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Argumentation LogicsLecture 6:
Argumentation with structured arguments (2)
Attack, defeat, preferences
Henry PrakkenChongqing
June 3, 2010
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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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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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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