Report

On the Semantics of Argumentation 1 Antonis Kakas Francesca Toni Paolo Mancarella Department of Computer Science University of Cyprus Department of Computing Imperial College Dipartimento di Informatica Universita di Pisa 20 April, 2012 London Argumentation Forum Contents 2 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 3 Abstract Argumentation: <Args, Attack> Args: a set of arguments Attack: a binary relation on Args Attack: the argument “a” attacks the argument “b” A attacks B iff a A and b B s.t. (a,b) Attack, for (a,b) any A,B P(Args) Acceptability semantics is defined via a relative Acceptability Relation between (sets of) arguments: Acc(,0): Given 0 the set can be accepted. Acceptability Semantics Informal Motivation 4 Acceptability Relation: Follow the “universal” intuition: An argument (or a set of arguments) can be accepted iff 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 RELATIVE notion. An argument (or a set of arguments) is acceptable iff it renders all its counter-arguments non-acceptable. Acceptability Semantics Informal Motivation 5 An argument (or a set of arguments), , is acceptable iff 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) A is non-acceptable iff there exists a set of arguments D that attacks A such that D is acceptable (relative to A). Acceptability Semantics Definition DEFINITION A 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’) iff Δ 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 iff Acc(Δ,{}) holds. Acceptability Semantics Some Results 7 Admissible implies 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-defeating iff 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 selfdefeating attacking sets by counter-attacking them back. This extends Admissibility Acceptability Semantics Extending Admissibility 8 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 a1 {a} is Acceptable a1 a a1 a2 a3 a1 Acceptability Semantics Self-defeat ↔ Reductio ad Absurdum 9 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 This S: renders one of its attacks acceptable is a kind of Reductio ad Absurdum Principle! Part2: Argumentation Logic 10 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 11 Consider Propositional Logic (PL) and its Natural Deduction (ND) proof system. Take out the Reductio ad Absurdum rule (¬I rule) from ND Direct Proofs, ├MRA . RA will be recovered at the semantic level by Acceptability Only Argumentation Args: Att: Framework for AL: <Args, Att> Sets of Propositional Formulae: ∆ (Direct proofs from ∆ and the given theory, T) A attacks ∆ : T ∪ ∪ ├MRA D defends A: D={¬} for some in A D={} when T ∪ ├MRA Argumentation Logic = Propositional Logic Sketch Proof 12 ¬Acc({},{}) Genuine RA derivation for Genuine RA derivations: [ . [’ . c() . [ . . ¬ ] T ¬ ├MRA ’ is necessary for the direct derivation of ] Technical Lemma: For classically consistent theories if there exists a RA derivation for the there exists a Genuine RA derivation for . Natural Deduction (RA) as Argumentation Example: ¬Acc({},{}) Genuine RA derivation for 13 Θ = {¬ (θεός θνητός), ¬ θνητός ¬ πεθάνει, πεθάνει} Θεός [θεός [¬ θνητός ¬ θνητός ¬ θνητός θνητός ¬πεθάνει πεθάνει ¬ (¬θνητός) ] θνητός θεός θνητός ¬(θεός θνητός) ] ¬ θεός The argument, ¬ θνητός, that can defend against the attack θνητός cannot do so as it is self-defeating. Hence the argument, θεός, is not acceptable. Natural Deduction (RA) as Argumentation Example: ¬Acc({},{}) Genuine RA derivation for 15 Θ = {¬ (θεός θνητός), ¬ θνητός -> ¬ πεθάνει, πεθάνει} [θεός [θνητός ¬ θνητός ¬πεθάνει πεθάνει Θεός θεός (copy) θεός θνητός ¬(Θεός θνητός) ] Attack ??? ¬ θνητός θνητός ] ¬ θεός {} Violates the Genuine property! θεός Argumentation Logic Results (1) 16 T classically consistent AL = PL (for the restricted language of ¬ and ) AL entails iff ¬Acc({¬},{}) holds. Interpretation Both of implication in AL differs from PL, e.g. 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 Logic Results (2) 17 AL extends PL when T is (classically) inconsistent Directly consistent AL does not trivialize AL entails iff ¬Acc({¬},{}) and Acc({},{}) hold. AL isolates out the non-relevant use of Reductio ad Absurdum Example: Logical Paradoxes (T = { φ , ¬φ } ) Directly Use inconsistent “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 ShavesHimself(barber) ¬ ShavedByBarber(barber) ¬ ShavesHimself(barber) ShavedByBarber(barber) Example – Classical Logic ¬ SH(P) SB(P) SB(b) SH(b) SH(P) ¬ SB(P) ¬ SB(b) ¬ SH(b) SB(b) |- SH(b) |- ¬ SB(b) ¬ SB(b) |- ¬ SH(b) |- SB(b) i.e. 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) SB(b) SH(b) ¬ ACC(SB(b)) ¬ ACC(¬ SB(b)) SB(b) is a non-acceptable argument ¬ SB(b) is a non-acceptable argument Law of excluded middle for SB(b)? The law (SB(b) or ¬ SB(b)) is non-acceptable. SH(P) ¬ SB(P) ¬ SB(b) ¬ SH(b) 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 21 Acceptability is a direct, minimal and natural formulation of the semantics of abstract argumentation. Let the “Formalism Tell” vs “Telling the Formalism”. Approach is a Synthesis of Labelling and Extension based approaches Is it “complete”? 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 Paraconsistent Logic -- Only genuine use of Reductio ad Absurdum Looking for the analogue of a “model theory” for AL.