Argumentation Logics Lecture 6 Argumentation with structured arguments

Argumentation Logics Lecture 6: Argumentation with structured arguments (2) Attack, defeat, preferences Henry Prakken Chongqing June 3, 2010 1

Overview n Argumentation with structured arguments: n n n Attack Defeat Preferences 2

Argumentation systems n An argumentation system is a tuple AS = (L, -, R, ) where: n n n L is a logical language - is a contrariness function from L to 2 L R = Rs Rd is a set of strict and defeasible inference rules is a partial preorder on Rd If -( ) then: n n if -( ) then is a contrary of ; if -( ) then and are contradictories _ _ n = , = 3

Knowledge bases n A knowledge base in AS = (L, -, R, = ’) is a pair (K, =<’) where K L and ’ is a partial preorder on K/Kn. Here: n n n Kn = (necessary) axioms Kp = ordinary premises Ka = assumptions 4

Structure of arguments n An argument A on the basis of (K, ’) in (L, -, R, ) is: n if K with n n A 1, . . . , An if there is a strict inference rule Conc(A 1), . . . , Conc(An) n n Conc(A) = { } Sub(A) = Def. Rules(A) = Conc(A) = { } Sub(A) = Sub(A 1) . . . Sub(An) {A} Def. Rules(A) = Def. Rules(A 1) . . . Def. Rules(An) A 1, . . . , An if there is a defeasible inference rule Conc(A 1), . . . , Conc(An) n n n Conc(A) = { } Sub(A) = Sub(A 1) . . . Sub(An) {A} Def. Rules(A) = Def. Rules(A 1) . . . Def. Rules(An) {A 1, . . . , An } 5

Admissible argument orderings n Let A be a set of arguments. A partial preorder a on A is admissible if: n n n 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 = A 1, . . . , An then n n for all 1 ≤ i ≤ n: A a Ai, for some 1 ≤ i ≤ n: Ai a A 6

Argumentation theories n An argumentation theory is a triple AT = (AS, KB, a) where: n n n AS is an argumentation system KB is a knowledge base in AS a is an admissible ordering on Args AT where n Args AT = {A | A is an argument on the basis of KB in AS} 7

Attack and defeat (with - = ¬ and Ka = ) n A rebuts B (on B’ ) if n n n A undercuts B (on B’ ) if n n Conc(A) = ¬Conc(B’ ) for some B’ Sub(B ); and B’ applies a defeasible rule to derive Conc(B’ ) Naming convention implicit Conc(A) = ¬B’ for some B’ Sub(B ); and B’ applies a defeasible rule A undermines B if n Conc(A) = ¬ for some Prem(B )/Kn; A defeats B iff for some B’ n n n 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’ 8

Example cont’d R: n r 1: n r 2: n r 3: n r 4: n r 5: n r 6: n r 7: n r 8: p q p, q r s t t ¬r 1 u v v, q ¬t p, v ¬s s ¬p Kn = {p}, Kp = {s, u} 9

Argument acceptability n Dung-style semantics and proof theory directly apply! 10

The ultimate status of conclusions n With grounded semantics: n n With preferred semantics: n n 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 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: n n n 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 11

Argument preference n n n 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: n S 1 <s S 2 if there exists an s 1 S 1 such that for all s 2 S 2: s 1 < s 2 12

Argument preference: some notation n An argument A is: n if K with n n n A 1, . . . , An if there is a strict inference rule Conc(A 1), . . . , Conc(An) n n n Def. Rules(A) = Last. Def. Rules(A) = Def. Rules(A 1) . . . Def. Rules(An) Last. Def. Rules(A) = Last. Def. Rules(A 1) . . . Last. Def. Rules(An) A 1, . . . , An if there is a defeasible inference rule Conc(A 1), . . . , Conc(An) n n Def. Rules(A) = Def. Rules(A 1) . . . Def. Rules(An) {A 1, . . . , An } Last. Def. Rules(A) = {A 1, . . . , An } 13

Example Rd : n r 1: p q n r 2: p r n r 3: s t Rs: q, r ¬t K: n p, s 14

Argument preference: two alternatives n Last-link comparison: n n A <a B iff Condition (1) or (2) of Def 5. 1. 10 holds, or Last. Defrules(B) <s Last. Defrules(A), or Last. Defrules(A/B) are empty and Prem(A) Prem(B) <s Weakest link comparison: n A <a B iff Condition (1) or (2) of Def 5. 1. 10 holds, or n n Prem(A) <s Prem(B), and If Defrules(B) , then Defrules(A) <s Defrules(B) 15

Last link vs. weakest link (1) R: n r 1: p q n r 2: p, q r n r 3: s t n r 4: t ¬r 1 n r 5: u v n r 6: v ¬t r 3 < r 6, r 5 < r 3 K: n p, s, u 16

Last link vs. weakest link (2) n n n d 1: In Scotland Scottish d 2: Scottish Likes Whisky d 3: Likes Fitness ¬Likes Whisky K: In Scotland, Likes Fitness d 1 < d 2, d 1 < d 3 17

Last link vs. weakest link (3) n n n d 1: Snores Misbehaves d 2: Misbehaves May be removed d 3: Professor ¬May be removed K: Snores, Professor d 1 < d 2, d 1 < d 3 18