 28
Explanatory relation Explanations An argument A explains an explanandum E if and only if A attacks Source ( E ). J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 17 / 28
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 � = ∅ . J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 17 / 28
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 r a ∈ Rules( A ), such that for all r b ∈ Rules( B ), we have r a ≤ r b J´ er´ emie Dauphin & Marcos Cramer ASPIC-END May 2, 2017 18 / 28
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