1. Commonsense Reasoning and Argumentation 14/15 HC 10: Structured argumentation (3) Henry Prakken 16 March 2015

2. Overview More about rationality postulates Related research The need for defeasible rules

3. 3 Subtleties concerning rebuttals (1) d1: Ring  Married d2: Party animal  Bachelor s1: Bachelor  ¬Married K n : Ring, Party animal d2 < d1

4. 4 Subtleties concerning rebuttals (2) d1: Ring  Married d2: Party animal  Bachelor s1: Bachelor  ¬Married s2: Married  ¬Bachelor K n : Ring, Party animal d2 < d1

5. 5 Subtleties concerning rebuttals (3) d1: Ring  Married d2: Party animal  Bachelor s1: Bachelor  ¬Married s2: Married  ¬Bachelor K n : Ring, Party animal

6. 6 Subtleties concerning rebuttals (4) R d = { ,      } R s = all deductively valid inference rules K n : d1: Ring  Married d2: Party animal  Bachelor n1: Bachelor  ¬Married Ring, Party animal

7. 7 Argumentation systems (with generalised negation) An argumentation system is a tuple AS = ( L, -, R,n) 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 n: R d  L is a naming convention for defeasible rules

8. 8 Generalised negation The – function generalises negation. If   - (  ) then: if   - (  ) then  is a contrary of  ; if   - (  ) then  and  are contradictories We write -  = ¬  if  does not start with a negation -  =  if is of the form ¬ 

9. 9 Attack and defeat (the general case) A undermines B (on  ) if Conc(A) = -  for some   Prem(B )/ K n ; A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’  Sub(B ) with a defeasible top rule; A undercuts B (on B’ ) if Conc(A) = -n(r) ’for some B’  Sub(B ) with defeasible top rule r A contrary-undermines/rebuts B (on  /B’ ) if Conc(A) is a contrary of  / Conc(B ’) A defeats B iff for some B’ A undermines B on  and either A contrary-undermines B’ on  or not A < a  ; or A rebuts B on B’ and either A contrary-rebuts B’ or not A < a B’ ; or A undercuts B on B’

10. 10 Consistency in ASPIC+ (with generalised negation) For any S  L S is directly consistent iff S does not contain two formulas  and –(  ) The strict closure Cl(S) of S is S + everything derivable from S with only R s. S is indirectly consistent iff Cl(S) is directly consistent. Parametrised by choice of strict rules

11. 11 Rationality postulates for ASPIC+ (with generalised negation) Closure under subarguments always satisfied Direct and indirect consistency: without preferences satisfied if R s closed under transposition or AS closed under contraposition; and K n is indirectly consistent; and AT is `well-formed’ with preferences satisfied if in addition  is ‘reasonable’ Weakest- and last link ordering are reasonable AT is well-formed if: If  is a contrary of  then (1)   K n and (2)  is not the consequent of a strict rule

12. Relation with other work (1) Assumption-based argumentation (Dung, Kowalski, Toni...) is special case of ASPIC+ (with generalised negation) with Only ordinary premises Only strict inference rules All arguments of equal priority …

13. Reduction of ASPIC+ defeasible rules to ABA rules (Dung & Thang, JAIR 2014) Assumptions: L consists of literals No preferences No rebuttals of undercutters p 1, …, p n  q becomes d i, p 1, …, p n,not¬q  q where: d i = n(p 1, …, p n  q) d i, not¬q are assumptions  = - (not  ),  = - (¬  ), ¬  = - (  ) 1-1 correspondence between grounded, preferred and stable extensions of ASPIC+ and ABA

14. 14 From defeasible to strict rules: example r1: Quaker  Pacifist r2: Republican  ¬Pacifist Pacifist Quaker  Pacifist Republican r1 r2

15. 15 From defeasible to strict rules: example s1: Appl(s1), Quaker, not¬Pacifist  Pacifist s2: Appl(s2), Republican, notPacifist  ¬Pacifist Pacifist QuakerAppl(s1)not¬Pacifist ¬Pacifist RepublicannotPacifistAppl(s2)

16. Can ASPIC+ preferences be reduced to ABA assumptions? d1: Bird  Flies d2: Penguin  ¬Flies d1 < d2 Becomes d1: Bird, not Penguin  Flies d2: Penguin  ¬Flies Only works in special cases, e.g. not with weakest-link ordering

17. Classical argumentation (Besnard & Hunter, …) Given L a propositional logical language and |- standard- logical consequence over L : An argument is a pair (S,p) such that S  L and p  L S |- p S is consistent No S’  S is such that S’ |- p Various notions of attack, e.g.: “Direct defeat”: argument (S,p) attacks argument (S’,p’) iff p |- ¬q for some q  S’ “Direct undercut”: argument (S,p) attacks argument (S’,p’) iff p |- ¬q and ¬q |- p for some q  S’ Only these two attacks satisfy consistency.

18. Relation with other work (2) Two variants of classical argumentation with premise attack (Amgoud & Cayrol, Besnard & Hunter) are special case of ASPIC+ with Only ordinary premises Only strict inference rules (all valid propositional or first-order inferences from finite sets) - = ¬ No preferences Arguments must have classically consistent premises …

19. Results on classical argumentation (Cayrol 1995; Amgoud & Besnard 2013) In classical argumentation with premise attack, only ordinary premises and no preferences: Preferred and stable extensions and maximal conflict-free sets coincide with maximal consistent subsets of the knowledge base So p is defensible iff there exists an argument for p The grounded extension coincides with the intersection of all maximal consistent subsets of the knowledge base So p is justified iff there exists an argument for p without counterargument Lindebaum’s lemma: Every consistent set is contained in a maximal consistent set

20. 20 Modelling default reasoning in classical argumentation Quakers are usually pacifist Republicans are usually not pacifist Nixon was a quaker and a republican

21. 21 A modelling in classical logic K p : Quaker  Pacifist Republican   ¬Pacifist Quaker, Republican Pacifist Quaker Quaker  Pacifist ¬Pacifist Republican Republican  ¬Pacifist ¬(Quaker  Pacifist)

22. 22 A modelling in classical logic K n : Quaker & ¬Ab1  Pacifist Republican & ¬Ab2   ¬Pacifist Quaker, Republican K p : ¬Ab1, ¬Ab2 (attackable) Pacifist Quaker¬Ab1 ¬Pacifist ¬Ab2Republican Quaker & ¬Ab1  Pacifist Republican & ¬Ab2  ¬Pacifist Ab1

23. 23 A modelling in classical logic Pacifist Quaker¬Ab1 ¬Pacifist ¬Ab2Republican Quaker & ¬Ab1  Pacifist Republican & ¬Ab2  ¬Pacifist Ab1 Ab2

24. Can defeasible reasoning be reduced to plausible reasoning? Is it natural to reduce all forms of attack to premise attack? My answer: no In classical argumentation: can the material implication represent defaults? My answer: no

25. Default contraposition in classical argumentation Heterosexuals are normally married. John is not married Assume when possible that things are normal What can we conclude about John’s sexual orientation?

26. Default contraposition in classical argumentation Heterosexuals are normally married H & ¬Ab  M John is not married (¬M) Assume when possible that things are normal ¬Ab The first default implies that non-married people are normally not heterosexual ¬M & ¬Ab  ¬H So John is not heterosexual

27. Default contraposition in classical argumentation (2) Men normally have no beard => Creatures with a beard are normally not men This type of sensor usually does not give false alarms => False alarms are usually not given by this type of sensor Witnesses interrogated by the police usually tell the truth => People interrogated by the police who do not speak the truth are usually not a witness Statisticians call these inferences “base rate fallacies”

28. The case of classical argumentation Birds usually fly Penguins usually don’t fly All penguins are birds Penguins are abnormal birds w.r.t. flying Tweety is a penguin

29. The case of classical argumentation Birds usually fly Bird & ¬Ab1  Flies Penguins usually don’t fly Penguin & ¬Ab2  ¬Flies All penguins are birds Penguin  Bird Penguins are abnormal birds w.r.t. flying Penguin  Ab1 Tweety is a penguin Penguin ¬Ab1 ¬Ab2

30. The case of classical argumentation Bird & ¬Ab1  Flies Penguin & ¬Ab2  ¬Flies Penguin  Bird Penguin  Ab1 Penguin ¬Ab1 ¬Ab2 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 KpKp KnKn

31. The case of classical argumentation Bird & ¬Ab1  Flies Penguin & ¬Ab2  ¬Flies Penguin  Bird Penguin  Ab1 Penguin ¬Ab1 ¬Ab2 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 - and for Ab1 and Ab2 But ¬Flies follows KpKp KnKn

32. The case of classical argumentation Bird & ¬Ab1  Flies Penguin & ¬Ab2  ¬Flies Penguin  Bird Penguin  Ab1 ObservedAsPenguin & ¬Ab3  Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3

33. The case of classical argumentation Bird & ¬Ab1  Flies Penguin & ¬Ab2  ¬Flies Penguin  Bird Penguin  Ab1 ObservedAsPenguin & ¬Ab3  Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3 - and for Ab1 and Ab2 and Ab3

34. The case of classical argumentation Bird & ¬Ab1  Flies Penguin & ¬Ab2  ¬Flies Penguin  Bird Penguin  Ab1 ObservedAsPenguin & ¬Ab3  Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3 - and for Ab1 and Ab2 and Ab3 ¬Ab3 > ¬Ab2 > ¬Ab1 makes ¬Flies follow But is this ordering natural?

35. 35 Contraposition of legal rules r1: Snores  Misbehaves r2: Misbehaves  May be removed r3: Professor  ¬May be removed K: Snores, Professor r1 < r2, r1 < r3, r3 < r2 May be removed Misbehaves Snores  May be removed Professor r1 r2 r3 This is the intuitive outcome R3 < R2

36. 36 Contraposition of legal rules r1: Snores  Misbehaves r2: Misbehaves  May be removed r3: Professor  ¬May be removed K: Snores, Professor r1 < r2, r1 < r3, r3 < r2 May be removed Misbehaves Snores  May be removed Professor r1 r2 r3 But with contraposition (and last or weakest link) we have this outcome

37. My conclusion Classical logic’s material implication is too strong for representing defeasible generalisations or legal rules => Models of legal argument (and many other kinds of argument) need defeasible inference rules Defeasible reasoning cannot be modelled as inconsistency handling in deductive logic John Pollock: Defeasible reasoning is the rule, deductive reasoning is the exception

38. Next lecture The lottery paradox Self-defeat and odd defeat loops Floating conclusions The need for dynamics