Argumentation in logic and interaction

1
Argumentation in logic and interaction

• Henry Prakken
• Chongqing, China
• October 9, 2009

2
Why do agents need argumentation?

• For their internal reasoning
• To draw conclusions given conflicting arguments
• For their interaction with other agents
• To persuade given a conflict of opinion

3
Overview

• Argumentation in logic
• semantics
• games
• Argumentation in dialogue
• distributed information
• dynamics

4
We should lower taxes
Lower taxes increase productivity
Increased productivity is good
5
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
6
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
USA lowered taxes but productivity decreased
7
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that
USA lowered taxes but productivity decreased
8
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that
Prof. P is not objective
Prof. P has political ambitions
People with political ambitions are not objective
USA lowered taxes but productivity decreased
9
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Prof. P says that
Prof. P is not objective
Prof. P has political ambitions
People with political ambitions are not objective
USA lowered taxes but productivity decreased
10
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Increased inequality is good
Prof. P says that
Prof. P is not objective
Prof. P has political ambitions
People with political ambitions are not objective
Increased inequality stimulates competition
USA lowered taxes but productivity decreased
Competition is good
11
We should lower taxes
We should not lower taxes
Lower taxes increase productivity
Increased productivity is good
Lower taxes increase inequality
Increased inequality is bad
Lower taxes do not increase productivity
Increased inequality is good
Prof. P says that
Prof. P is not objective
Prof. P has political ambitions
People with political ambitions are not objective
Increased inequality stimulates competition
USA lowered taxes but productivity decreased
Competition is good
12
1. An argument is In if all arguments defeating
it are Out. 2. An argument is Out if it is
defeated by an argument that is In.
Grounded semantics minimises node colouring
Preferred semantics maximises node colouring
Dung 1995
A
B
E
D
C
13
Structured arguments

• Arguments are trees where
• Nodes are formulas
• Links are applications of inference rules
• Strict rules (?1, ..., ?n ? ?) or
• Defeasible rules (?1, ..., ?n ? ?)
• Defeat
• on premise
• on defeasible inference
• Rebutting or undercutting
• See e.g. www.cs.uu.nl/research/techreps/UU-CS-2009
  -019.html

14
What are good semantics? Odd and even defeat
loops
D
E
D
E
A
B
A
B
C
C
F
15
R W says that p ? p
A Alice says that Bob is unreliable, so Bob is
unreliable
Exception W is unreliable
B Bob says that Carole is unreliable, so Carole
is unreliable
D
E
C Carole says that Alice is unreliable, so
Alice is unreliable
D Bob says that John was the killer, so John was
the killer
A
B
E Eric says that John was not the killer, so
John was not the killer
C
16
R W says that p ? p
A Alice says that Bob is unreliable, so Bob is
unreliable
Exception W is unreliable
B Bob says that Carole is unreliable, so Carole
is unreliable
D
E
C Carole says that Fred is unreliable, so Fred
is unreliable
F Fred says that Alice is unreliable, so Alice
is unreliable
A
B
D Bob says that John was the killer, so John was
the killer
C
F
E Eric says that John was not the killer, so
John was not the killer
17
R W says that p ? p
A Alice says that Bob is unreliable, so Bob is
unreliable
Exception W is unreliable
B Bob says that Carole is unreliable, so Carole
is unreliable
D
E
C Carole says that Fred is unreliable, so Fred
is unreliable
F Fred says that Alice is unreliable, so Alice
is unreliable
A
B
D Bob says that John was the killer, so John was
the killer
C
F
E Eric says that John was not the killer, so
John was not the killer
18
1. An argument is In if all arguments defeating
it are Out. 2. An argument is Out if it is
defeated by an argument that is In.
D
E
D
E
A
B
A
B
C
C
F
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1. An argument is In if all arguments defeating
it are Out. 2. An argument is Out if it is
defeated by an argument that is In.
D
E
D
E
A
B
A
B
C
C
F
20
1. An argument is In if all arguments defeating
it are Out. 2. An argument is Out if it is
defeated by an argument that is In.
3. An argument is acceptable if it is In in all
labellings
D
E
D
E
E is not acceptable
E is acceptable
A
B
A
B
C
C
F
21
S defends A if all defeaters of A are defeated by
a member of S S is admissible if it is
conflict-free and defends all its members
A,C,E is admissible
D
E
A
B
C
F
22
S defends A if all defeaters of A are defeated by
a member of S S is admissible if it is
conflict-free and defends all its members
A,C,E is admissible
D
E
B,D,F is admissible
A
B
C
F
23
S defends A if all defeaters of A are defeated by
a member of S S is admissible if it is
conflict-free and defends all its members
E is admissible
D
E
A
B
C
24
S defends A if all defeaters of A are defeated by
a member of S S is admissible if it is
conflict-free and defends all its members
E is admissible
D
E
but B,D is not
A
B
C
25
S defends A if all defeaters of A are defeated by
a member of S S is admissible if it is
conflict-free and defends all its members
E is admissible
D
E
but B,D is not
A
B
C
and A,B,D is not
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Proof theory

• Argument games between proponent and opponent
• Proponent starts with an argument
• Then each party replies with a suitable defeater
• A winning criterion
• E.g. the other player cannot move
• Acceptability status corresponds to existence of
  a winning strategy.

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A sound and complete game for grounded semantics

• The rules
• Each move replies to previous move
• (Proponent does not repeat moves)
• Proponent moves (strict) defeaters, opponent
  moves defeaters
• A player wins iff the other player cannot move
• Result A is in the grounded extension iff
  proponent has a winning strategy in a game about
  A.

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A defeat graph
A
F
B
C
E
D
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A game tree
move
P A
A
F
B
C
E
D
30
A game tree
move
P A
A
F
O F
B
C
E
D
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A game tree
P A
A
F
O F
B
P E
C
E
move
D
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A game tree
P A
A
F
O B
O F
move
B
P E
C
E
D
33
A game tree
P A
A
F
O B
O F
B
P C
P E
C
E
move
D
34
A game tree
P A
A
F
O B
O F
B
P C
P E
C
E
O D
move
D
35
A game tree
P A
A
F
O B
O F
B
P C
P E
P E
C
E
move
O D
D
36
Interaction

• Argument games verify status of argument (or
  statement) given a single theory (knowledge base)
• But real argumentation dialogues have
• Distributed information
• Dynamics

37
Game for grounded semantics unsound in
distributed settings
Knowledge bases
Inference rules
p ? q s ? ?q r ? ?s r ? ?p
Paul p, r
P1 q since p
Olga s
38
Game for grounded semantics unsound in
distributed settings
Knowledge bases
Inference rules
p ? q s ? ?q r ? ?s r ? ?p
Paul p, r
P1 q since p
Olga s
O1 ?q since s
39
Game for grounded semantics unsound in
distributed settings
Knowledge bases
Inference rules
p ? q s ? ?q r ? ?s r ? ?p
Paul p, r
P1 q since p
Olga s, r
O1 ?q since s
P2 ?s since r
40
Game for grounded semantics unsound in
distributed settings
Knowledge bases
Inference rules
p ? q s ? ?q r ? ?s r ? ?p
Paul p, r
P1 q since p
Olga s, r
O1 ?q since s
P2 ?s since r
O2 ?p since r
41
Need for other speech acts
Knowledge bases
Inference rules
p ? q r ? p s ? ?r
Paul r
P1 q since p
Olga s
42
Need for other speech acts
Knowledge bases
Inference rules
p ? q r ? p s ? ?r
Paul r
P1 q since p
Olga s
O1 why p?
43
Need for other speech acts
Knowledge bases
Inference rules
p ? q r ? p s ? ?r
Paul r
P1 q since p
Olga s
O1 why p?
P2 p since r
44
Need for other speech acts
Knowledge bases
Inference rules
p ? q r ? p s ? ?r
Paul r
P1 q since p
Olga s
O1 why p?
P2 p since r
O2 ?r since s
45
A real argumentation dialogue

• I claim that we should lower taxes
• Why?
• Since lower taxes increase productivity, which is
  good
• I disagree. We should not lower taxes, since that
  would increase inequality, which is bad.
• Besides, lower taxes will not increase
  productivity
• Why not?
• Since the USA recently lowered their taxes but
  productivity decreased.
• OK, I admit that lower taxes do not always
  increase productivity I retract my claim.

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Some properties that can be studied

• Correspondence with participants beliefs
• If union of beliefs implies p, can/will agreement
  on p result?
• If participants agree on p, does union of beliefs
  imply p?
• Disregarding vs. assuming agent
  strategies/tactics
• In general it will be hard to enforce agreement