Argumentation Logics
Lecture 5:
Argumentation with structured
arguments (1)
argument structure
Henry Prakken
Chongqing
June 2, 2010
Contents
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Structured argumentation:
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Arguments
Argument schemes
2
Merits of Dung (1995)
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Framework for nonmonotonic logics
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Comparison and properties
Guidance for development
From intuitions to theoretical notions
But should not be used for KR
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The structure of arguments:
two approaches
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Both approaches: arguments are inference trees
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Assumption-based approaches (Dung-Kowalski-Toni, Besnard &
Hunter, …)
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Sound reasoning from uncertain premises
Arguments attack each other on their assumptions (premises)
Rule-based approaches (Pollock, Vreeswijk, …)
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Risky (‘defeasible’) reasoning from certain premises
Arguments attack each other on applications of defeasible
inference rules
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Aspic framework: overview
Argument structure:
 Trees where
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Nodes are wff of a logical language L
Links are applications of inference rules
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Rs = Strict rules (1, ..., 1  ); or
Rd= Defeasible rules (1, ..., 1  )
Reasoning starts from a knowledge base K  L
Defeat: attack on conclusion, premise or
inference, + preferences
Argument acceptability based on Dung (1995)
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Argumentation systems

An argumentation system is a tuple AS = (L, -,R,)
where:
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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
If   -() then:
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if   -() then  is a contrary of ;
if   -() then  and  are contradictories
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 = _,  = _ 
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Knowledge bases
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A knowledge base in AS = (L, -,R,= ’) is a
pair (K, =<’) where K  L and ’ is a partial
preorder on K/Kn. Here:
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Kn = (necessary) axioms
Kp = ordinary premises
Ka = assumptions
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Structure of
arguments
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An argument A on the basis of (K, ’) in (L, -,R, ) is:
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 if   K with
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A1, ..., An   if there is a strict inference rule Conc(A1), ...,
Conc(An)  
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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)  
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Conc(A) = {}
Sub(A) = Sub(A1)  ...  Sub(An)  {A}
DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An  }
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P
Q1,R1,R2  K
Q1
Q1, Q2  P
Q2
R1
R1, R2  Q2
R2
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Example
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}
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Types of arguments
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An argument A is:
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Strict if DefRules(A) = 
Defeasible if not
Firm if Prem(A)  Kn
Plausible if not firm
S |-  means there is a strict argument A s.t.
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Conc(A) = 
Prem(A)  S
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Domain-specific vs. inference
general inference rules Flies
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R1: Bird  Flies
R2: Penguin  Bird
Penguin  K
Bird
Penguin
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Rd = {,     }
Rs = all deductively
valid inference rules
Bird  Flies  K
Penguin  Bird  K
Penguin  K
Penguin
Flies
Bird
Bird Flies
Penguin  Bird
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Argument(ation) schemes:
general form
Premise 1,
…,
Premise n
Therefore (presumably), conclusion
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Defeasible inference rules!
But also critical questions
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Negative answers are counterarguments
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Expert testimony
(Walton 1996)
E is expert on D
E says that P
P is within D
Therefore (presumably), P is the case
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Critical questions:
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Is E biased?
Is P consistent with what other experts say?
Is P consistent with known evidence?
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Witness testimony
W says P
W was in the position to observe P
Therefore (presumably), P
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Critical questions:
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Is W sincere?
Does W’s memory function properly?
Did W’s senses function properly?
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Arguments from consequences
Action A brings about G,
G is good
Therefore (presumably), A should be done
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Critical questions:
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Does A also have bad consequences?
Are there other ways to bring about G?
...
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Temporal persistence
(Forward)
P is true at T1 and T2 > T1
Therefore (presumably), P is
still true at T2
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Critical questions:
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Was P known to be false between T1 and T2?
Is the gap between T1 and T2 too long?
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Temporal persistence
(Backward)
P is true at T1 and T2 < T1
Therefore (presumably), P was
already true at T2
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Critical questions:
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Was P known to be false between T1 and T2?
Is the gap between T1 and T2 too long?
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X murdered Y
dmp
Y murdered in
house at 4:45
V murdered in L at T &
S was in L at T 
S murdered V
X in 4:45{X in 4:30}
X in 4:45
accrual
X in 4:45{X in 5:00}
backw
temp pers
forw
temp pers
X left 5:00
X in 4:30
accrual
X in 4:30{W1}
testimony
W1: “X in 4:30”
X in 4:30{W2}
testimony
testimony
W2: “X in 4:30”
W3: “X left 5:00”
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