Argumentation Logics Lecture 5: Argumentation with structured arguments (1) argument structure Henry Prakken Chongqing June 2, 2010

2 Contents Structured argumentation: Arguments Argument schemes (Attack and defeat)

3 Merits of Dung (1995) Framework for nonmonotonic logics Comparison and properties Guidance for development From intuitions to theoretical notions But should not be used for practical applications

4 AB C D E

5 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 not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased

6 Steps in argumentation Construct arguments (from a knowledge base) Determine which arguments attack each other Determine which attacking arguments defeat each other (with preferences) Determine the dialectical status of all arguments (justified, defensible or overruled)

7 ASPIC Framework for rule- based argumentation Inspired by John Pollock ( ) Developed by Gerard Vreeswijk (1993,1997) Leila Amgoud, Martin Caminada, Henry Prakken,... ( )

8 Aspic framework: overview Argument structure: Trees where Nodes are wff of a logical language L Links are applications of inference rules R s = Strict rules (  1,...,  1   ); or R d = Defeasible rules (  1,...,  1   ) Reasoning starts from a knowledge base K  L Attack: on conclusion, premise or inference Defeat: attack + preferences Dialectical status based on Dung (1995)

9 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  = _ ,  = _  Example: classical negation as a contrariness function: - (  ) = {  } if does not start with a negation - (  ) = { ,  }

10 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

11 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   }

12 Q1Q2 P R1R2 R1, R2  Q2 Q1, Q2  P Q1,R1,R2  K

13 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 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 Implicit!

14 Example 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}

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

16 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

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

18 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

19 Witness testimony Critical questions: Is W sincere? Does W’s memory function properly? Did W’s senses function properly? W says P W was in the position to observe P Therefore (presumably), P

20 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

21 Temporal persistence (Forward) Critical questions: Was P known to be false between T1 and T2? Is the gap between T1 and T2 too long? P is true at T1 and T2 > T1 Therefore (presumably), P is still true at T2

22 Temporal persistence (Backward) Critical questions: Was P known to be false between T1 and T2? Is the gap between T1 and T2 too long? P is true at T1 and T2 < T1 Therefore (presumably), P was already true at T2

23 X murdered Y Y murdered in house at 4:45 X in 4:45 X in 4:45 {X in 4:30} X in 4:45 {X in 5:00} X left 5:00 W3: “X left 5:00”W1: “X in 4:30” W2: “X in 4:30” X in 4:30 {W1} X in 4:30 {W2} X in 4:30 accrual testimony forw temp pers backw temp pers dmp accrual V murdered in L at T & S was in L at T  S murdered V