Commonsense Reasoning 10/11 HC 9 Structured argumentation (1)

1
Commonsense Reasoning 10/11HC 9Structured
argumentation (1)

• Henry Prakken
• December 22, 2010

2
Overview

• Structured argumentation
• Arguments
• Attack
• Defeat

3
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
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, ..., ?n ? ?) or
• Rd Defeasible rules (?1, ..., ?n ? ?)
• Reasoning starts from a knowledge base K ? L
• Defeat attack on conclusion, premise or
  inference, preferences
• Argument acceptability based on Dung (1995)

5
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
• If ? ? -(?) then
• if ? ? -(?) then ? is a contrary of ?
• if ? ? -(?) then ? and ? are contradictories
• ? _?, ? _?

6
Knowledge bases

• A knowledge base in AS (L, -,R,?) is a pair
  (K, ?) where K ? L and K is a partition Kn ? Kp
  ? Ka where
• Kn necessary premises
• Kp ordinary premises
• Ka assumptions
• Moreover, ? is a partial preorder on K/Kn.

7
Structure of arguments

• An argument A on the basis of (K, ?) in (L, -,R,
  ?) is
• ? if ? ? K with
• Conc(A) ?
• Sub(A) ?
• DefRules(A) ?
• A1, ..., An ? ? if there is a strict inference
  rule Conc(A1), ..., Conc(An) ? ?
• 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 ? ?

8
Rs Rd p,q ? s p ? t u,v ? w s,r,t ? v
Kn q Kp p,u Ka r
A1 p A5 A1 ? t A2 q A6 A1,A2 ? s A3
r A7 A5,A3,A6 ? v A4 u A8 A7,A4 ? w
w
u, v ? w ? Rs
p
v
u
p, q ? s ? Rs
s,r,t ? v w ? Rd
a
s
r
t
p ? t ? Rd
p
n
p
p
q
p
9
Types of arguments

• An argument A is
• 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.
• Conc(A) ?
• Prem(A) ? S

10
Rs Rd p,q ? s p ? t u,v ? w s,r,t ? v
Kn q Kp p,u Ka r
A1 p A5 A1 ? t A2 q A6 A1,A2 ? s A3
r A7 A5,A3,A6 ? v A4 u A8 A7,A4 ? w
w
p
v
u
a
An argument A is - Strict if DefRules(A) ? -
Defeasible if not strict - Firm if Prem(A) ? Kn
- Plausible if not firm
s
r
t
p
n
p
p
q
p
11
Example

• R
• r1 p ? q
• r2 p,q ? r
• r3 s ? t
• r4 t ? r1
• r5 u ? v
• r6 v,q ? t
• r7 p,v ? s
• r8 s ? p
• Kn p, Kp s,u

12
Admissible argument orderings

• Let A be a set of arguments. A partial preorder
  ?a on A is admissible if
• If A is firm and strict and B is defeasible or
  plausible then B lta A
• If A A1, ..., An ? ? then
• for all 1 ? i ? n A ?a Ai,
• for some 1 ? i ? n Ai ?a A

13
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 ArgsAT where
• ArgsAT A A is an argument on the basis of KB
  in AS

14
Attack and defeat(with only contradictories and
Ka ?)

• A undermines B (on ?) if
• Conc(A) -? for some ? ? Prem(B )/ Kn
• A rebuts B (on B ) if
• Conc(A) -Conc(B ) for some B ? Sub(B ) and
• B applies a defeasible rule to derive Conc(B )
• A undercuts B (on B ) if
• Conc(A) -B for some B ? Sub(B ) and
• B applies a defeasible rule
• A defeats B iff for some B
• A undermines B on ? and not A lta ? or
• A rebuts B on B and not A lta B or
• A undercuts B on B

Naming convention implicit
15
Example contd

• R
• r1 p ? q
• r2 p,q ? r
• r3 s ? t
• r4 t ? r1
• r5 u ? v
• r6 v,q ? t
• r7 p,v ? s
• r8 s ? p
• Kn p, Kp s,u

16
Argument acceptability

• Dung-style semantics and proof theory directly
  apply!

17
The ultimate status of conclusions

• With grounded 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
• With preferred semantics
• 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 otherwise (?)
• 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 ?

18
Domain-specific vs. inference general inference
rules
Flies

• d1 Bird ? Flies
• s1 Penguin ? Bird
• Penguin ? K
• Rd ?, ? ? ? ? ?
• Rs all valid inference
• rules of prop. l.
• Bird ? Flies ? K
• Penguin ? Bird ? K
• Penguin ? K

Bird
Penguin
Flies
Bird
Bird? Flies
Penguin
Penguin ? Bird
19
Argument(ation) schemes general form

• But also critical questions

Premise 1, , Premise n Therefore
(presumably), conclusion
20
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?

21
Arguments from consequences

• Critical questions
• Does A also have bad consequences?
• Are there other ways to bring about G?
• ...

Action A brings about G, G is good Therefore
(presumably), A should be done
22
Argument schemes in ASPIC

• Argument schemes are defeasible inference rules
• Critical questions are pointers to
  counterarguments
• Some point to undermining attacks
• Some point to rebutting attacks
• Some point to undercutting attacks

23
Witness testimony

• Critical questions
• Is W sincere?
• Does Ws memory function properly?
• Did Ws senses function properly?

W says P W was in the position to observe
P Therefore (presumably), P
24
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 gt T1 Therefore
(presumably), P is still true at T2
25
Temporal persistence(Backward)

• 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 lt T1 Therefore
(presumably), P was already true at T2
26
X murdered Y
d.m.p.
Y murdered in house at 445
X in 445
V murdered in L at T S was in L at T ? S
murdered V
accrual
X in 445X in 430
X in 445X in 500
backw temp pers
forw temp pers
X left 500
X in 430
accrual
X in 430W1
X in 430W2
testimony
testimony
testimony
W2 X in 430
W3 X left 500
W1 X in 430