Henry Prakken Chongqing June 4, 2010

1. Argumentation LogicsLecture 7:Argumentation with structured arguments (3) Rationality postulates, Self-defeat Henry Prakken Chongqing June 4, 2010

2. Overview • Argumentation with structured arguments: • Rationality postulates • Self-defeat • Odd and even defeat cycles

3. Rationality postulates(Caminada & Amgoud 2007) • Let E be any stable, preferred or grounded extension: • If B Sub(A) and AE then BE • The set {| = Conc(A) for some A E } is • closed under RS; • directly and indirectly consistent.

4. Rationality postulatesfor ASPIC system Closure under subarguments always satisfied Direct and indirect consistency: without ‘real’ preferences satisfied if Rs closed under transposition, or AS closed under contraposition (and some further conditions) with ‘real’ preferences satisfied if in additiona is weakest or last-link ordering

5. Subtleties concerning rebuttals (1) d1: Ring Married d2: Party animal Bachelor s1: Bachelor  ¬Married K: Ring, Party animal

6. Subtleties concerning rebuttals (2) d1: Ring Married d2: Party animal Bachelor s1: Bachelor  ¬Married s2: Married  ¬Bachelor K: Ring, Party animal

7. Subtleties concerning rebuttals (3) Rd = {,     } Rs = all deductively valid inference rules K: d1: Ring  Married d2: Party animal  Bachelor n1: Bachelor  ¬Married Ring, Party animal

8. Parallel ‘self-defeat’ q q p p

9. q,r  p Serial self-defeat A’ p A’ A

10. ¬r1 A is unreliable A: “A is unreliable” r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable

11. ¬r1 J is the killer A is unreliable A: “J is the killer” A: “A is unreliable”

12. ¬r1 J is the killer A is unreliable A: “J is the killer” A: “A is unreliable”

13. ¬r1 J is the not killer J is the killer A is unreliable B: “J is not the killer” A: “J is the killer” A: “A is unreliable”

14. R: W says that p  p A: Alice says that Bob is unreliable, so Bob is unreliable Exception: W is unreliable B: Bob says that Carole is unreliable, so Carole is unreliable E D C: Carole says that Alice is unreliable, so Alice is unreliable D: Bob says that John was the killer, so John was the killer A B E: Eric says that John was not the killer, so John was not the killer C

15. R: W says that p  p A: Alice says that Bob is unreliable, so Bob is unreliable Exception: W is unreliable B: Bob says that Carole is unreliable, so Carole is unreliable E D C: Carole says that Fred is unreliable, so Fred is unreliable F: Fred says that Alice is unreliable, so Alice is unreliable A B D: Bob says that John was the killer, so John was the killer F C E: Eric says that John was not the killer, so John was not the killer

16. R: W says that p  p A: Alice says that Bob is unreliable, so Bob is unreliable Exception: W is unreliable B: Bob says that Carole is unreliable, so Carole is unreliable E D C: Carole says that Fred is unreliable, so Fred is unreliable F: Fred says that Alice is unreliable, so Alice is unreliable A B D: Bob says that John was the killer, so John was the killer F C E: Eric says that John was not the killer, so John was not the killer

17. 1. An argument is In if all arguments defeating it are Out. 2. An argument is Out if it is defeated by an argument that is In. E D E D A B A B C F C

18. 1. An argument is In if all arguments defeating it are Out. 2. An argument is Out if it is defeated by an argument that is In. E D E D A B A B C F C

19. 1. An argument is In if all arguments defeating it are Out. 2. An argument is Out if it is defeated by an argument that is In. 3. An argument is justified if it is In in all labellings E D E D E is justified E is not justified A B A B C F C

20. S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members {A,C,E} is admissible … E D A B F C

21. S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members {A,C,E} is admissible … E D {B,D,F} is admissible … A B F C

22. S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members {E} is admissible … E D A B C

23. S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members {E} is admissible … E D but {B,D} is not … A B C

24. S defends A if all defeaters of A are defeated by a member of S S is admissible if it is conflict-free and defends all its members {E} is admissible … E D but {B,D} is not … A B C and {A,B,D} is not

25. A problem(?) with grounded semantics We have: We want(?): A B A B C C D D

26. A problem(?) with grounded semantics A B C A = Frederic Michaud is French since he has a French name B = Frederic Michaud is Dutch since he is a marathon skater C = F.M. likes the EU since he is European (assuming he is not Dutch or French) D = F.M. does not like the EU since he looks like a person who does not like the EU D

27. A problem(?) with grounded semantics E A B C A = Frederic Michaud is French since Alice says so B = Frederic Michaud is Dutch since Bob says so C = F.M. likes the EU since he is European (assuming he is not Dutch or French) D = F.M. does not like the EU since he looks like a person who does not like the EU D E = Alice and Bob are unreliable since they contradict each other