Argumentation Logics Lecture 7: Argumentation with structured arguments (3) Henry Prakken Chongqing June 4, 2010

2 Overview Argumentation with structured arguments: Rationality postulates Self-defeat Floating conclusions Legal proof is defeasible Can be modelled with argumentation logics But dynamics is also important

3 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)

4 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 + preference ordering on arguments Dialectical status based on Dung (1995)

5 Rationality postulates (Caminada & Amgoud 2007) Let E be any stable, preferred or grounded extension: 1. If B  Sub(A) and A  E then B  E 1. Always satisfied in ASPIC 2. The set {  |  = Conc(A) for some A  E} is closed under R S and consistent. 1. Only satisfied in ASPIC with further conditions on strict rules and argument ordering

6 Example violation of consistency d1: Ring  Married d2: Party animal  Bachelor s1: Bachelor  ¬Married K : {Ring, Party animal} d2 < d1 With both the last-link and weakest-link ordering, both “Married” and “¬Married” are justified conclusions.

7 Solution: add ‘transposition’ of strict rules d1: Ring  Married d2: Party animal  Bachelor s1: Bachelor  ¬Married s2: Married  ¬Bachelor K : {Ring, Party animal} d2 < d1 With both the last-link and weakest-link ordering, “Married” is a justified and “¬Married” is an overruled conclusion.

8 Subtleties concerning rebuttals (3) R d = { ,      } R s = all deductively valid inference rules K: d1: Ring  Married d2: Party animal  Bachelor n1: Bachelor  ¬Married Ring, Party animal

9 Serial self-defeat p  A’ q,r  p A’A

10 r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable ¬r1 A is unreliable A: “A is unreliable”

11 r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable ¬r1 A is unreliable A: “A is unreliable” “A is unreliable” and “¬r1” cannot have a status

12 ¬r1 A is unreliable A: “A is unreliable” J is the killer A: “J is the killer” r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable

13 ¬r1 A is unreliable A: “A is unreliable” J is the killer A: “J is the killer” r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable

14 ¬r1 A is unreliable A: “A is unreliable” J is the killer A: “J is the killer” J is the not killer B: “J is not the killer” r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable Grounded semantics

15 ¬r1 A is unreliable A: “A is unreliable” J is the killer A: “J is the killer” J is not the killer B: “J is not the killer” r1: W says that p  p r2: W is unreliable  ¬r1 k1: Alice says that Alice is unreliable Preferred semantics

16 A problem(?) with grounded semantics We have: We want(?): AB C D AB C D

17 A similar “problem” floating conclusions d1: x was born in Netherlands  x is Dutch d2: x has Chinese name  X is Chinese d3: x is Dutch  x likes badminton d4: x is Chinese  x likes badminton k1: Wah-chi was born in the Netherlands k2: Wah-chi has a Chinese name

18 Mei li is Chinese Mei li has a Chinese name Mei li is Dutch Mei li was born in The Netherlands  is justified iff all extensions contain an argument with conclusion  (but it does not have to be the same argument) Mei li likes badminton In grounded semantics  is defensible, in preferred semantics  is justified

19 Floating conclusions: still invalid? (Horty) Witness John says: the suspect shot the victim to death If a witness says P then usually P is the case So, the suspect shot the victim to death So, the suspect killed the victim Witness Bob says: the suspect stabbed the victim to death If a witness says P then usually P is the case So, the suspect stabbed the victim to death So, the suspect killed the victim One solution: add an undercutter “if two witnesses contradict each other, then they are both unreliable”

20 Floating conclusions: Don’t ignore dynamics Any judge would ask further questions Did you hear anything? Where did you stand? How dark was it? The law’s way of dealing with dynamics: Procedures for fair and effective dispute resolution

21 Uncertainty in legal proof Legal proof of facts is (almost?) never conclusive Witnesses can be unreliable Documents can be forged DNA tests have an error margin Confessions might be false Experts sometimes disagree … So legal proof is defeasible

22 Applying commonsense generalisations Critical questions: are there exceptions to the generalisation? exceptional classes of people may have other reasons to flea Illegal immigrants Customers of prostitutes … P If P then usually Q Therefore (presumably), Q People who flea from a crime scene usually have consciousness of guilt Consc of Guilt Fleas If Fleas then usually Consc of Guilt

23 Expert testimony Critical questions: Is E biased? Do other experts disagree? Are E’s statements based on evidence? E is expert on P E says that P Therefore (presumably), P is the case

24 Witness testimony Critical questions: Is W sincere? Is W’s memory OK? Were W’s senses OK? Witness W says P Therefore (presumably), P

25 Explanation (Abduction) Critical questions: Could there be another reason why Q has been observed? Does P cause something else which we know to be false? … P causes Q Q has been observed Therefore (presumably), P is the case

26 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

27 Final remarks Legal proof is dialectic Considering pro and con Quality of investigation influences quality of proof If you don’t search for counterevidence, you will not find it … The structure and nature of arguments guides this search