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# On the Semantics of Argumentation

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1. 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

2. 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

3. 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.

4. 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.

5. 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).

6. 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.

7. 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

8. 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

9. 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!

10. 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.

11. 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

12. 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 

13. Natural Deduction (RA) as ArgumentationExample: ¬Acc({},{}) Genuine RA derivation for  • Θ = {¬ (θεόςθνητός), ¬ 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.

14. Natural Deduction (RA) as ArgumentationExample: ¬Acc({},{}) Genuine RA derivation for  • Θ = {¬ (θεόςθνητός), ¬ θνητός -> ¬ πεθάνει, πεθάνει} [θεός ¬ θνητός ¬πεθάνει πεθάνει ] ¬ θεός Θεός [θνητός θεός (copy) θεός θνητός ¬(Θεός θνητός) ] Attack ??? θνητός ¬θνητός {} θεός Violates the Genuine property!

15. 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 ab and ¬(ab) 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 }

16. 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).

17. 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)

18. 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).

19. 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?

20. 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.