Argumentation Logics
Lecture 6:
Argumentation with structured
arguments (2)
Attack, defeat, preferences
Henry Prakken
Chongqing
June 3, 2010
Overview

Argumentation with structured
arguments:



Attack
Defeat
Preferences
2
Argumentation systems

An argumentation system is a tuple AS = (L, -,R,)
where:





L is a logical language
- is a contrariness function from L to 2L
R = Rs Rd is a set of strict and defeasible inference rules
 is a partial preorder on Rd
Example: classical negation as a contrariness
function:


-()
= {} if does not start with a negation
-() = {, }
3
Knowledge bases

A knowledge base in AS = (L, -,R,= ’) is a
pair (K, ’) where K  L and ’ is a partial
preorder on K/Kn. Here:



Kn = (necessary) axioms
Kp = ordinary premises
Ka = assumptions
4
Structure of
arguments

An argument A on the basis of (K, ’) in (L, -,R, ) is:

 if   K with




A1, ..., An   if there is a strict inference rule Conc(A1), ...,
Conc(An)  




Conc(A) = {}
Sub(A) = 
DefRules(A) = 
Conc(A) = {}
Sub(A) = Sub(A1)  ...  Sub(An)  {A}
DefRules(A) = DefRules(A1)  ...  DefRules(An)
A1, ..., An   if there is a defeasible inference rule Conc(A1), ...,
Conc(An)  



Conc(A) = {}
Sub(A) = Sub(A1)  ...  Sub(An)  {A}
DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An  }
5
Rs = all valid inference
rules of propositional
and first-order logic
Rd = {,     }
Kp = { (1) Information I concerns health of person P
(2) Person P does not agree with publication of information I
(3) i is innformation concerning health of person p 
i is information concerning private life of person p
(4) (i is information concerning private life of person p &
Person p does not agree with publication of information i) 
It is forbidden to publish information i }
-elimination
Forbidden to publish I
,       Rd
not shown!
(i concerns health of p &
p does not agree with publication
of p ) Forbidden to publish i
1,2,3,4  K
I concerns private life of P &
P does not agree with
publication of I
I concerns private
life of P
P does not agree with
publication of I
,   &   Rs
i concerns health of p 
i concerns private life of p
,       Rs
I concerns
health of P
6
Domain-specific vs. inference
general inference rules Flies



R1: Bird  Flies
R2: Penguin  Bird
Penguin  K
Bird
Penguin





Rd = {,     }
Rs = all deductively
valid inference rules
Bird  Flies  K
Penguin  Bird  K
Penguin  K
Penguin
Flies
Bird
Bird Flies
Penguin  Bird
7
Argument(ation) schemes:
general form
Premise 1,
…,
Premise n
Therefore (presumably), conclusion


Defeasible inference rules!
But also critical questions

Negative answers are counterarguments
8
Expert testimony
(Walton 1996)
E is expert on D
E says that P
P is within D
Therefore (presumably), P is the case

Critical questions:



Is E biased?
Is P consistent with what other experts say?
Is P consistent with known evidence?
9
Arguments from consequences
Action A brings about G,
G is good
Therefore (presumably), A should be done

Critical questions:



Does A also have bad consequences?
Are there other ways to bring about G?
...
10
Argumentation theories

An argumentation theory is a triple AT = (AS,KB, a)
where:

AS is an argumentation system
KB is a knowledge base in AS

a is an (admissible) ordering on Args


AT
where
Args AT = {A | A is an argument on the basis of KB in AS}
11
Attack and defeat
(with - = ¬ and Ka = )

A rebuts B (on B’ ) if



A undercuts B (on B’ ) if




Conc(A) = ¬Conc(B’ ) for some B’  Sub(B ); and
B’ applies a defeasible rule
to derive
Conc(B’ )
Naming
convention
implicit
Conc(A) = ¬B’ for some B’  Sub(B ); and
B’ applies a defeasible rule
A undermines B if

Conc(A) = ¬ for some   Prem(B )/Kn;
A defeats B iff for some B’



A rebuts B on B’ and not A <a B’ ; or
A undermines B and not A <a B ; or
A undercuts B on B’
12
We should lower taxes
Lower taxes
increase
productivity
Increased
productivity
is good
13
We should lower taxes
Lower taxes
increase
productivity
Increased
productivity
is good
We should not lower taxes
Lower taxes
increase
inequality
Increased
inequality
is bad
14
We should lower taxes
Lower taxes
increase
productivity
We should not lower taxes
Increased
productivity
is good
Lower taxes
increase
inequality
Increased
inequality
is bad
Lower taxes do
not increase
productivity
USA lowered
taxes but
productivity
decreased
15
We should lower taxes
Lower taxes
increase
productivity
Prof. P says
that …
We should not lower taxes
Increased
productivity
is good
Lower taxes
increase
inequality
Increased
inequality
is bad
Lower taxes do
not increase
productivity
USA lowered
taxes but
productivity
decreased
16
We should lower taxes
Lower taxes
increase
productivity
Prof. P says
that …
People with
political
ambitions
are biased
We should not lower taxes
Increased
productivity
is good
Prof. P is
biased
Prof. P has
political
ambitions
Lower taxes
increase
inequality
Increased
inequality
is bad
Lower taxes do
not increase
productivity
USA lowered
taxes but
productivity
decreased
17
Example cont’d
R:
 r1:
 r2:
 r3:
 r4:
 r5:
 r6:
 r7:
 r8:
pq
p,q  r
st
t  ¬r1
uv
v,q  ¬t
p,v  ¬s
s  ¬p
Kn = {p}, Kp = {s,u}
Naming convention for undercutters:
negate the name of the inference rule
18
Argument acceptability

Dung-style semantics and proof theory
directly apply!
19
The dialectical status of
conclusions

With grounded semantics:




With preferred semantics:




A is justified if A  g.e.
A is overruled if A  g.e. and A is defeated by g.e.
A is defensible otherwise
A is justified if A  p.e for all p.e.
A is defensible if A  p.e. for some but not all p.e.
A is overruled if A  p.e for no p.e.
In all semantics:



 is justified if  is the conclusion of some justified argument
(Alternative: if all extensions contain an argument for )
 is defensible if  is not justified and  is the conclusion of
some defensible argument
 is overruled if  is not justified or defensible and there
exists an overruled argument for 
20
Argument preference
(informal)

a can be defined in any way
a could be defined in terms of  (on
Rd) and/or ’ (on K)

Origins of  and ’: domain-specific!

21
Argument preference:
two alternatives
(Informal, ordering on K ignored)
 Last-link comparison:


A <a B iff the last defeasible rule of B is
strictly preferred over the last defeasible
rule of A
Weakest link comparison:

A <a B iff the weakest defeasible rule of B
is strictly preferred over the last defeasible
rules of A
22
Last link vs. weakest link (1)
R:
 r1: p  q
 r2: p,q  r
 r3: s  t
 r4: t  ¬r1
 r5: u  v
 r6: v  ¬t
r3 < r6, r5 < r3
K:

p,s,u
23
Last link vs. weakest link (2)





r1: In Scotland  Scottish
r2: Scottish  Likes Whisky
r3: Likes Fitness  ¬Likes Whisky
K: In Scotland, Likes Fitness
r1 < r2, r1 < r3
24
Last link vs. weakest link (3)





r1: Snores  Misbehaves
r2: Misbehaves  May be removed
r3: Professor  ¬May be removed
K: Snores, Professor
r1 < r2, r1 < r3
25