1. Argumentation Logics Lecture 6: Argumentation with structured arguments (2) Attack, defeat, preferences Henry Prakken Chongqing June 3, 2010

2. 2 Overview Argumentation with structured arguments: Attack Defeat Preferences

3. 3 Argumentation systems An argumentation system is a tuple AS = ( L, -, R,  ) where: L is a logical language - is a contrariness function from L to 2 L R = R s  R d is a set of strict and defeasible inference rules  is a partial preorder on R d If   - (  ) then: if   - (  ) then  is a contrary of  ; if   - (  ) then  and  are contradictories  = _ ,  = _ 

4. 4 Knowledge bases A knowledge base in AS = ( L, -, R, =  ’) is a pair ( K, =< ’) where K  L and  ’ is a partial preorder on K / K n. Here: K n = (necessary) axioms K p = ordinary premises K a = assumptions

5. 5 Structure of arguments An argument A on the basis of ( K,  ’) in ( L, -, R,  ) is:  if   K with Conc(A) = {  } Sub(A) =  DefRules(A) =  A 1,..., A n   if there is a strict inference rule Conc(A 1 ),..., Conc(A n )   Conc(A) = {  } Sub(A) = Sub(A 1 ) ...  Sub(A n )  {A} DefRules(A) = DefRules(A 1 ) ...  DefRules(A n ) A 1,..., A n   if there is a defeasible inference rule Conc(A 1 ),..., Conc(A n )   Conc(A) = {  } Sub(A) = Sub(A 1 ) ...  Sub(A n )  {A} DefRules(A) = DefRules(A 1 ) ...  DefRules(A n )  {A 1,..., A n   }

6. 6 Admissible argument orderings Let A be a set of arguments. A partial preorder  a on A is admissible if: If A is firm and strict and B is defeasible or plausible then B < a A; If A  K a and B  K a then A < a B; If A = A 1,..., A n   then for all 1 ≤ i ≤ n: A  a A i, for some 1 ≤ i ≤ n: A i  a A

7. 7 Argumentation theories An argumentation theory is a triple AT = (AS,KB,  a ) where: AS is an argumentation system KB is a knowledge base in AS  a is an admissible ordering on Args AT where Args AT = {A | A is an argument on the basis of KB in AS}

8. 8 Attack and defeat (with - = ¬ and K a =  ) A rebuts B (on B’ ) if Conc(A) = ¬Conc(B’ ) for some B’  Sub(B ); and B’ applies a defeasible rule to derive Conc(B’ ) A undercuts B (on B’ ) if Conc(A) = ¬B’ for some B’  Sub(B ); and B’ applies a defeasible rule A undermines B if Conc(A) = ¬  for some   Prem(B )/ K n ; A defeats B iff for some B’ A rebuts B on B’ and not A < a B’ ; or A undermines B and not A < a B ; or A undercuts B on B’ Naming convention implicit

9. 9 Example cont’d R : r1: p  q r2: p,q  r r3: s  t r4: t  ¬r1 r5: u  v r6: v,q  ¬t r7: p,v  ¬s r8: s  ¬p K n = { p}, K p = { s,u}

10. 10 Argument acceptability Dung-style semantics and proof theory directly apply!

11. 11 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 

12. 12 Argument preference Defined in terms of  (on R d ) and  ’ (on K ) Origins of  and  ’: domain-specific! Ordering < s on sets in terms of an ordering  (or  ’) on their elements: S1 < s S2 if there exists an s1  S1 such that for all s2  S2: s1 < s2

13. 13 Argument preference: some notation An argument A is:  if   K with DefRules(A) =  LastDefRules(A) =  A 1,..., A n   if there is a strict inference rule Conc(A 1 ),..., Conc(A n )   DefRules(A) = DefRules(A 1 ) ...  DefRules(A n ) LastDefRules(A) = LastDefRules(A 1 ) ...  LastDefRules(A n ) A 1,..., A n   if there is a defeasible inference rule Conc(A 1 ),..., Conc(A n )   DefRules(A) = DefRules(A 1 ) ...  DefRules(A n )  {A 1,..., A n   } LastDefRules(A) = {A 1,..., A n   }

14. 14 Example R d : r1: p  q r2: p  r r3: s  t R s : q, r  ¬t K: p,s

15. 15 Argument preference: two alternatives Last-link comparison: A < a B iff Condition (1) or (2) of Def 5.1.10 holds, or LastDefrules(B) < s LastDefrules(A), or LastDefrules(A/B) are empty and Prem(A) < s Prem(B) Weakest link comparison: A < a B iff Condition (1) or (2) of Def 5.1.10 holds, or Prem(A) < s Prem(B), and If Defrules(B)  , then Defrules(A) < s Defrules(B)

16. 16 Last link vs. weakest link (1) R : r1: p  q r2: p,q  r r3: s  t r4: t  ¬r1 r5: u  v r6: v  ¬t r3 < r6, r5 < r3 K: p,s,u

17. 17 Last link vs. weakest link (2) d1: In Scotland  Scottish d2: Scottish  Likes Whisky d3: Likes Fitness  ¬Likes Whisky K: In Scotland, Likes Fitness d1 < d2, d1 < d3

18. 18 Last link vs. weakest link (3) d1: Snores  Misbehaves d2: Misbehaves  May be removed d3: Professor  ¬May be removed K: Snores, Professor d1 < d2, d1 < d3