Argumentation Logics Lecture 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 2 L R = R s  R d is a set of strict and defeasible inference rules  is a partial preorder on R d Example: classical negation as a contrariness function: - (  ) = {  } if does not start with a negation - (  ) = { ,  }

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 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 R s = all valid inference rules of propositional and first-order logic R d = { ,      } K p = { (1) Information I concerns health of person P (2) Person P does not agree with publication of information I (3) i is innformation concerning health of person p  i is information concerning private life of person p (4) (i is information concerning private life of person p & Person p does not agree with publication of information i)  It is forbidden to publish information i } Forbidden to publish I (i concerns health of p & p does not agree with publication of p )  Forbidden to publish i I concerns private life of P & P does not agree with publication of I I concerns private life of P P does not agree with publication of I I concerns health of P i concerns health of p  i concerns private life of p 1,2,3,4  K ,       R s ,       R d ,    &   R s  -elimination not shown!

7 Domain-specific vs. inference general inference rules R1: Bird  Flies R2: Penguin  Bird Penguin  K R d = { ,      } R s = all deductively valid inference rules Bird  Flies  K Penguin  Bird  K Penguin  K Flies Bird Penguin Flies Bird Bird  Flies Penguin Penguin  Bird

8 Argument(ation) schemes: general form Defeasible inference rules! But also critical questions Negative answers are counterarguments Premise 1, …, Premise n Therefore (presumably), conclusion

9 Expert testimony (Walton 1996) Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence? E is expert on D E says that P P is within D Therefore (presumably), P is the case

10 Arguments from consequences Critical questions: Does A also have bad consequences? Are there other ways to bring about G?... Action A brings about G, G is good Therefore (presumably), A should be done

11 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}

12 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

13 We should lower taxes Lower taxes increase productivity Increased productivity is good

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

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

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

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

18 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} Naming convention for undercutters: negate the name of the inference rule

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

20 The dialectical 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 if A  p.e for no p.e. 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 

21 Argument preference (informal)  a can be defined in any way  a could be defined in terms of  (on R d ) and/or  ’ (on K ) Origins of  and  ’: domain-specific!

22 Argument preference: two alternatives (Informal, ordering on K ignored) Last-link comparison: A < a B iff the last defeasible rule of B is strictly preferred over the last defeasible rule of A Weakest link comparison: A < a B iff the weakest defeasible rule of B is strictly preferred over the last defeasible rules of A

23 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

24 Last link vs. weakest link (2) r1: In Scotland  Scottish r2: Scottish  Likes Whisky r3: Likes Fitness  ¬Likes Whisky K: In Scotland, Likes Fitness r1 < r2, r1 < r3

25 Last link vs. weakest link (3) r1: Snores  Misbehaves r2: Misbehaves  May be removed r3: Professor  ¬May be removed K: Snores, Professor r1 < r2, r1 < r3