Download presentation
Presentation is loading. Please wait.
1
Argumentation Logics Lecture 6: Argumentation with structured arguments (2) Attack, defeat, preferences Henry Prakken Chongqing June 3, 2010
2
2 Overview Argumentation with structured arguments: Attack Defeat Preferences
3
3 Argumentation systems An argumentation system is a tuple AS = ( L, -, R, ) where: L is a logical language - is a contrariness function from L to 2 L R = R s R d is a set of strict and defeasible inference rules is a partial preorder on R d If - ( ) then: if - ( ) then is a contrary of ; if - ( ) then and are contradictories = _ , = _
4
4 Knowledge bases A knowledge base in AS = ( L, -, R, = ’) is a pair ( K, =< ’) where K L and ’ is a partial preorder on K / K n. Here: K n = (necessary) axioms K p = ordinary premises K a = assumptions
5
5 Structure of arguments An argument A on the basis of ( K, ’) in ( L, -, R, ) is: if K with Conc(A) = { } Sub(A) = DefRules(A) = A 1,..., A n if there is a strict inference rule Conc(A 1 ),..., Conc(A n ) Conc(A) = { } Sub(A) = Sub(A 1 ) ... Sub(A n ) {A} DefRules(A) = DefRules(A 1 ) ... DefRules(A n ) A 1,..., A n if there is a defeasible inference rule Conc(A 1 ),..., Conc(A n ) Conc(A) = { } Sub(A) = Sub(A 1 ) ... Sub(A n ) {A} DefRules(A) = DefRules(A 1 ) ... DefRules(A n ) {A 1,..., A n }
6
6 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 K a and B K a then A < a B; If A = A 1,..., A n then for all 1 ≤ i ≤ n: A a A i, for some 1 ≤ i ≤ n: A i a A
7
7 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}
8
8 Attack and defeat (with - = ¬ and K a = ) 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 )/ K n ; 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
9
9 Example cont’d R : 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 ¬p K n = { p}, K p = { s,u}
10
10 Argument acceptability Dung-style semantics and proof theory directly apply!
11
11 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
12
12 Argument preference Defined in terms of (on R d ) 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
13
13 Argument preference: some notation An argument A is: if K with DefRules(A) = LastDefRules(A) = A 1,..., A n if there is a strict inference rule Conc(A 1 ),..., Conc(A n ) DefRules(A) = DefRules(A 1 ) ... DefRules(A n ) LastDefRules(A) = LastDefRules(A 1 ) ... LastDefRules(A n ) A 1,..., A n if there is a defeasible inference rule Conc(A 1 ),..., Conc(A n ) DefRules(A) = DefRules(A 1 ) ... DefRules(A n ) {A 1,..., A n } LastDefRules(A) = {A 1,..., A n }
14
14 Example R d : r1: p q r2: p r r3: s t R s : q, r ¬t K: p,s
15
15 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)
16
16 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 ¬t r3 < r6, r5 < r3 K: p,s,u
17
17 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
18
18 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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.