Argumentation Logics Lecture 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: 1. If B  Sub(A) and A  E then B  E 2. The set {  |  = Conc(A) for some A  E } is closed under R S ; directly and indirectly consistent.

4 Rationality postulates for ASPIC system Closure under subarguments always satisfied Direct and indirect consistency: without ‘real’ preferences satisfied if R s 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) R d = { ,      } R s = all deductively valid inference rules K: d1: Ring  Married d2: Party animal  Bachelor n1: Bachelor  ¬Married Ring, Party animal

8 Parallel ‘self-defeat’ p pp qq q

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

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

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

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

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

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

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

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

17 AB C DE AB DE CF 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.

18 AB C DE AB DE CF 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.

19 AB C DE AB DE CF 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 is not justifiedE is justified 3. An argument is justified if it is In in all labellings

20 AB DE CF 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 …

21 AB DE CF 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 … {B,D,F} is admissible …

22 AB C DE 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 …

23 AB C DE 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 … but {B,D} is not …

24 AB C DE 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 … but {B,D} is not … and {A,B,D} is not

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

26 A problem(?) with grounded semantics AB C D 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

27 A problem(?) with grounded semantics AB C D 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 E E = Alice and Bob are unreliable since they contradict each other