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Argumentation LogicsLecture 7:

Argumentation with structured arguments (3)

Henry PrakkenChongqing

June 4, 2010

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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

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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)

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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)

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Rationality postulates(Caminada & Amgoud 2007)

Let E be any stable, preferred or grounded extension:

1. If B Sub(A) and A E then B E1. Always satisfied in ASPIC

2. The set {| = Conc(A) for some A E} is closed under RS and consistent.

1. Only satisfied in ASPIC with further conditions on strict rules and argument ordering

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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.

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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.

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Subtleties concerning rebuttals (3)

Rd = {, }Rs = all deductively valid inference rulesK: d1: Ring Married d2: Party animal Bachelor n1: Bachelor ¬Married Ring, Party animal

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Serial self-defeat

p

A’

q,r p

A’ A

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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”

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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

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¬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

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¬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

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¬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

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¬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

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A problem(?) with grounded semantics

We have: We want(?):

A B

C

D

A B

C

D

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A similar “problem”floating conclusions

d1: x was born in Netherlands x is Dutch

d2: x has Chinese name X is Chinesed3: x is Dutch x likes badmintond4: x is Chinese x likes badmintonk1: Wah-chi was born in the Netherlandsk2: Wah-chi has a Chinese name

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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

Mei li likes badminton

In grounded semantics is defensible, in preferred semantics is justified

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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”

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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

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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

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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 …

PIf P then usually QTherefore (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

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Expert testimony

Critical questions: Is E biased? Do other experts disagree? Are E’s statements based on evidence?

E is expert on PE says that PTherefore (presumably), P is the case

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Witness testimony

Critical questions: Is W sincere? Is W’s memory OK? Were W’s senses OK?

Witness W says PTherefore (presumably), P

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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 observedTherefore (presumably), P is the case

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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 > T1Therefore (presumably), P isstill true at T2

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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