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

2 Contents Structured argumentation: Arguments Argument schemes

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 KR

4 The structure of arguments: two approaches Both approaches: arguments are inference trees Assumption-based approaches (Dung-Kowalski-Toni, Besnard & Hunter, …) Sound reasoning from uncertain premises Arguments attack each other on their assumptions (premises) Rule-based approaches (Pollock, Vreeswijk, …) Risky (‘defeasible’) reasoning from certain premises Arguments attack each other on applications of defeasible inference rules

5 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 Defeat: attack on conclusion, premise or inference, + preferences Argument acceptability based on Dung (1995)

6 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  = _ ,  = _ 

7 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

8 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   }

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

10 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}

11 Types of arguments An argument A is: Strict if DefRules(A) =  Defeasible if not 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

12 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

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

14 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

15 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

16 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

17 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

18 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

19 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