Tackling Defeasible Reasoning in Bochum: the Research Group for - PowerPoint PPT Presentation

Extension-based Approaches: Default Logic (Reiter) • idea: split the factual part of the knowledge base (Gelfond, Lifschitz, Przymusinska, 1991) Republican Republican Base 1 Republican ∨ Democrat political Base 2 Democrat Democrat • two extensions: 1. Republican, political 2. Democrat, political 13/43

2. If somebody writes legibly then usually the right hand is not broken. wl rhb 3. He writes legibly. wl With disjunctive default logic we get two extensions: 1. wl, rhb, lhb 2. wl, rhb Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb ∨ rhb 14/43

3. He writes legibly. wl With disjunctive default logic we get two extensions: 1. wl, rhb, lhb 2. wl, rhb Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb ∨ rhb 2. If somebody writes legibly then usually the right hand is not broken. wl ⇒ ¬ rhb 14/43

With disjunctive default logic we get two extensions: 1. wl, rhb, lhb 2. wl, rhb Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb ∨ rhb 2. If somebody writes legibly then usually the right hand is not broken. wl ⇒ ¬ rhb 3. He writes legibly. wl 14/43

Problematic Example for Disjunctive Defaults Consider the following example: 1. Either his left hand or his right hand is broken. lhb ∨ rhb 2. If somebody writes legibly then usually the right hand is not broken. wl ⇒ ¬ rhb 3. He writes legibly. wl With disjunctive default logic we get two extensions: 1. wl, ¬ rhb, lhb 2. wl, rhb 14/43

General Stratagem So far: Manipulate the database! 1. produce new defeasible rules from the given ones 2. produce new factual knowledge bases when confronted with disjunctive information 15/43

Enters: the Argumentative Approach • Instead of manipulating the knowledge base and reasoning on top of the manipulated database, • we will, in what follows, use a more direct approach to the modeling of Reasoning by Cases in the context of defeasible reasoning, following the inference scheme: A ∨ B A ⇒ · · · ⇒ C B ⇒ · · · ⇒ C C or, more generally: A ∨ B A | ∼ C B | ∼ C C • This will allow us to have more control over defeating conditions … • … and to avoid pitfalls as the ones demonstrated above. 16/43

• an argument a 2 Arg A with Conc a 2 C , and • an argument a 3 Arg B with Conc a 3 C , we introduce a new RbC-Argument a 1 a 2 a 3 C . A New Type of Argument: RbC-Arguments Basic idea: Given • an argument a 1 ∈ Arg ( T ) for which Conc ( a 1 ) = A ∨ B , 17/43

• an argument a 3 Arg B with Conc a 3 C , we introduce a new RbC-Argument a 1 a 2 a 3 C . A New Type of Argument: RbC-Arguments Basic idea: Given • an argument a 1 ∈ Arg ( T ) for which Conc ( a 1 ) = A ∨ B , • an argument a 2 ∈ Arg ( ⟨D , K ∪ { A }⟩ ) with Conc ( a 2 ) = C , and 17/43

we introduce a new RbC-Argument a 1 a 2 a 3 C . A New Type of Argument: RbC-Arguments Basic idea: Given • an argument a 1 ∈ Arg ( T ) for which Conc ( a 1 ) = A ∨ B , • an argument a 2 ∈ Arg ( ⟨D , K ∪ { A }⟩ ) with Conc ( a 2 ) = C , and • an argument a 3 ∈ Arg ( ⟨D , K ∪ { B }⟩ ) with Conc ( a 3 ) = C , 17/43

A New Type of Argument: RbC-Arguments Basic idea: Given • an argument a 1 ∈ Arg ( T ) for which Conc ( a 1 ) = A ∨ B , • an argument a 2 ∈ Arg ( ⟨D , K ∪ { A }⟩ ) with Conc ( a 2 ) = C , and • an argument a 3 ∈ Arg ( ⟨D , K ∪ { B }⟩ ) with Conc ( a 3 ) = C , we introduce a new RbC-Argument ⟨ a 1 , [ a 2 ] , [ a 3 ] � C ⟩ . 17/43

• We say that a 1 a n are hypothetical sub-arguments of a , in signs: a 1 a n HSub a . • For each a i , Hyp a i A i . More general: RbC-Argument Definition Where • a 0 ∈ Arg ( T ) with Conc ( a 0 ) = ∨ n i = 1 A i and • a i ∈ Arg ( ⟨D , K ∪ { A i }⟩ ) \ Arg ( T ) (1 ≤ i ≤ n ), ⟨ a 0 , [ a 1 ] , . . . , [ a n ] � ∨ n i = 1 Conc ( A i ) ⟩ is an RbC-argument. 18/43

• For each a i , Hyp a i A i . More general: RbC-Argument Definition Where • a 0 ∈ Arg ( T ) with Conc ( a 0 ) = ∨ n i = 1 A i and • a i ∈ Arg ( ⟨D , K ∪ { A i }⟩ ) \ Arg ( T ) (1 ≤ i ≤ n ), ⟨ a 0 , [ a 1 ] , . . . , [ a n ] � ∨ n i = 1 Conc ( A i ) ⟩ is an RbC-argument. • We say that a 1 , . . . , a n are hypothetical sub-arguments of a , in signs: a 1 , . . . , a n ∈ HSub ( a ) . 18/43

More general: RbC-Argument Definition Where • a 0 ∈ Arg ( T ) with Conc ( a 0 ) = ∨ n i = 1 A i and • a i ∈ Arg ( ⟨D , K ∪ { A i }⟩ ) \ Arg ( T ) (1 ≤ i ≤ n ), ⟨ a 0 , [ a 1 ] , . . . , [ a n ] � ∨ n i = 1 Conc ( A i ) ⟩ is an RbC-argument. • We say that a 1 , . . . , a n are hypothetical sub-arguments of a , in signs: a 1 , . . . , a n ∈ HSub ( a ) . • For each a i , Hyp ( a i ) = A i . 18/43

We have for instance the arguments: • a 1 p q r Arg T • a 2 q s v Arg q • a 3 r u v Arg r • a 4 a 1 a 2 a 3 v Arg T . q s v a2: p q r v a1: a3: r u v Let T = ⟨D , K⟩ consist of D = { p ⇒ q ∨ r , q ⇒ s , s ⇒ v , r ⇒ u , u ⇒ v , t ⇒ ¬ s } and K = { p , t } . 19/43

Let T = ⟨D , K⟩ consist of D = { p ⇒ q ∨ r , q ⇒ s , s ⇒ v , r ⇒ u , u ⇒ v , t ⇒ ¬ s } and K = { p , t } . We have for instance the arguments: • a 1 = ⟨⟨ p ⟩ ⇒ q ∨ r ⟩ ∈ Arg ( T ) • a 2 = ⟨⟨ q ⟩ ⇒ s ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { q }⟩ ) • a 3 = ⟨⟨ r ⟩ ⇒ u ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { r }⟩ ) • a 4 = ⟨ a 1 , [ a 2 ] , [ a 3 ] � v ⟩ ∈ Arg ( T ) . q s v a2: p q ∨ r v a1: r u v a3: 19/43

What about attacks? 19/43

• a 5 t s Arg T • a 1 = ⟨⟨ p ⟩ ⇒ q ∨ r ⟩ ∈ Arg ( T ) • a 2 = ⟨⟨ q ⟩ ⇒ s ⇒ v ⟩ ∈ Arg ( ⟨S , D , K ∪ { q }⟩ ) • a 3 = ⟨⟨ r ⟩ ⇒ u ⇒ v ⟩ ∈ Arg ( ⟨S , D , K ∪ { r }⟩ ) • a 4 = ⟨ a 1 , [ a 2 ] , [ a 3 ] � v ⟩ ∈ Arg ( T ) q s v a2: p q ∨ r v a1: r u v a3: 20/43

• a 1 = ⟨⟨ p ⟩ ⇒ q ∨ r ⟩ ∈ Arg ( T ) • a 2 = ⟨⟨ q ⟩ ⇒ s ⇒ v ⟩ ∈ Arg ( ⟨S , D , K ∪ { q }⟩ ) • a 3 = ⟨⟨ r ⟩ ⇒ u ⇒ v ⟩ ∈ Arg ( ⟨S , D , K ∪ { r }⟩ ) • a 4 = ⟨ a 1 , [ a 2 ] , [ a 3 ] � v ⟩ ∈ Arg ( T ) • a 5 = ⟨⟨ t ⟩ ⇒ ¬ s ⟩ ∈ Arg ( T ) a5: t ¬ s q s v a2: p q ∨ r v a1: r u v a3: 20/43

• a 6 q s Arg q • a 1 = ⟨⟨ p ⟩ ⇒ q ∨ r ⟩ ∈ Arg ( T ) • a 2 = ⟨⟨ q ⟩ ⇒ s ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { q }⟩ ) • a 3 = ⟨⟨ r ⟩ ⇒ u ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { r }⟩ ) • a 4 = ⟨ a 1 , [ a 2 ] , [ a 3 ] � v ⟩ ∈ Arg ( T ) q s v a2: p q ∨ r v a1: r u v a3: 21/43

• a 1 = ⟨⟨ p ⟩ ⇒ q ∨ r ⟩ ∈ Arg ( T ) • a 2 = ⟨⟨ q ⟩ ⇒ s ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { q }⟩ ) • a 3 = ⟨⟨ r ⟩ ⇒ u ⇒ v ⟩ ∈ Arg ( ⟨D , K ∪ { r }⟩ ) • a 4 = ⟨ a 1 , [ a 2 ] , [ a 3 ] � v ⟩ ∈ Arg ( T ) • a 6 = ⟨⟨ q ⟩ ⇒ ¬ s ⟩ ∈ Arg ( ⟨D , K ∪ { q }⟩ ) q ¬ s a6: q s v a2: p q ∨ r v a1: a3: r u v 21/43

1. a Arg T and b Arg T HArg T or 2. a HArg T and b HArg T and Hyp a Hyp b . Argumentation frameworks are now triples Arg T HArg T Attacks T . Altogether attacks are defined as follows: Attacks T Arg T Arg T Arg T HArg T HArg T HArg T where a rebuts b b 1 b n B iff Conc a B or B Conc a and Attacks again (non-nested case) Let HArg ( T ) be the set of all arguments a for which there is a b ∈ Arg ( T ) for which a ∈ HSub ( b ) . 22/43

1. a Arg T and b Arg T HArg T or 2. a HArg T and b HArg T and Hyp a Hyp b . Altogether attacks are defined as follows: Attacks T Arg T Arg T Arg T HArg T HArg T HArg T where a rebuts b b 1 b n B iff Conc a B or B Conc a and Attacks again (non-nested case) Let HArg ( T ) be the set of all arguments a for which there is a b ∈ Arg ( T ) for which a ∈ HSub ( b ) . Argumentation frameworks are now triples ⟨ Arg ( T ) , HArg ( T ) , Attacks ( T ) ⟩ . 22/43

1. a Arg T and b Arg T HArg T or 2. a HArg T and b HArg T and Hyp a Hyp b . where a rebuts b b 1 b n B iff Conc a B or B Conc a and Attacks again (non-nested case) Let HArg ( T ) be the set of all arguments a for which there is a b ∈ Arg ( T ) for which a ∈ HSub ( b ) . Argumentation frameworks are now triples ⟨ Arg ( T ) , HArg ( T ) , Attacks ( T ) ⟩ . Altogether attacks are defined as follows: Attacks ( T ) ⊆ ( Arg ( T ) × Arg ( T )) ∪ ( Arg ( T ) × HArg ( T )) ∪ ( HArg ( T ) × HArg ( T )) 22/43

2. a HArg T and b HArg T and Hyp a Hyp b . Attacks again (non-nested case) Let HArg ( T ) be the set of all arguments a for which there is a b ∈ Arg ( T ) for which a ∈ HSub ( b ) . Argumentation frameworks are now triples ⟨ Arg ( T ) , HArg ( T ) , Attacks ( T ) ⟩ . Altogether attacks are defined as follows: Attacks ( T ) ⊆ ( Arg ( T ) × Arg ( T )) ∪ ( Arg ( T ) × HArg ( T )) ∪ ( HArg ( T ) × HArg ( T )) where a rebuts b = ⟨ b 1 , . . . , b n ⇒ B ⟩ iff Conc ( a ) = ¬ B or B = ¬ Conc ( a ) and 1. a ∈ Arg ( T ) and b ∈ Arg ( T ) ∪ HArg ( T ) or 22/43

Attacks again (non-nested case) Let HArg ( T ) be the set of all arguments a for which there is a b ∈ Arg ( T ) for which a ∈ HSub ( b ) . Argumentation frameworks are now triples ⟨ Arg ( T ) , HArg ( T ) , Attacks ( T ) ⟩ . Altogether attacks are defined as follows: Attacks ( T ) ⊆ ( Arg ( T ) × Arg ( T )) ∪ ( Arg ( T ) × HArg ( T )) ∪ ( HArg ( T ) × HArg ( T )) where a rebuts b = ⟨ b 1 , . . . , b n ⇒ B ⟩ iff Conc ( a ) = ¬ B or B = ¬ Conc ( a ) and 1. a ∈ Arg ( T ) and b ∈ Arg ( T ) ∪ HArg ( T ) or 2. a ∈ HArg ( T ) and b ∈ HArg ( T ) and Hyp ( a ) = Hyp ( b ) . 22/43

Unrestricted Rebut

joint work: Jesse Heyninck and Christian Straßer 22/43

• unrestricted rebut: only the first requirement • pro: natural (Caminada) • contra: leads to trouble for many semantics such as preferred, stable, etc Status quo • in ASPIC + only restricted rebut: a rebuts b iff 1. the conclusion of a is contrary to the conclusion of b 2. and b has a defeasible top rule 23/43

Status quo • in ASPIC + only restricted rebut: a rebuts b iff 1. the conclusion of a is contrary to the conclusion of b 2. and b has a defeasible top rule • unrestricted rebut: only the first requirement • pro: natural (Caminada) • contra: leads to trouble for many semantics such as preferred, stable, etc 23/43

• Sub-argument closure: where a and b Sub a , b . • Closure under strict rules: where a 1 a n Arg T and Conc a 1 Conc a n B , also a 1 a n B Arg T • Consistency: Conc a a is consistent. • Rationality postulates: Enters: Caminada et al. (COMMA 2014) • for grounded semantics unrestricted rebut works just fine (really?) 24/43

• Closure under strict rules: where a 1 a n Arg T and Conc a 1 Conc a n B , also a 1 a n B Arg T • Consistency: Conc a a is consistent. Enters: Caminada et al. (COMMA 2014) • for grounded semantics unrestricted rebut works just fine (really?) • Rationality postulates: • Sub-argument closure: where a ∈ E and b ∈ Sub ( a ) , b ∈ E . 24/43

• Consistency: Conc a a is consistent. Enters: Caminada et al. (COMMA 2014) • for grounded semantics unrestricted rebut works just fine (really?) • Rationality postulates: • Sub-argument closure: where a ∈ E and b ∈ Sub ( a ) , b ∈ E . • Closure under strict rules: where a 1 , . . . , a n ∈ E ∩ Arg ( T ) and Conc ( a 1 ) , . . . , Conc ( a n ) ⊢ B , also ⟨ a 1 , . . . , a n → B ⟩ ∈ E ∩ Arg ( T ) 24/43

Enters: Caminada et al. (COMMA 2014) • for grounded semantics unrestricted rebut works just fine (really?) • Rationality postulates: • Sub-argument closure: where a ∈ E and b ∈ Sub ( a ) , b ∈ E . • Closure under strict rules: where a 1 , . . . , a n ∈ E ∩ Arg ( T ) and Conc ( a 1 ) , . . . , Conc ( a n ) ⊢ B , also ⟨ a 1 , . . . , a n → B ⟩ ∈ E ∩ Arg ( T ) • Consistency: { Conc ( a ) | a ∈ E} is consistent. 24/43

• first select unattacked arguments • remove the arguments attacked by the selected arguments • select unattacked arguments • remove the arguments attacked by the selected arguments • and so on … until fixed point is reached Grounded Semantics c d a b e f h i g 25/43

• remove the arguments attacked by the selected arguments • select unattacked arguments • remove the arguments attacked by the selected arguments • and so on … until fixed point is reached Grounded Semantics c d a b e f h i g • first select unattacked arguments 25/43

• select unattacked arguments • remove the arguments attacked by the selected arguments • and so on … until fixed point is reached Grounded Semantics c d a b e f h i g • first select unattacked arguments • remove the arguments attacked by the selected arguments 25/43

• remove the arguments attacked by the selected arguments • and so on … until fixed point is reached Grounded Semantics c d a b e f h i g • first select unattacked arguments • remove the arguments attacked by the selected arguments • select unattacked arguments 25/43

• and so on … until fixed point is reached Grounded Semantics c d a b e f h i g • first select unattacked arguments • remove the arguments attacked by the selected arguments • select unattacked arguments • remove the arguments attacked by the selected arguments 25/43

Grounded Semantics c d a b e f h i g • first select unattacked arguments • remove the arguments attacked by the selected arguments • select unattacked arguments • remove the arguments attacked by the selected arguments • and so on … until fixed point is reached 25/43

• Non-interference 1 Where T and T are argumentation theories and A is a formula such that Atoms A Atoms then: T A iff A Non-Interference For a set of formulas F let Atoms ( F ) be the set of all propositional atoms in F . 1 Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic relevance criterion. 26/43

Non-Interference For a set of formulas F let Atoms ( F ) be the set of all propositional atoms in F . • Non-interference 1 Where T = ⟨D , K⟩ and T ′ = ⟨D ′ , K ′ ⟩ are argumentation theories and A is a formula such that Atoms ( D ∪ K ∪ { A } ) ∩ Atoms ( D ′ ∪ K ′ ) = ∅ then: T | ∼ A iff ⟨D ∪ D ′ , K ∪ K ′ ⟩ | ∼ A . 1 Caminada, Carnielli, Dunne (JLC, 2012). Avron (2016) calls this the basic relevance criterion. 26/43

• Clearly: a p is in the grounded extension. • Now, take the knowledge base: p s s . • (Let the strict rules be closed under classical logic.) • Now, a is attacked by s s p . • As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . 27/43

• Now, take the knowledge base: p s s . • (Let the strict rules be closed under classical logic.) • Now, a is attacked by s s p . • As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. 27/43

• (Let the strict rules be closed under classical logic.) • Now, a is attacked by s s p . • As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. • Now, take the knowledge base: {⊤ ⇒ p , ⊤ ⇒ s , ⊤ ⇒ ¬ s } . 27/43

• Now, a is attacked by s s p . • As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. • Now, take the knowledge base: {⊤ ⇒ p , ⊤ ⇒ s , ⊤ ⇒ ¬ s } . • (Let the strict rules be closed under classical logic.) 27/43

• As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. • Now, take the knowledge base: {⊤ ⇒ p , ⊤ ⇒ s , ⊤ ⇒ ¬ s } . • (Let the strict rules be closed under classical logic.) • Now, a is attacked by ⟨⟨⊤ ⇒ s ⟩ , ⟨⊤ ⇒ ¬ s ⟩ → ¬ p ⟩ . 27/43

• Thus, Non-Interference doesn’t hold for unrestricted rebut. The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. • Now, take the knowledge base: {⊤ ⇒ p , ⊤ ⇒ s , ⊤ ⇒ ¬ s } . • (Let the strict rules be closed under classical logic.) • Now, a is attacked by ⟨⟨⊤ ⇒ s ⟩ , ⟨⊤ ⇒ ¬ s ⟩ → ¬ p ⟩ . • As a consequence, a is not in the grounded extension. 27/43

The problem with unrestricted rebut in ASPIC − • Take the knowledge base: {⊤ ⇒ p } . • Clearly: a = ⟨⊤ ⇒ p ⟩ is in the grounded extension. • Now, take the knowledge base: {⊤ ⇒ p , ⊤ ⇒ s , ⊤ ⇒ ¬ s } . • (Let the strict rules be closed under classical logic.) • Now, a is attacked by ⟨⟨⊤ ⇒ s ⟩ , ⟨⊤ ⇒ ¬ s ⟩ → ¬ p ⟩ . • As a consequence, a is not in the grounded extension. • Thus, Non-Interference doesn’t hold for unrestricted rebut. 27/43

Prima Facie solution • sort out inconsistent arguments (Wu, 2012: this works in ASPIC + ) • however, this doesn’t work with unrestricted rebut 28/43

Problem: • b c is inconsistent and thus filtered out • this leaves a and c in but a c out of the grounded extension. • Failure of closure! Prima Facie solution ii Let {⊤ ⇒ 1 p , p ⇒ 1 q , ⊤ ⇒ 2 ¬ ( p ∧ q ) } be our knowledge base. a b c We have, e.g., the following arguments: b ⊕ c a ⊕ c a ⊕ b • a = ⟨⊤ ⇒ 1 p ⟩ • b = ⟨ a ⇒ 1 q ⟩ • a ⊕ b = ⟨ a , b → p ∧ q ⟩ • c = ⟨⊤ ⇒ 2 ¬ ( p ∧ q ) ⟩ • a ⊕ c = ⟨ a , c → ¬ q ⟩ • b ⊕ c = ⟨ b , c → ¬ p ⟩ 29/43

Prima Facie solution ii Let {⊤ ⇒ 1 p , p ⇒ 1 q , ⊤ ⇒ 2 ¬ ( p ∧ q ) } be our knowledge base. a b c We have, e.g., the following arguments: b ⊕ c a ⊕ c a ⊕ b • a = ⟨⊤ ⇒ 1 p ⟩ • b = ⟨ a ⇒ 1 q ⟩ Problem: • a ⊕ b = ⟨ a , b → p ∧ q ⟩ • b ⊕ c is inconsistent and • c = ⟨⊤ ⇒ 2 ¬ ( p ∧ q ) ⟩ thus filtered out • a ⊕ c = ⟨ a , c → ¬ q ⟩ • this leaves a and c in but • b ⊕ c = ⟨ b , c → ¬ p ⟩ a ⊕ c out of the grounded extension. 29/43 • Failure of closure!

n • A 1 A n 1 A i , or df i n • A 1 A n 1 A i . df i • Concs a Conc b b Sub a df Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a . Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., 30/43

n • A 1 A n 1 A i . df i • Concs a Conc b b Sub a df Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a . Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., ∧ n • { A 1 , . . . , A n } = df i = 1 A i , or 30/43

• Concs a Conc b b Sub a df Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a . Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., ∧ n • { A 1 , . . . , A n } = df i = 1 A i , or ∨ n • { A 1 , . . . , A n } = df i = 1 A i . 30/43

Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a . Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., ∧ n • { A 1 , . . . , A n } = df i = 1 A i , or ∨ n • { A 1 , . . . , A n } = df i = 1 A i . • Concs ( a ) = df { Conc ( b ) | b ∈ Sub ( a ) } 30/43

Definition a gen-rebuts b iff b is defeasible and Conc a for some Concs b . Definition a gen-defeats b iff a gen-rebuts c for some c Sub b and c a . Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., ∧ n • { A 1 , . . . , A n } = df i = 1 A i , or ∨ n • { A 1 , . . . , A n } = df i = 1 A i . • Concs ( a ) = df { Conc ( b ) | b ∈ Sub ( a ) } 30/43

Enters: ASPIC ⊖ : generalized unrestricted rebut • lifting of the contrariness operator to (finite) sets of formulas, e.g., ∧ n • { A 1 , . . . , A n } = df i = 1 A i , or ∨ n • { A 1 , . . . , A n } = df i = 1 A i . • Concs ( a ) = df { Conc ( b ) | b ∈ Sub ( a ) } Definition a gen-rebuts b iff b is defeasible and Conc ( a ) = ∆ for some ∆ ⊆ Concs ( b ) . Definition a gen-defeats b iff a gen-rebuts c for some c ∈ Sub ( b ) and c ⪯ a . 30/43

Back to the example a b c b ⊕ c a ⊕ c a ⊕ b 31/43

Rationality Where the strict rules are obtained from classical logic, for weakest link we get • sub-argument closure • closure under strict rules • consistency • non-interference 32/43

Comparative Studies

Jesse Heyninck, Christian Straßer: Relations between assumption-based approaches in nonmonotonic logic and formal argumentation (NMR 2016, Cape Town, also available on Arxiv) 32/43

The landscape • ABA: assumption-based argumentation (Dung, Kowalski, Toni) • ALs: adaptive logics (Batens) • DACR: default assumptions (Makinson) • KLM: preferential semantics (Shoham, Kraus/Lehman/Magidor) 33/43