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## On the Semantics of Argumentation

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**On the Semantics of Argumentation**Antonis Kakas Francesca Toni Paolo Mancarella Department of Computer ScienceDepartment of Computing University of CyprusImperial College Dipartimento di Informatica Universita di Pisa 20 April, 2012 London Argumentation Forum**Contents**• Part 1: Acceptability Semantics for Abstract Argumentation • Generalizing old work on the Argumentation Based Acceptability Semantics for Logic Programming • Part 2: Argumentation Logic • Recent work on the application of Acceptability Semantics to reformulate (Propositional) Logic in terms of Argumentation**Acceptability Semantics for Abstract Argumentation**• Abstract Argumentation: <Args, Attack> • Args: a set of arguments • Attack: a binary relation on Args • (a,b) Attack: the argument “a” attacks the argument “b” • A attacks B iff a A and b B s.t. (a,b) Attack, for any A,B P(Args) • Acceptability semantics is defined via a relativeAcceptability Relation between (sets of) arguments: • Acc(,0): Given 0 the set can be accepted.**Acceptability SemanticsInformal Motivation**• Acceptability Relation: Follow the “universal” intuition: An argument (or a set of arguments) can be acceptediff all its counter-arguments can be rejected. • Can we formalize directly this intuition? • How are we to understand the “Rejection of Argument”? • As “Can not be Accepted”? • The argument can play a role in rejecting its counter-arguments • Hence Acceptance is a RELATIVEnotion. An argument (or a set of arguments) is acceptableiff it renders all its counter-arguments non-acceptable.**Acceptability SemanticsInformal Motivation**An argument (or a set of arguments), ,is acceptableiff all its counter-arguments, A, are rendered non-acceptable. • How do we understand “non-acceptable” or more generally “non-acceptable relative to ”? • Admissibility answers this by “ attacks (back) A”. • This is an approximation of the negation of acceptable! Negation of Acceptance: An argument (or a set of arguments) Ais non-acceptableiffthere exists a set of arguments D that attacks A such that D is acceptable (relative to A).**Acceptability SemanticsDefinition**DEFINITIONA set is acceptable relative to 0, i.e. Acc(, 0) holds, iff(i) is a subset of 0 or (ii) for any set A that attacks there exists a set D that attacks A – D defends against A - such that D is acceptable relative to 0 A, i.e.Acc(D, 0 A) holds. FACC: 2P(Args)xP(Args) 2P(Args)xP(Args)(P(Args) is the power set 2Argsof Arg) FACC(acc)(Δ,Δ’)iffΔ Δ’, or for any A s.t. A attacks Δ: there exists D s.t. D attacks A and acc(D, Δ’ Δ A). • The operator FACC is monotonic. • Acceptability,Acc(-,-), is defined as the least fixed point of FACC Definition of SEMANTICS: Δ is acceptable iffAcc(Δ,{}) holds.**Acceptability SemanticsSome Results**• Admissibleimplies Acceptable • Acyclic AF: Acceptability Semantics = Grounded Semantics • In general, it captures the well known semantic notions. Does it give anything else? • Captures semantic notions of self-defeating (set of) argument(s): • S is self-defeatingiff there exists an attacking set, A, against S such that ¬Acc(A, {}) and Acc(A, S) hold. • Hence S renders one of its attacks acceptable! • Acceptable sets do not need to defend against such self-defeating attacking sets by counter-attacking them back. • This extends Admissibility**Acceptability SemanticsExtending Admissibility**• Example of Self-Defeating: Odd Loops • Elements of (any length) odd loops are not acceptable. • But also arguments that are attacked only by elements of (isolated) odd loops are acceptable. a {a} is Acceptable a1 a1 a a1 a2 a3 a1**Acceptability SemanticsSelf-defeat ↔Reductio ad Absurdum**• Self-defeat emerges implicitly as a semantic notion from the minimal formulation of the acceptability semantics. • C.f. other semantics where this is explicit and syntactic. • Self-defeating S: renders one of its attacks acceptable • This is a kind of Reductio ad Absurdum Principle!**Part2: Argumentation Logic**• Can we understand Reductio ad Absurdum in Logic as a case of self-defeating under acceptability? • Can this help to formulate (Propositional) Logic in terms of Argumentation? • Originally, logic was developed to formulated human argumentation. • PL can be reformulated as a realization of abstract argumentation under an acceptability type semantics. • Argumentation Logic • Naturally extends PL for (classically) inconsistent theories.**Argumentation Logic**• Consider Propositional Logic (PL) and its Natural Deduction (ND) proof system. • Take out the Reductio ad Absurdum rule (¬I rule) from ND • Only Direct Proofs, ├MRA . • RA will be recovered at the semantic level by Acceptability • Argumentation Framework for AL: <Args, Att> • Args:Sets of Propositional Formulae: ∆ (Direct proofs from ∆ and the given theory, T) • Att:A attacks ∆ : T├MRA D defends A: D={¬} for some in A D={} when T├MRA**Argumentation Logic = Propositional LogicSketch Proof**• ¬Acc({},{}) Genuine RA derivation for • Genuine RA derivations: [ . [’ . c() . [ . . ¬ ] ] • Technical Lemma: For classically consistent theories if there exists a RA derivation for the there exists a Genuine RA derivation for . T ¬ ├MRA ’ is necessary for the direct derivation of **Natural Deduction (RA) as ArgumentationExample:**¬Acc({},{}) Genuine RA derivation for • Θ = {¬ (θεόςθνητός), ¬ θνητός ¬ πεθάνει, πεθάνει} Θεός [θεός ¬ (¬θνητός) θνητός θεός θνητός ¬(θεός θνητός) ] ¬ θεός [¬θνητός ¬πεθάνει πεθάνει ] θνητός ¬ θνητός ¬ θνητός The argument, ¬θνητός, that can defend against the attack θνητόςcannot do so as it is self-defeating. Hence the argument, θεός, is not acceptable.**Natural Deduction (RA) as ArgumentationExample:**¬Acc({},{}) Genuine RA derivation for • Θ = {¬ (θεόςθνητός), ¬ θνητός -> ¬ πεθάνει, πεθάνει} [θεός ¬ θνητός ¬πεθάνει πεθάνει ] ¬ θεός Θεός [θνητός θεός (copy) θεός θνητός ¬(Θεός θνητός) ] Attack ??? θνητός ¬θνητός {} θεός Violates the Genuine property!**Argumentation LogicResults (1)**• T classically consistent • AL = PL (for the restricted language of ¬ and ) • AL entails iff¬Acc({¬},{}) holds. • Interpretation of implicationin AL differs from PL, e.g. • Both ab and ¬(ab) are acceptable w.r.t. to T={¬a} • AL distinguishes two forms of Inconsistency of T • Classically inconsistent but directly consistent (under├MRA) • Violation of rule of «Excluded Middle». • For some,φ, neither φ nor ¬φ is acceptable, e.g. T = { φ , ¬φ } • Directly inconsistent • For some φ, T has a direct argument for φ and ¬φ, e.g. T= { φ , ¬φ }**Argumentation LogicResults (2)**• AL extends PL when T is (classically) inconsistent • Directly consistent • AL does nottrivialize • AL entails iff ¬Acc({¬},{}) and Acc({},{}) hold. • AL isolates outthe non-relevant use of Reductio ad Absurdum • Example: Logical Paradoxes (T = { φ , ¬φ } ) • Directly inconsistent • Use “Belief Revision Type” approach • 1. Close T under direct consequence: C(T), • 2. Maximally directly consistent subsets of C(T).**Example of Argumentation Logic**• “A barber shaves anyone that does not shave himself” • ¬ShavesHimself(Person)ShavedByBarber(Person) • ShavesHimself(Person) ¬ShavedByBarber(Person) • Self-reference: When Person = barber • ShavedByBarber(barber)ShavesHimself(barber) • ¬ShavedByBarber(barber) ¬ShavesHimself(barber)**Example – Classical Logic**• ¬ SH(P) SB(P) SH(P) ¬ SB(P) • SB(b) SH(b) ¬ SB(b) ¬ SH(b) • SB(b) |- SH(b) |- ¬ SB(b) i.e. SB(b) |- • ¬ SB(b) |- ¬ SH(b) |- SB(b) i.e. ¬ SB(b)|- • Problem arises due to the excluded middle law • SB(P) or ¬ SB(P) , for any person P, even for P=barber. • This makes the theory inconsistent and therefore non meaningful (even for any other person than the barber). • Problem arises as SB(b)must take a truth value (in model theory).**Example – Argumentation Logic**• ¬ SH(P) SB(P) SH(P) ¬ SB(P) • SB(b) SH(b) ¬ SB(b) ¬ SH(b) • ¬ ACC(SB(b)) SB(b) is a non-acceptable argument • ¬ ACC(¬ SB(b)) ¬ SB(b) is a non-acceptable argument • Law of excluded middle for SB(b)? • The law (SB(b) or ¬ SB(b)) is non-acceptable. • Each one of SB(b) and ¬ SB(b) is directly inconsistent and so non-acceptable • The negation of the law, ¬ (SB(b) or ¬ SB(b)), is acceptable (?) • Gives up the law of excluded middle! • Give up (two valued) model theory?**Conclusions**• Acceptability is a direct, minimal and natural formulation of the semantics of abstract argumentation. • Approach is a Synthesis of Labelling and Extension based approaches • Is it “complete”? • Let the “Formalism Tell” vs “Telling the Formalism”. • Acceptability “tells us” (encapsulates) a Reductio ad Absurdum Principle in argumentation. • This enables a reformulation of PL in terms of argumentation => Argumentation Logic (AL) • AL is a conservative extension of PL into a type of Relevance Para-consistent Logic-- Only genuine use of Reductio ad Absurdum • Looking for the analogue of a “model theory” for AL.