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Argumentation Logics Lecture 5: Argumentation with structured arguments (1) argument structure Henry Prakken Chongqing June 2, 2010

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2 Contents Structured argumentation: Arguments Argument schemes

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3 Merits of Dung (1995) Framework for nonmonotonic logics Comparison and properties Guidance for development From intuitions to theoretical notions But should not be used for KR

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4 The structure of arguments: two approaches Both approaches: arguments are inference trees Assumption-based approaches (Dung-Kowalski-Toni, Besnard & Hunter, …) Sound reasoning from uncertain premises Arguments attack each other on their assumptions (premises) Rule-based approaches (Pollock, Vreeswijk, …) Risky (‘defeasible’) reasoning from certain premises Arguments attack each other on applications of defeasible inference rules

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5 Aspic framework: overview Argument structure: Trees where Nodes are wff of a logical language L Links are applications of inference rules R s = Strict rules ( 1,..., 1 ); or R d = Defeasible rules ( 1,..., 1 ) Reasoning starts from a knowledge base K L Defeat: attack on conclusion, premise or inference, + preferences Argument acceptability based on Dung (1995)

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6 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 = _ , = _

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7 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

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8 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 }

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9 Q1Q2 P R1R2 R1, R2 Q2 Q1, Q2 P Q1,R1,R2 K

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10 Example 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}

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11 Types of arguments An argument A is: Strict if DefRules(A) = Defeasible if not Firm if Prem(A) K n Plausible if not firm S |- means there is a strict argument A s.t. Conc(A) = Prem(A) S

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12 Domain-specific vs. inference general inference rules R1: Bird Flies R2: Penguin Bird Penguin K R d = { , } R s = all deductively valid inference rules Bird Flies K Penguin Bird K Penguin K Flies Bird Penguin Flies Bird Bird Flies Penguin Penguin Bird

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13 Argument(ation) schemes: general form Defeasible inference rules! But also critical questions Negative answers are counterarguments Premise 1, …, Premise n Therefore (presumably), conclusion

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14 Expert testimony (Walton 1996) Critical questions: Is E biased? Is P consistent with what other experts say? Is P consistent with known evidence? E is expert on D E says that P P is within D Therefore (presumably), P is the case

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15 Witness testimony Critical questions: Is W sincere? Does W’s memory function properly? Did W’s senses function properly? W says P W was in the position to observe P Therefore (presumably), P

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16 Arguments from consequences Critical questions: Does A also have bad consequences? Are there other ways to bring about G?... Action A brings about G, G is good Therefore (presumably), A should be done

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17 Temporal persistence (Forward) Critical questions: Was P known to be false between T1 and T2? Is the gap between T1 and T2 too long? P is true at T1 and T2 > T1 Therefore (presumably), P is still true at T2

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18 Temporal persistence (Backward) Critical questions: Was P known to be false between T1 and T2? Is the gap between T1 and T2 too long? P is true at T1 and T2 < T1 Therefore (presumably), P was already true at T2

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19 X murdered Y Y murdered in house at 4:45 X in 4:45 X in 4:45 {X in 4:30} X in 4:45 {X in 5:00} X left 5:00 W3: “X left 5:00”W1: “X in 4:30” W2: “X in 4:30” X in 4:30 {W1} X in 4:30 {W2} X in 4:30 accrual testimony forw temp pers backw temp pers dmp accrual V murdered in L at T & S was in L at T S murdered V

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