Commonsense Reasoning and Argumentation 14/15 HC 8 Structured argumentation (1) Henry Prakken March 2, 2015.

Commonsense Reasoning and Argumentation 14/15

HC 8Structured argumentation (1)

Henry PrakkenMarch 2, 2015

2

Overview

Structured argumentation: Arguments Attack Defeat

A B

C D E

4

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

Lower taxes do not increase productivity

Prof. P says that …

Prof. P has political ambitions

People with political ambitions are not objective

Prof. P is not objective

Increased inequality is good

Increased inequality stimulates competition

Competition is good

USA lowered taxes but productivity decreased

5

Two accounts of the fallibility of

arguments 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 John Pollock

Nicholas Rescher

Robert Kowalski

Tony Hunter

6

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)










Attack on conclusion









Attack on premise …










… often becomes attack on intermediate conclusion













Attack on inference


























Competition is good

USA lowered taxes but productivity decreased Indirect

defence

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 and defeasible 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 ¬

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

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.

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Structure of arguments

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 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 there is a defeasible 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 }

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Rs: Rd:

p,q s p tu,v w s,r,t v

Kn = {q} Kp = {p,r,u}

w

v u

s r t

p q p

p

pnp

p

u, v w Rs

p, q s Rs

s,r,t v Rd

p t Rd

A1 = p A5 = A1 t

A2 = q A6 = A1,A2 s

A3 = r A7 = A5,A3,A6 v

A4 = u A8 = A7,A4 w

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Types of arguments An argument A is:

Strict if DefRules(A) = Defeasible if not strict Firm if Prem(A) Kn Plausible if not firm

S |- means there is a strict argument A s.t.

Conc(A) = Prem(A) S

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Rs: Rd:


w

v u

s r t

p q p

p

pnp

p

A1 = p A5 = A1 t

A2 = q A6 = A1,A2 s

A3 = r A7 = A5,A3,A6 v

A4 = u A8 = A7,A4 w

Kn = {q} Kp = {p,r,u}

An argument A is:- Strict if DefRules(A) = - Defeasible if not strict- Firm if Prem(A) Kn - Plausible if not firm

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Example

R: d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬sKn = {p}, Kp = {s,u,x}

n(p q ) = d1

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Attack 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 attacks B iff A undermines or rebuts or undercuts B.

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Rs: Rd:


w

v u

s r t

p q p

p

pnp

p

A1 = p A5 = A1 t

A2 = q A6 = A1,A2 s

A3 = r A7 = A5,A3,A6 v

A4 = u A8 = A7,A4 w

Kn = {q} Kp = {p,r,u}

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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 a finite argument on the basis of K in

AS } 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 indirectly consistent premises (to be defined later)

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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 B on B’ = and not A < ; or A rebuts B on B’ and not A < B’ ; or A undercuts B on B’

Direct vs. subargument attack/defeatPreference-dependent vs. preference-independent attacks

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Example cont’d

R: d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬sKn = {p}, Kp = {s,u,x}

n(p q ) = d1

27

Abstract argumentation frameworks corresponding to

SAFs

An abstract argumentation framework corresponding to a SAF = (Args,C, ) is a pair (Args,D) where D is the defeat relation on Args defined by C and .

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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














Competition is good


C

A B

E

D

A B

C D E

A B

C D E

A’

A B

C D E

A’

P1 P2 P3 P4

P8 P9P7P5 P6

33

D

C3

BA

B3

D4

A3

C3 D3

B2A1

C1

B1

C2

A2

B3

D4

A3

C3 D3

B2A1

C1

B1

C2

A2

D4 <a B2

B3

D4

A3

C3 D3

B2A1

C1

B1

C2

A2

D4 <a B2

Commonsense Reasoning and Argumentation 14/15 HC 8 Structured argumentation (1) Henry Prakken March 2, 2015.

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