1
Argumentation LogicsLecture 7:
Argumentation with structured arguments (3)
Henry PrakkenChongqing
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
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)
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 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
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)
Rd = {, }Rs = all deductively valid inference rulesK: 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(?):
A B
C
D
A B
C
D
17
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
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
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 …
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
23
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
24
Witness testimony
Critical questions: Is W sincere? Is W’s memory OK? Were W’s senses OK?
Witness W says PTherefore (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 observedTherefore (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 > T1Therefore (presumably), P isstill 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
Top Related