1. Henry Prakken Guangzhou (China) 10 April 2018
Spring School on Argumentation in AI & Law Day 1 – lecture 2 Structured argumentation
Henry Prakken
Guangzhou (China)
10 April 2018

2. Overview
Structured argumentation:
Arguments
Attack
Defeat

3. From abstract to structured argumentation
Never forget that this is an abstraction of more concrete argumentation structures.
P.M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n–person games.
Artificial Intelligence, 77:321–357, 1995.

4. From abstract to structured argumentation
Dung defines status of arguments given a set of arguments and a defeat relation
But we must also have a theory on
How can we construct arguments?
How can we attack (and defeat) arguments?

5. Attack on conclusion Employer is liable Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
No safety instructions
The next slides introduce structured argumentation.
Attack on conclusion

6. Attack on premise Employer is liable Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
No safety instructions
Employee had no work-related injury
Attack on premise
Injury caused by poor physical condition

7. Attack on inference Employer is liable Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
No safety instructions
Employee had no work-related injury
Colleague says so
Colleague is not credible
Attack on inference
Injury caused by poor physical condition
C is friend of claimant

8. Employee secured the stairs
Employer is liable
Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
Employee secured the stairs
No safety instructions
Employee had no work-related injury
Colleague says so
Colleague is not credible
Camera evidence
Injury caused by poor physical condition
Indirect defence
C is friend of claimant

9. Employee secured the stairs
Employer is liable
Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
Employee secured the stairs
No safety instructions
Employee had no work-related injury
Colleague says so
Colleague is not credible
Camera evidence
Injury caused by poor physical condition
C is friend of claimant

10. From abstract to structured argumentation
Never forget that this is an abstraction of more concrete argumentation structures.
P.M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n–person games.
Artificial Intelligence, 77:321–357, 1995.

11. Employee secured the stairsB
Employer is liable
Employer is not liable
Employer breached duty of care
Employee had work-related injury
Employee was careless
Rule 2
Rule 1
Employee did not secure stairs
Employee secured the stairs
No safety instructions
Employee had no work-related injury
Colleague says so
Colleague is not credible
Camera evidence
Injury caused by poor physical condition E
C is friend of claimant C D

12. Two accounts of the fallibility of arguments
Tony Hunter
Nicholas Rescher
Plausible Reasoning: all fallibility located in the premises
Assumption-based argumentation (Kowalski, Dung, Toni,…
Classical argumentation (Cayrol, Besnard, Hunter, …)
Defeasible reasoning: all fallibility located in the defeasible inferences
Pollock, Loui, Vreeswijk, Prakken & Sartor, …
ASPIC+ combines these accounts
Robert Kowalski
John Pollock

13. ASPIC+ framework: overview
Argument structure:
Inference graphs 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)
If no formula is used more than once, then an argument is a tree.
Intuitively, if a rule is strict, then if you accept its premises, you must accept its conclusion no matter what. If a rule is defeasible, and you accept its premises, then you must accept its conclusion if you cannot find good reasons not to accept it.

14. Argumentation systems (with symmetric negation)
An argumentation system is a triple AS = (L,R,n) where:
L is a logical language with negation (¬)
R = Rs Rd is a set of strict (1,…, n  ) and defeasible (1,…, n  ) inference rules
n: Rd  L is a naming convention for defeasible rules
Notation:
- = ¬ if  does not start with a negation
- =  if  is of the form ¬
If phi starts with a negation, then –phi strips off the negation. 14

15. Knowledge bases
A knowledge base in AS = (L,R,n) is a set K  L where K is a partition Kn  Kp with:
Kn = necessary premises
Kp = ordinary premises 15

16. Argumentation theories
An argumentation theory is a pair AT = (AS, K) where AS is an argumentation system and K a knowledge base in AS. 16

17. Structure of arguments
TopRule(A)
Gerard Vreeswijk
An argument A on the basis of an argumentation theory is:
 if   K with
Prem(A) = {}, Conc(A) = , Sub(A) = {}, DefRules(A) = 
A1, ..., An   if A1, ..., An are arguments such that there is a strict inference rule Conc(A1), ..., Conc(An)  
Prem(A) = Prem(A1)  ...  Prem(An)
Conc(A) = 
Sub(A) = Sub(A1)  ...  Sub(An)  {A}
DefRules(A) = DefRules(A1)  ...  DefRules(An)
A1, ..., An   if A1, ..., An are arguments such that there is a defeasible inference rule Conc(A1), ..., Conc(An)  
DefRules(A) = DefRules(A1)  ...  DefRules(An)  {A1, ..., An  }
TopRule(A)
1 declares all premises to be the base cases of an argument.
Then 2/3 say that if you have a set of arguments, and you have an inference rule with their conclusions as antecedents, then you can combine them into a new argument.

18. Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,r,u}
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  Rd p s r t
What are the types of the arguments?
p  t  Rd p n p p q p

19. Types of arguments An argument A is: Strict if DefRules(A) = 
Defeasible if not strict
Firm if Prem(A)  Kn
Plausible if not firm 19

20. - Strict if DefRules(A) =  - Defeasible if not strict
Rs: Rd:
p,q  s p  t
u,v  w s,r,t  v
Kn = {q} Kp = {p,r,u}
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 p
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

21. Attack A undermines B (on ) if A rebuts B (on B’ ) if
Conc(A) = - for some   Prem(B )/ Kn;
A rebuts B (on B’ ) if
Conc(A) = -Conc(B’ ) for some B’  Sub(B) with a defeasible top rule
A undercuts B (on B’ ) if
Conc(A) = -n(r ) ’for some B’  Sub(B ) with defeasible top rule r
A attacks B iff A undermines or rebuts or undercuts B.

22. Rs: Rd: p,q  s p  t u,v  w s,r,t  v Kn = {q} Kp = {p,r,u}
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 p s r t p n p p q p

23. Structured argumentation frameworks
Let AT = (AS,K) be an argumentation theory
A structured argumentation framework (SAF) defined by AT is a triple (Args,C, a) where
Args = {A | A is an argument on the basis of AT}
C is the attack relation on Args
a is an ordering on Args (A <a B iff A a B and not B a A)
A c-SAF is a SAF in which all arguments have consistent premises 23

24. Defeat Direct vs. subargument attack/defeat
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 ) with a defeasible top rule
A undercuts B (on B’ ) if
Conc(A) = -n(r) ’for some B’  Sub(B ) with defeasible top rule r
A defeats B iff for some B’
A undermines or rebuts B on B’ and not A <a B’ ; or
A undercuts B on B’
Which if B’ =  means
and not A <a 
Direct vs. subargument attack/defeat
Preference-dependent vs. preference-independent attacks 24 24

25. Abstract argumentation frameworks corresponding to SAFs
An abstract argumentation framework corresponding to a (c-)SAF = (Args,C, a) is a pair (Args,D) where D is the defeat relation on Args defined by C and a. 25 25

26. Argument preference In general its origin is undefined
General constraint: A <a B if B is strict-and-firm and A is defeasible or plausible.
Sometimes defined in terms of partial preorders  (on Rd) and ’ (on Kp)
Origins of  and ’: domain-specific!
Some possible criteria:
Probabilistic strength
Legal priority rules
Importance of legal or moral values …
Partial preorder = transitive + reflexive

27. Which inference rules should we choose?
A tradition in AI: the inference rules encode domain-specific knowledge
A philosophically more well-founded approach: the inference rules express general patterns of reasoning
Strict rules (+ axioms): a sound and complete proof theory of a ‘standard’ logic for L
Defeasible rules: argument schemes. 27

28. Domain-specific vs. inference general inference rules
Flies
d1: Bird  Flies
s1: Penguin  Bird
Penguin  K
Rd = {,     }
Rs includes {S   | S |-PL  and
S is finite}
Bird  Flies  K
Penguin  Bird  K
Bird
Penguin
Flies
Bird
Bird Flies
Penguin
Penguin  Bird 28

29. Deriving the strict rules from a monotonic logic
For any logic L with (monotonic) consequence notion that is compact and satisfies Cut |-L define
S  p  Rs iff
S is finite and S |-L p
Works well only for logics that are compact (what is implied by an infinite set is implied by at least one finite subset) and satisfies the Cut rule (if P implies Q and Q implies R then P implies R). For logics that do not satisfy the Cut rule, chaining of strict rules should be forbidden.

30. Argument(ation) schemes: general form
Douglas Walton
But also critical questions
Premise 1,
… ,
Premise n
Therefore (presumably), conclusion 30

31. Logical vs. dialogical aspects of argument schemes
Some critical questions ask “why this premise?”
Other critical questions ask “is there no exception?”
But burden of proof is on respondent to show that there are exceptions!
One cannot ask such questions; one can only state counterarguments

32. 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

33. Reasoning with defeasible generalisations
But defaults can have exceptions
And there can be conflicting defaults P
If P then normally/usually/typically Q
So (presumably), Q
- What experts say is usually true
- People with political ambitions are usually not objective about security
- People with names typical from country C usually have nationality C
- People who flee from a crime scene when the police arrives are
normally involved in the crime
- Chinese tea is usually very good
Now that generalisations are in the object language, we can reason about them:
properties
provenance 33 33

34. Legal rule application
IF conditions THEN legal consequence
conditions
So, legal consequence

35. Legal rule application: critical questions
Is the rule valid?
Is the rule applicable to this case?
Must the rule be applied?
Is there a statutory exception?
Does applying the rule violate its purpose?
Does applying the rule have bad consequences?
Is there a principle that overrides the rule?
Example of statutory exception: the Dutch Civil Code states that a CC rule on creditor-debtor relations shall not apply if its application is manifestly unreasonable.
Rule: Vehicles are not allowed in the park
Example violating purpose: disallowing a war memorial jeep in a park violates purpose of promoting peace and tranquility in the park
Example of bad consequences: not allowing an ambulance in the park.
Example of violating principle: no person shall profit form his own wrongdoing.

36. Analogy Relevantly similar cases should be decided in the same way
This case is relevantly similar to that precedent
Therefore (presumably), this case should have
The same outcome as the precedent
Critical questions:
Are there also relevant differences between the cases?
Are there conflicting precedents?
Stealing electricity vs copying software

37. Arguments from consequences
Critical questions:
Does A also have bad (good) consequences?
Are there other ways to bring about G?
...
Action A causes G,
G is good (bad)
Therefore (presumably), A should (not) be done 37

38. Example (arguments pro and con an action)
We should make spam a criminal offence
We should not make spam a criminal offence
Making spam a criminal offence reduces spam
Reduction of spam
is good
Making spam a criminal offence increases workload of police and judiciary
Increased workload of police and judiciary is bad

39. Example (arguments pro alternative actions)
We should make spam a criminal offence
We should make spam civilly unlawful
Making spam a criminal offence reduces spam
Reduction of spam
is good
Making spam civilly unlawful reduces spam
Reduction of spam is good
Don’t only compare performing the action with not performing the action, but also with alternative ways of realising the good consequences.

40. Refinement: promoting or demoting legal/societal values
Critical questions:
Are there other ways to cause G?
Does A also cause something else that promotes or demotes other values?
...
Action A causes G,
G promotes (demotes) legal/societal value V
Therefore (presumably), A should (not) be done 40

41. Example (arguments pro and con an action)
We should save DNA of all citizens
We should not save DNA of all citizens
Saving DNA of all citizens leads to solving more crimes
Solving more crimes promotes security
Saving DNA of all citizens makes more private data publicly accessible
Making more private data publicly available demotes privacy 41

42. Example (arguments pro alternative actions)
We should save DNA of all citizens
We should have more police
Saving DNA of all citizens leads to solving more crimes
Solving more crimes promotes security
Having more police leads to solving more crimes
Solving more crimes promotes security 42