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Henry Prakken Chongqing June 3, 2010

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

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Henry Prakken Chongqing June 3, 2010

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

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

  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 2L R = Rs Rd is a set of strict and defeasible inference rules  is a partial preorder on Rd If   -() then: if   -() then  is a contrary of ; if   -() then  and  are contradictories  = _,  = _ 

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

  5. Structure of arguments • An argumentA on the basis of (K, ’) in (L, -,R, ) is: •  if K with • Conc(A) = {} • Sub(A) =  • DefRules(A) = • A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(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)  • Conc(A) = {} • Sub(A) = Sub(A1)  ...  Sub(An)  {A} • DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An}

  6. Admissible argument orderings Let Abe a set of arguments. A partial preorder aon A is admissible if: If A is firm and strict and B is defeasible or plausible then B<aA; If A Ka and B Ka then A<aB; If A = A1, ..., An then for all 1 ≤ i ≤ n: AaAi, for some 1 ≤ i ≤ n: AiaA

  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 ATwhere Args AT= {A | A is an argument on the basis of KB in AS}

  8. Attack and defeat(with - = ¬ and Ka = ) • ArebutsB (on B’ ) if • Conc(A) = ¬Conc(B’ ) for some B’ Sub(B ); and • B’ applies a defeasible rule to derive Conc(B’ ) • AundercutsB (on B’ ) if • Conc(A) = ¬B’ for some B’ Sub(B ); and • B’ applies a defeasible rule • AunderminesB if • Conc(A) = ¬ for some   Prem(B )/Kn; • A defeatsB iff for some B’ • A rebuts B on B’ and not A <aB’ ; or • A undermines B and not A <aB ; or • A undercuts B on B’ Naming convention implicit

  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 Kn = {p}, Kp = {s,u}

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

  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. Argument preference • Defined in terms of  (on Rd) and ’ (on K) • Origins of  and ’: domain-specific! • Ordering <s onsets 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. Argument preference: some notation An argumentA is:  if K with DefRules(A) = LastDefRules(A) = A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)   DefRules(A) = DefRules(A1)  ...  DefRules(An) LastDefRules(A) =LastDefRules(A1)  ...  LastDefRules(An) A1, ..., An if there is a defeasible inference rule Conc(A1), ..., Conc(An)  DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An} LastDefRules(A) ={A1, ..., An}

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

  15. Argument preference: two alternatives Last-link comparison: A <aB iffCondition (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 <aB iffCondition (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. 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. 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. 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

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