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Commonsense Reasoning and Argumentation 14/15 HC 10: Structured argumentation (3) Henry Prakken 16 March 2015

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Overview More about rationality postulates Related research The need for defeasible rules

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3 Subtleties concerning rebuttals (1) d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married K n : Ring, Party animal d2 < d1

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4 Subtleties concerning rebuttals (2) d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor K n : Ring, Party animal d2 < d1

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5 Subtleties concerning rebuttals (3) d1: Ring Married d2: Party animal Bachelor s1: Bachelor ¬Married s2: Married ¬Bachelor K n : Ring, Party animal

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6 Subtleties concerning rebuttals (4) R d = { , } R s = all deductively valid inference rules K n : d1: Ring Married d2: Party animal Bachelor n1: Bachelor ¬Married Ring, Party animal

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7 Argumentation systems (with generalised negation) An argumentation system is a tuple AS = ( L, -, R,n) 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 n: R d L is a naming convention for defeasible rules

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8 Generalised negation The – function generalises negation. If - ( ) then: if - ( ) then is a contrary of ; if - ( ) then and are contradictories We write - = ¬ if does not start with a negation - = if is of the form ¬

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9 Attack and defeat (the general case) 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 contrary-undermines/rebuts B (on /B’ ) if Conc(A) is a contrary of / Conc(B ’) A defeats B iff for some B’ A undermines B on and either A contrary-undermines B’ on or not A < a ; or A rebuts B on B’ and either A contrary-rebuts B’ or not A < a B’ ; or A undercuts B on B’

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10 Consistency in ASPIC+ (with generalised negation) For any S L S is directly consistent iff S does not contain two formulas and –( ) The strict closure Cl(S) of S is S + everything derivable from S with only R s. S is indirectly consistent iff Cl(S) is directly consistent. Parametrised by choice of strict rules

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11 Rationality postulates for ASPIC+ (with generalised negation) Closure under subarguments always satisfied Direct and indirect consistency: without preferences satisfied if R s closed under transposition or AS closed under contraposition; and K n is indirectly consistent; and AT is `well-formed’ with preferences satisfied if in addition is ‘reasonable’ Weakest- and last link ordering are reasonable AT is well-formed if: If is a contrary of then (1) K n and (2) is not the consequent of a strict rule

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Relation with other work (1) Assumption-based argumentation (Dung, Kowalski, Toni...) is special case of ASPIC+ (with generalised negation) with Only ordinary premises Only strict inference rules All arguments of equal priority …

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Reduction of ASPIC+ defeasible rules to ABA rules (Dung & Thang, JAIR 2014) Assumptions: L consists of literals No preferences No rebuttals of undercutters p 1, …, p n q becomes d i, p 1, …, p n,not¬q q where: d i = n(p 1, …, p n q) d i, not¬q are assumptions = - (not ), = - (¬ ), ¬ = - ( ) 1-1 correspondence between grounded, preferred and stable extensions of ASPIC+ and ABA

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14 From defeasible to strict rules: example r1: Quaker Pacifist r2: Republican ¬Pacifist Pacifist Quaker Pacifist Republican r1 r2

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15 From defeasible to strict rules: example s1: Appl(s1), Quaker, not¬Pacifist Pacifist s2: Appl(s2), Republican, notPacifist ¬Pacifist Pacifist QuakerAppl(s1)not¬Pacifist ¬Pacifist RepublicannotPacifistAppl(s2)

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Can ASPIC+ preferences be reduced to ABA assumptions? d1: Bird Flies d2: Penguin ¬Flies d1 < d2 Becomes d1: Bird, not Penguin Flies d2: Penguin ¬Flies Only works in special cases, e.g. not with weakest-link ordering

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Classical argumentation (Besnard & Hunter, …) Given L a propositional logical language and |- standard- logical consequence over L : An argument is a pair (S,p) such that S L and p L S |- p S is consistent No S’ S is such that S’ |- p Various notions of attack, e.g.: “Direct defeat”: argument (S,p) attacks argument (S’,p’) iff p |- ¬q for some q S’ “Direct undercut”: argument (S,p) attacks argument (S’,p’) iff p |- ¬q and ¬q |- p for some q S’ Only these two attacks satisfy consistency.

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Relation with other work (2) Two variants of classical argumentation with premise attack (Amgoud & Cayrol, Besnard & Hunter) are special case of ASPIC+ with Only ordinary premises Only strict inference rules (all valid propositional or first-order inferences from finite sets) - = ¬ No preferences Arguments must have classically consistent premises …

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Results on classical argumentation (Cayrol 1995; Amgoud & Besnard 2013) In classical argumentation with premise attack, only ordinary premises and no preferences: Preferred and stable extensions and maximal conflict-free sets coincide with maximal consistent subsets of the knowledge base So p is defensible iff there exists an argument for p The grounded extension coincides with the intersection of all maximal consistent subsets of the knowledge base So p is justified iff there exists an argument for p without counterargument Lindebaum’s lemma: Every consistent set is contained in a maximal consistent set

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20 Modelling default reasoning in classical argumentation Quakers are usually pacifist Republicans are usually not pacifist Nixon was a quaker and a republican

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21 A modelling in classical logic K p : Quaker Pacifist Republican ¬Pacifist Quaker, Republican Pacifist Quaker Quaker Pacifist ¬Pacifist Republican Republican ¬Pacifist ¬(Quaker Pacifist)

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22 A modelling in classical logic K n : Quaker & ¬Ab1 Pacifist Republican & ¬Ab2 ¬Pacifist Quaker, Republican K p : ¬Ab1, ¬Ab2 (attackable) Pacifist Quaker¬Ab1 ¬Pacifist ¬Ab2Republican Quaker & ¬Ab1 Pacifist Republican & ¬Ab2 ¬Pacifist Ab1

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23 A modelling in classical logic Pacifist Quaker¬Ab1 ¬Pacifist ¬Ab2Republican Quaker & ¬Ab1 Pacifist Republican & ¬Ab2 ¬Pacifist Ab1 Ab2

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Can defeasible reasoning be reduced to plausible reasoning? Is it natural to reduce all forms of attack to premise attack? My answer: no In classical argumentation: can the material implication represent defaults? My answer: no

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Default contraposition in classical argumentation Heterosexuals are normally married. John is not married Assume when possible that things are normal What can we conclude about John’s sexual orientation?

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Default contraposition in classical argumentation Heterosexuals are normally married H & ¬Ab M John is not married (¬M) Assume when possible that things are normal ¬Ab The first default implies that non-married people are normally not heterosexual ¬M & ¬Ab ¬H So John is not heterosexual

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Default contraposition in classical argumentation (2) Men normally have no beard => Creatures with a beard are normally not men This type of sensor usually does not give false alarms => False alarms are usually not given by this type of sensor Witnesses interrogated by the police usually tell the truth => People interrogated by the police who do not speak the truth are usually not a witness Statisticians call these inferences “base rate fallacies”

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The case of classical argumentation Birds usually fly Penguins usually don’t fly All penguins are birds Penguins are abnormal birds w.r.t. flying Tweety is a penguin

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The case of classical argumentation Birds usually fly Bird & ¬Ab1 Flies Penguins usually don’t fly Penguin & ¬Ab2 ¬Flies All penguins are birds Penguin Bird Penguins are abnormal birds w.r.t. flying Penguin Ab1 Tweety is a penguin Penguin ¬Ab1 ¬Ab2

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The case of classical argumentation Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 Penguin ¬Ab1 ¬Ab2 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 KpKp KnKn

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The case of classical argumentation Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 Penguin ¬Ab1 ¬Ab2 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 - and for Ab1 and Ab2 But ¬Flies follows KpKp KnKn

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The case of classical argumentation Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 ObservedAsPenguin & ¬Ab3 Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3

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The case of classical argumentation Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 ObservedAsPenguin & ¬Ab3 Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3 - and for Ab1 and Ab2 and Ab3

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The case of classical argumentation Bird & ¬Ab1 Flies Penguin & ¬Ab2 ¬Flies Penguin Bird Penguin Ab1 ObservedAsPenguin & ¬Ab3 Penguin ObservedAsPenguin ¬Ab1 ¬Ab2 ¬Ab3 Arguments: - for Flies using ¬Ab1 - for ¬Flies using ¬Ab2 and ¬Ab3 - for Penguin using ¬Ab3 - and for Ab1 and Ab2 and Ab3 ¬Ab3 > ¬Ab2 > ¬Ab1 makes ¬Flies follow But is this ordering natural?

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35 Contraposition of legal rules r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed K: Snores, Professor r1 < r2, r1 < r3, r3 < r2 May be removed Misbehaves Snores May be removed Professor r1 r2 r3 This is the intuitive outcome R3 < R2

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36 Contraposition of legal rules r1: Snores Misbehaves r2: Misbehaves May be removed r3: Professor ¬May be removed K: Snores, Professor r1 < r2, r1 < r3, r3 < r2 May be removed Misbehaves Snores May be removed Professor r1 r2 r3 But with contraposition (and last or weakest link) we have this outcome

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My conclusion Classical logic’s material implication is too strong for representing defeasible generalisations or legal rules => Models of legal argument (and many other kinds of argument) need defeasible inference rules Defeasible reasoning cannot be modelled as inconsistency handling in deductive logic John Pollock: Defeasible reasoning is the rule, deductive reasoning is the exception

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Next lecture The lottery paradox Self-defeat and odd defeat loops Floating conclusions The need for dynamics

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