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Argumentation LogicsLecture 6:

Argumentation with structured arguments (2)

Attack, defeat, preferences

Henry PrakkenChongqing

June 3, 2010

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Overview Argumentation with structured

arguments: Attack Defeat Preferences

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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

= _, = _

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Knowledge bases A knowledge base in AS = (L, -,R,= ’) is

a pair (K, =<’) where K L and ’ is a partial preorder on K/Kn. Here: Kn = (necessary) axioms Kp = ordinary premises Ka = assumptions

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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

}

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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 <a A; If A Ka and B Ka then A <a B; If A = A1, ..., An then

for all 1 ≤ i ≤ n: A a Ai, for some 1 ≤ i ≤ n: Ai a A

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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 Args AT where

Args AT = {A | A is an argument on the basis of KB in AS}

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Attack and defeat(with - = ¬ and Ka = )

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 undermines B if Conc(A) = ¬ for some Prem(B )/Kn;

A defeats B iff for some B’ A rebuts B on B’ and not A <a B’ ; or A undermines B and not A <a B ; or A undercuts B on B’

Naming convention implicit

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Example cont’dR: 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 ¬pKn = {p}, Kp = {s,u}

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Argument acceptability Dung-style semantics and proof

theory directly apply!

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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

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Argument preference

Defined in terms of (on Rd) and ’ (on K)

Origins of and ’: domain-specific!

Ordering <s on sets in terms of an ordering (or ’) on their elements: S1 <s S2 if there exists an s1 S1 such that

for all s2 S2: s1 < s2

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Argument preference: some notation

An argument A is: if K with

DefRules(A) = LastDefRules(A) =

A1, ..., An if there is a strict inference rule Conc(A1), ..., Conc(An)

DefRules(A) = DefRules(A1) ... DefRules(An) LastDefRules(A) = LastDefRules(A1) ...

LastDefRules(An) A1, ..., An if there is a defeasible inference rule

Conc(A1), ..., Conc(An) DefRules(A) = DefRules(A1) ... DefRules(An) {A1, ...,

An } LastDefRules(A) = {A1, ..., An }

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Example Rd: r1: p q r2: p r r3: s t

Rs: q, r ¬t

K: p,s

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Argument preference: two alternatives

Last-link comparison: A <a B iff Condition (1) or (2) of Def 5.1.10

holds, or LastDefrules(B) <s LastDefrules(A), or LastDefrules(A/B) are empty and Prem(A) <s

Prem(B) Weakest link comparison:

A <a B iff Condition (1) or (2) of Def 5.1.10 holds, or

Prem(A) <s Prem(B), and If Defrules(B) , then Defrules(A) <s Defrules(B)

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Last link vs. weakest link (1)

R: r1: p q r2: p,q r r3: s t r4: t ¬r1 r5: u v r6: v ¬tr3 < r6, r5 < r3K: p,s,u

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Last link vs. weakest link (2)

d1: In Scotland Scottish d2: Scottish Likes Whisky d3: Likes Fitness ¬Likes Whisky

K: In Scotland, Likes Fitness d1 < d2, d1 < d3

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Last link vs. weakest link (3)

d1: Snores Misbehaves d2: Misbehaves May be removed d3: Professor ¬May be removed

K: Snores, Professor d1 < d2, d1 < d3