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Commonsense Reasoning and Argumentation 14/15 HC 8 Structured argumentation (1) Henry Prakken March 2, 2015

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2 Overview Structured argumentation: Arguments Attack Defeat

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AB C D E

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4 We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased

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5 Two accounts of the fallibility of arguments Plausible Reasoning: all fallibility located in the premises Assumption-based argumentation (Kowalski, Dung, Toni,… Classical argumentation (Cayrol, Besnard, Hunter, …) Defeasible reasoning: all fallibility located in the defeasible inferences Pollock, Loui, Vreeswijk, Prakken & Sartor, … ASPIC+ combines these accounts John Pollock Nicholas Rescher Robert Kowalski Tony Hunter

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6 Aspic+ framework: overview Argument structure: Directed acyclic graphs where Nodes are wff of a logical language L Links are applications of inference rules R s = Strict rules ( 1,..., n ); or R d = Defeasible rules ( 1,..., n ) 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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We should lower taxes Lower taxes increase productivity Increased productivity is good

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Attack on conclusion

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity USA lowered taxes but productivity decreased Attack on premise …

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … USA lowered taxes but productivity decreased … often becomes attack on intermediate conclusion

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective USA lowered taxes but productivity decreased Attack on inference

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective USA lowered taxes but productivity decreased

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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased Indirect defence

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14 Argumentation systems (with symmetric negation) An argumentation system is a triple AS = ( L, R,n) where: L is a logical language with negation (¬) R = R s R d is a set of strict and defeasible inference rules n: R d L is a naming convention for defeasible rules Notation: - = ¬ if does not start with a negation - = if is of the form ¬

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15 Knowledge bases A knowledge base in AS = ( L, R,n) is a set K L where K is a partition K n K p with: K n = necessary premises K p = ordinary premises

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16 Argumentation theories An argumentation theory is a pair AT = (AS, K) where AS is an argumentation system and K a knowledge base in AS.

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17 Structure of arguments An argument A on the basis of an argumentation theory is: if K with Prem(A) = { }, Conc(A) = , Sub(A) = { }, DefRules(A) = A 1,..., A n if there is a strict inference rule Conc(A 1 ),..., Conc(A n ) Prem(A) = Prem(A 1 ) ... Prem(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 ) Prem(A) = Prem(A 1 ) ... Prem(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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18 R s :R d : p,q sp t u,v ws,r,t v K n = {q} K p = {p,r,u} w vu s r t pqp p pnp p u, v w R s p, q s R s s,r,t v R d p t R d A1 = pA5 = A1 t A2 = qA6 = A1,A2 s A3 = r A7 = A5,A3,A6 v A4 = uA8 = A7,A4 w

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19 Types of arguments An argument A is: Strict if DefRules(A) = Defeasible if not strict 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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20 R s :R d : p,q sp t u,v ws,r,t v w vu s r t p q p p pnp p A1 = pA5 = A1 t A2 = qA6 = A1,A2 s A3 = r A7 = A5,A3,A6 v A4 = uA8 = A7,A4 w K n = {q} K p = {p,r,u} An argument A is: - Strict if DefRules(A) = - Defeasible if not strict - Firm if Prem(A) Kn - Plausible if not firm

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21 Example R : d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬s K n = { p}, K p = { s,u,x} n(p q ) = d1

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22 Attack A undermines B (on ) if Conc(A) = - for some Prem(B )/ K n ; A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’ Sub(B) with a defeasible top rule A undercuts B (on B’ ) if Conc(A) = -n(r ) ’for some B’ Sub(B ) with defeasible top rule r A attacks B iff A undermines or rebuts or undercuts B.

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23 R s :R d : p,q sp t u,v ws,r,t v w vu s r t p q p p pnp p A1 = pA5 = A1 t A2 = qA6 = A1,A2 s A3 = r A7 = A5,A3,A6 v A4 = uA8 = A7,A4 w K n = {q} K p = {p,r,u}

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24 Structured argumentation frameworks Let AT = (AS, K ) be an argumentation theory A structured argumentation framework (SAF) defined by AT is a triple (Args,C, a ) where Args = {A | A is a finite argument on the basis of K in AS } C is the attack relation on Args a is an ordering on Args (A < a B iff A a B and not B a A) A c-SAF is a SAF in which all arguments have indirectly consistent premises (to be defined later)

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25 Defeat A undermines B (on ) if Conc(A) = - for some Prem(B )/ K n ; A rebuts B (on B’ ) if Conc(A) = -Conc(B’ ) for some B’ Sub(B ) with a defeasible top rule A undercuts B (on B’ ) if Conc(A) = -n(r) ’for some B’ Sub(B ) with defeasible top rule r A defeats B iff for some B’ A undermines B on B’ = and not A < ; or A rebuts B on B’ and not A < B’ ; or A undercuts B on B’ Direct vs. subargument attack/defeat Preference-dependent vs. preference-independent attacks

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26 Example cont’d R : d1: p q d2: s t d3: t ¬d1 d4: u v d5: v,x ¬t d6: s ¬p s1: p,q r s2: v ¬s K n = { p}, K p = { s,u,x} n(p q ) = d1

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27 Abstract argumentation frameworks corresponding to SAFs An abstract argumentation framework corresponding to a SAF = (Args,C, ) is a pair (Args,D) where D is the defeat relation on Args defined by C and .

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28 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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We should lower taxes Lower taxes increase productivity Increased productivity is good We should not lower taxes Lower taxes increase inequality Increased inequality is bad Lower taxes do not increase productivity Prof. P says that … Prof. P has political ambitions People with political ambitions are not objective Prof. P is not objective Increased inequality is good Increased inequality stimulates competition Competition is good USA lowered taxes but productivity decreased C AB E D

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AB C D E

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AB C D E A’

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AB C D E P1 P2P3P4 P8P9P7P5P6

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D C3 BA

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B3 D4 A3 C3D3 B2 A1 C1 B1 C2 A2

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B3 D4 A3 C3D3 B2 A1 C1 B1 C2 A2 D4 < a B2

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B3 D4 A3 C3D3 B2 A1 C1 B1 C2 A2 D4 < a B2

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