Commonsense Reasoning and Argumentation 13/14 HC 8 Structured argumentation (1)

1. Commonsense Reasoning and Argumentation 13/14HC 8Structured argumentation (1) Henry Prakken March 3, 2014 (updates March 5, 2014)

2. Overview • Structured argumentation: • Arguments • Attack • Defeat

3. A B E D C

4. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

5. We should lower taxes Lower taxes increase productivity Increased productivity is good

6. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad

7. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity USA lowered taxes but productivity decreased

8. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … USA lowered taxes but productivity decreased

9. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Prof. P has political ambitions

10. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Prof. P has political ambitions

11. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

12. We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good

13. Two accounts of the fallibility of arguments • Plausible Reasoning: all fallibility located in the premises • Assumption-based argumentation (Kowalski, Dung, Toni,… • Classical argumentation (Cayrol, Besnard, Hunter, …) • Defeasible reasoning: all fallibility located in the defeasible inferences • Pollock, Loui, Vreeswijk, Prakken & Sartor, DeLP, … • ASPIC+ combines these accounts

14. Aspic+ framework: overview Argument structure: • Trees where • Nodes are wff of a logical language L • Links are applications of inference rules • Rs = Strict rules (1, ..., n  ); or • Rd= Defeasible rules (1, ..., n  ) • Reasoning starts from a knowledge base K L • Defeat: attack on conclusion, premise or inference, + preferences • Argument acceptability based on Dung (1995)

15. Argumentation systems (with symmetric negation) An argumentation system is a triple AS = (L,R,n) where: L is a logical language closed under negation (¬) R = Rs Rd is a set of strict and defeasible inference rules n: RdL is a naming convention for defeasible rules Notation: - = ¬ if  does not start with a negation - =  if is of the form ¬

16. Knowledge bases A knowledge base in AS = (L,R,n) is a set K L where K is a partition Kn Kpwith: Kn = necessary premises Kp = ordinary premises

17. Argumentation theories An argumentation theory is a pair AT = (AS, K) where AS is an argumentation system and K a knowledge base in AS.

18. Structure of arguments • An argumentA on the basis of an argumentation theory is: •  if K with • Prem(A) = {}, Conc(A) = , Sub(A) = {}, DefRules(A) = • A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)   • Prem(A) = Prem(A1)  ...  Prem(An) • Conc(A) =  • Sub(A) = Sub(A1)  ...  Sub(An)  {A} • DefRules(A) = DefRules(A1)  ...  DefRules(An) • A1, ..., An if there is a defeasible inference rule Conc(A1), ..., Conc(An)  • Prem(A) = Prem(A1)  ...  Prem(An) • Conc(A) =  • Sub(A) = Sub(A1)  ...  Sub(An)  {A} • DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An}

19. Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,r,u} A1 = p A5 = A1  t A2 = q A6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = u A8 = A7,A4  w w u, v  w Rs p v u p, q  s Rs s,r,t  v Rd p t r s p  t Rd p n p p q p

20. Types of arguments An argument A is: Strict if DefRules(A) =  Defeasible if not strict Firm if Prem(A)  Kn Plausible if not firm S |-  means there is a strict argument A s.t. Conc(A) =  Prem(A)  S

21. Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,r,u} A1 = p A5 = A1  t A2 = q A6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = u A8 = A7,A4  w w p v u p An argument A is: - Strict if DefRules(A) =  - Defeasible if not strict - Firm if Prem(A)  Kn - Plausible if not firm t r s p n p p q p

22. Example R: • d1: p  q • d2: s  t • d3: t  ¬d1 • d4: u  v • d5: v,q  ¬t • d6: s  ¬p • s1: p,q  r • s2: p,v  ¬s Kn = {p}, Kp = {s,u}

23. Attack • AunderminesB (on ) if • Conc(A) = -for some   Prem(B )/ Kn; • ArebutsB (on B’ ) if • Conc(A) = -Conc(B’ ) for some B’ Sub(B) with a defeasible top rule • AundercutsB (on B’ ) if • Conc(A) = -n(r )’for some B’ Sub(B ) with defeasible top rule r • A attacksB iff A undermines or rebuts or undercuts B.

24. Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,r,u} A1 = p A5 = A1  t A2 = q A6 = A1,A2  s A3 = r A7 = A5,A3,A6  v A4 = u A8 = A7,A4  w w p v u p t r s p n p p q p

25. Structured argumentation frameworks Let AT = (AS,K) be an argumentation theory A structured argumentation framework (SAF) defined by AT is a triple (Args,C, a) where Args = {A | A is a finite argument on the basis of K in AS } C is the attack relation on Args ais an ordering on Args (A <aB iff AaB and not BaA) A c-SAF is a SAF in which all arguments have indirectly consistent premises (to be defined later)

26. Defeat AunderminesB (on ) if Conc(A) = - for some   Prem(B )/ Kn; ArebutsB (on B’ ) if Conc(A) = -Conc(B’ ) for some B’ Sub(B ) with a defeasible top rule AundercutsB (on B’ ) if Conc(A) = -n(r)’for some B’ Sub(B ) with defeasible top rule r A defeatsB iff for some B’ A undermines B on B’ = and not A <; or A rebuts B on B’and not A <B’ ; or A undercuts B on B’ Direct vs. subargument attack/defeat Preference-dependent vs. preference-independent attacks 26

27. Example cont’d R: • d1: p  q • d2: s  t • d3: t  ¬r1 • d4: u  v • d5: v,q  ¬t • d6: s  ¬p • s1: p,q  r • s2: p,v  ¬s Kn = {p}, Kp = {s,u}

28. Abstract argumentation frameworks corresponding to SAFs An abstract argumentation framework corresponding to a SAF = (Args,C, ) is a pair (Args,D) where D is the defeat relation on Args defined by C and . 28

29. The ultimate status of conclusions • With grounded semantics: • A is justified if A  g.e. • A is overruled if A  g.e. and A is defeated by g.e. • A is defensible otherwise • With preferred semantics: • A is justified if A  p.e for all p.e. • A is defensible if A  p.e. for some but not all p.e. • A is overruled otherwise (?) • In all semantics: •  is justified if  is the conclusion of some justified argument (Alternative: if all extensions contain an argument for ) •  is defensible if  is not justified and  is the conclusion of some defensible argument •  is overruled if  is not justified or defensible and there exists an overruled argument for 

30. A B We should lower taxes We should not lower taxes Lower taxes increase productivity Increased productivity is good Lower taxes increase inequality Increased inequality is bad Increased inequality is good Lower taxes do not increase productivity Prof. P says that … Prof. P is not objective E People with political ambitions are not objective USA lowered taxes but productivity decreased Increased inequality stimulates competition Prof. P has political ambitions Competition is good C D

31. A B E D C

32. A B A’ E D C

33. P1 P2 P3 P4 A B A’ E D C P5 P6 P7 P8 P9

35. A3 A2 B3 B2 D4 A1 C3 B1 D3 C2 C1

36. A3 A2 B3 B2 D4 A1 C3 B1 D3 C2 D4 <a B2 C1

37. A3 A2 B3 B2 D4 A1 C3 B1 D3 C2 D4 <a B2 C1

39. 1. An argument is In iff all arguments that defeat it are Out. 2. An argument is Out iff some argument that defeats it is In. P.M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n–person games. Artificial Intelligence, 77:321–357, 1995. Dung 1. An argument is In iff a. all arguments that defeat it are Out; and b. All its immediate subarguments are In 2. An argument is Out iff a. some argument that defeats it is In; or b. Some of its immediate subarguments are Out Pollock J.L. Pollock, Justification and defeat. Artificial Intelligence, 67:377–407, 1994.