Argumentation Logics Lecture 7: Argumentation with structured arguments (3)

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

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

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)

Aspic framework: overview Argument structure: Trees where Nodes are wff of a logical language L Links are applications of inference rules Rs = Strict rules (1, ..., 1  ); or Rd= 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)

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

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.

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.

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

q,r  p Serial self-defeat A’ p A’ A

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

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

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

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

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

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

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

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

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”

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

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

Applying commonsense generalisations Consc of Guilt • 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 If Fleas then usually Consc of Guilt Fleas People who flea from a crime scene usually have consciousness of guilt

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

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

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

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

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

Argumentation Logics Lecture 7: Argumentation with structured arguments (3)