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Commonsense Reasoning and Argumentation 15/16HC

10 Structured argumentation (3)

- Henry Prakken
- 30 March 2016

Overview

- Recap
- More about rationality postulates
- Related research
- The need for defeasible rules

Aspic framework overview

- Argument structure
- Directed acyclic 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)

Deriving the strict rules from a monotonic logic

- For any logic L with (monotonic) consequence

notion -L define

S ? p ? Rs iff S is finite and S -L p

Argument(ation) schemes general form

- But also critical questions

Premise 1, , Premise n Therefore

(presumably), conclusion

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

Two example argument orderings

- (Informal Kp ?, no strict-and-firm arguments)
- Weakest link ordering
- Compares the weakest defeasible rule of each

argument - Last-link ordering
- Compares the last defeasible rules of each

argument

Comparing ordered sets (elitist ordering, strict

version)

- Ordering lts on sets in terms of an ordering ? (or

?) on their elements - If S2 ? then not S2 lts S1
- If S1 ? ? and S2 ? then S1 lts S2
- Else S1 lts S2 if there exists an s1 ? S1 such

that for all s2 ? S2 s1 lt s2

Consistency in ASPIC(with symmetric negation)

- For any S ? L
- S is directly consistent iff S does not contain

two formulas ? and ? - The strict closure Cl(S) of S is S everything

derivable from S with only Rs. - S is indirectly consistent iff Cl(S) is directly

consistent. - Parametrised by choice of strict rules

9

Rationality postulates(Caminada Amgoud 2007)

- Let E be any Dung-extension and
- Conc(E) ?? Conc(A) for some A ? E
- An AT satisfies
- subargument closure iff B ? E whenever A ? E and

B ? Sub(A) - direct consistency iff Conc(E) is directly

consistent - strict closure iff Cl(Conc(E)) Conc(E)
- indirect consistency iff Conc(E) is indirectly

consistent

Rationality postulatesfor ASPIC (whether

consistent premises or not)

- Closure under subarguments always satisfied
- Strict closure, direct and indirect consistency
- without preferences satisfied if
- Rs closed under transposition or AS closed under

contraposition and - Kn is indirectly consistent
- with preferences satisfied if in addition lta is

reasonable - If A is plausible or defeasible and B is

strict-and-firm then A lt B - If A B ? ? then A and B have the same

strength - (Complicated condition)
- Weakest- and last link ordering are reasonable

Subtleties concerning rebuttals (1)

- d1 Ring ? Married
- d2 Party animal ? Bachelor
- s1 Bachelor ? Married
- Kn Ring, Party animal

d2 lt d1

Subtleties concerning rebuttals (2)

- d1 Ring ? Married
- d2 Party animal ? Bachelor
- s1 Bachelor ? Married
- s2 Married ? Bachelor
- Kn Ring, Party animal

d2 lt d1

Subtleties concerning rebuttals (3)

- d1 Ring ? Married
- d2 Party animal ? Bachelor
- s1 Bachelor ? Married
- s2 Married ? Bachelor
- Kn Ring, Party animal

Subtleties concerning rebuttals (4)

- Rd ?, ? ? ? ? ?
- Rs all deductively valid inference rules
- Kn
- d1 Ring ? Married
- d2 Party animal ? Bachelor
- n1 Bachelor ? Married
- Ring, Party animal

The lottery paradox (Kyburg 1960)

- Assume
- A lottery with 1 million tickets and 1 prize.
- The probability that some ticket wins is 1
- The probability that a given ticket Ti wins is

0.000001. - Suppose
- a highly probable belief is justified and
- what can be deduced from a set of justified

beliefs is justified. - Then 1,2,3 is inconsistent

Solutions to the lottery paradox in the literature

- Reject the conjunction principle for justified

beliefs (Kyburg) - Reject that what is highly probable is justified

(Pollock?) - Reject consistency for justified beliefs
- But retain restricted forms of consistency and

deductive closure (Makinson)

The lottery paradox in ASPIC

X1 will win and the other tickets will not win

- Define ? is justified iff some argument for ? is

in all S-extensions

Kp ?T1,,?T1.000.000 Kn X1 xor xor

X1.000.000 (Rs S ? ? S -PL ? and S is

finite Rd ?

Kp ?T1, ?T2, ?T3 Kn X1 xor X2 xor X3

Option 1 C1 A1 But then for all i Ci

Ai So none of A1,A2,A3 are in all extensions

Violates principle that highly probable beliefs

are justified

C1

T1

X1 xor X2 xor X3

?T3

?T2

?T1

B

A3

A2

A1

Kp ?T1, ?T2, ?T3 Kn X1 xor X2 xor X3

Excluded by third condition on lt

Option 2 C1 lt A1 But then for all i Ci lt

Ai So A1,A2,A3,B,C1,C2,C3 ? E for any extension

E Violates direct and indirect consistency

C1

T1

X1 xor X2 xor X3

?T3

?T2

?T1

B

A3

A2

A1

Argumentation systems (with generalised negation)

- An argumentation system is a tuple AS (L,

-,R,n) 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 - n Rd ? L is a naming convention for defeasible

rules

See reader (next four slides not discussed during

lecture)

Generalised negation

- The function generalises negation.
- If ? ? -(?) then
- if ? ? -(?) then ? is a contrary of ?
- if ? ? -(?) then ? and ? are contradictories
- We write
- -? ? if ? does not start with a negation
- -? ? if is of the form ?

Attack and defeat(the general case)

- 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 contrary-undermines/rebuts B (on ?/B ) if

Conc(A) is a contrary of ? / Conc(B ) - A defeats B iff for some B
- A undermines B on ? and either A

contrary-undermines B on ? or not A lta ? or - A rebuts B on B and either A contrary-rebuts B

or not A lta B or - A undercuts B on B

Consistency in ASPIC(with generalised negation)

- For any S ? L
- S is directly consistent iff S does not contain

two formulas ? and (?) - The strict closure Cl(S) of S is S everything

derivable from S with only Rs. - S is indirectly consistent iff Cl(S) is directly

consistent. - Parametrised by choice of strict rules

24

Rationality postulatesfor ASPIC (with

generalised negation)

- Closure under subarguments always satisfied
- Direct and indirect consistency
- without preferences satisfied if
- Rs closed under transposition or AS closed under

contraposition and - Kn is indirectly consistent and
- AT is well-formed
- with preferences satisfied if in addition ? is

reasonable - Weakest- and last link ordering are reasonable

AT is well-formed if If ? is a contrary of ?

then (1) ? ? Kn and (2) ? is not the

consequent of a strict rule

Relation with other work (1)

- Assumption-based argumentation (Dung, Kowalski,

Toni ...) is special case of ASPIC (with

generalised negation) with - Only ordinary premises
- Only strict inference rules
- All arguments of equal priority

Reduction of ASPIC defeasible rules to ABA rules

(Dung Thang, JAIR 2014)

1-1 correspondence between grounded, preferred

and stable extensions of ASPIC and ABA

- Assumptions
- L consists of literals
- No preferences
- No rebuttals of undercutters

p1, , pn ? q becomes di, p1, , pn,notq ?

q where di n(p1, , pn ? q) di, notq are

assumptions ? -(not?), ? -(?), ? -(?)

From defeasible to strict rules example

- r1 Quaker ? Pacifist
- r2 Republican ? Pacifist

Pacifist

?Pacifist

r1

r2

Quaker

Republican

From defeasible to strict rules example

- s1 Appl(s1), Quaker, notPacifist ? Pacifist
- s2 Appl(s2), Republican, notPacifist ? Pacifist

Pacifist

Pacifist

Quaker

Appl(s1)

notPacifist

Republican

notPacifist

Appl(s2)

Can ASPIC preferences be reduced to ABA

assumptions?

d1 Bird ? Flies d2 Penguin ? Flies d1 lt

d2 Becomes d1 Bird, not Penguin ? Flies d2

Penguin ? Flies

Only works in special cases, e.g. not with

weakest-link ordering

Classical argumentation (Besnard Hunter, )

- Given L a propositional logical language and -

standard-logical consequence over L - An argument is a pair (S,p) such that
- S ? L and p ? L
- S - p
- S is consistent
- No S ? S is such that S - p
- Various notions of attack, e.g.
- Direct defeat argument (S,p) attacks argument

(S,p) iff p - q for some q ? S - Direct undercut argument (S,p) attacks

argument (S,p) iff p - q and q - p for

some q ? S - Only these two attacks satisfy indirect

consistency.

Relation with other work (2)

- Two variants of classical argumentation with

premise attack (Amgoud Cayrol, Besnard

Hunter) are special case of ASPIC with - Only ordinary premises
- Only strict inference rules (all valid

propositional or first-order inferences from

finite sets) - -
- No preferences
- Arguments must have classically consistent

premises

Results on classical argumentation (Cayrol 1995

Amgoud Besnard 2013)

Lindebaums lemma Every consistent set is

contained in a maximal consistent set

- In classical argumentation with premise attack,

only ordinary premises and no preferences - Preferred and stable extensions and maximal

conflict-free sets coincide with maximal

consistent subsets of the knowledge base - So p is defensible iff there exists an argument

for p - The grounded extension coincides with the

intersection of all maximal consistent subsets of

the knowledge base - So p is justified iff there exists an argument

for p without counterargument

Can defeasible reasoning be reduced to plausible

reasoning?

- Is it natural to reduce all forms of attack to

premise attack? - My answer no
- In classical argumentation can the material

implication represent defaults? - My answer no

The case of classical argumentation

- Birds usually fly
- Penguins usually dont fly
- All penguins are birds
- Penguins are abnormal birds w.r.t. flying
- Tweety is a penguin

The case of classical argumentation

- Birds usually fly
- Bird Ab1 ? Flies
- Penguins usually dont fly
- Penguin Ab2 ? Flies
- All penguins are birds
- Penguin ? Bird
- Penguins are abnormal birds w.r.t. flying
- Penguin ? Ab1
- Tweety is a penguin
- Penguin
- Ab1
- Ab2

The case of classical argumentation

- Bird Ab1 ? Flies
- Penguin Ab2 ? Flies
- Penguin ? Bird
- Penguin ? Ab1
- Penguin
- Ab1
- Ab2

Kn

Kp

Arguments - for Flies using Ab1 - for Flies

using Ab2

The case of classical argumentation

- Bird Ab1 ? Flies
- Penguin Ab2 ? Flies
- Penguin ? Bird
- Penguin ? Ab1
- Penguin
- Ab1
- Ab2

Kn

Kp

Arguments - for Flies using Ab1 - for Flies

using Ab2 - and for Ab1 and Ab2 But Flies

follows

The case of classical argumentation

- Bird Ab1 ? Flies
- Penguin Ab2 ? Flies
- Penguin ? Bird
- Penguin ? Ab1
- ObservedAsPenguin Ab3 ? Penguin
- ObservedAsPenguin
- Ab1
- Ab2
- Ab3

- Arguments
- - for Flies using Ab1
- for Flies using Ab2 and Ab3
- for Penguin using Ab3

The case of classical argumentation

- Bird Ab1 ? Flies
- Penguin Ab2 ? Flies
- Penguin ? Bird
- Penguin ? Ab1
- ObservedAsPenguin Ab3 ? Penguin
- ObservedAsPenguin
- Ab1
- Ab2
- Ab3

- Arguments
- - for Flies using Ab1
- for Flies using Ab2 and Ab3
- for Penguin using Ab3
- - and for Ab1 and Ab2 and Ab3

The case of classical argumentation

- Bird Ab1 ? Flies
- Penguin Ab2 ? Flies
- Penguin ? Bird
- Penguin ? Ab1
- ObservedAsPenguin Ab3 ? Penguin
- ObservedAsPenguin
- Ab1
- Ab2
- Ab3

- Arguments
- - for Flies using Ab1
- for Flies using Ab2 and Ab3
- for Penguin using Ab3
- - and for Ab1 and Ab2 and Ab3
- Ab3 gt Ab2 gt Ab1 makes Flies follow
- But is this ordering natural?

My conclusion

- Classical logics material implication is too

strong for representing defeasible

generalisations in argumentation - gt General models of argumentation need

defeasible inference rules - Defeasible reasoning cannot be modelled as

inconsistency handling in deductive logic

John Pollock Defeasible reasoning is the rule,

deductive reasoning is the exception

Next lecture

- Self-defeat
- Odd defeat loops