The Attack as Intuitionistic Negation

We translate the argumentation networks A = (S,R) into a theory ∆A of intuitionistic logic, retaining S as the domain and using intuitionistic negation to model the attack R in A: the attack xRy is translated to x → ¬y. The intuitionistic models of ∆A characterise the complete extensions of A. The reduction of argumentation networks to intuitionistic logic yields, in addition to a representation theorem, some additional benefits: it allows us to give semantics to higher level attacks, where an attack “xRy” can itself attack another attack “uRv”; one can make higher level metastatements W on (S,R) and such meta-statements can attack and be attacked in the domain. 1 Background and orientation {sec1} This paper is a continuation of [1] but it is self-contained and is independent of [1], except that it expands the ideas of [1]. Given a finite abstract argumentation network A = (S,R), where S 6= ∅ is the set of arguments and R ⊆ S×S is the attack relation, we would like to view the set S as atomic propositions of the intuitionistic propositional calculus and translate the attack relation xRy as x→ ¬y, where “→” represents intuitionistic implication and “¬” represents intuitionistic negation. For each A we write a 1 ar X iv :1 51 0. 00 07 7v 1 [ cs .L O ] 3 0 Se p 20 15 theory ∆A such that all the complete extensions of A correspond to all the intuitionistic models of ∆A. The reduction of argumentation networks to intuitionistic logic yields, in addition to a representation theorem, some additional benefits. • It allows us to give semantics to higher level attacks, where an attack “xRy” can itself attack another attack “uRv”. • One can make higher level meta-statements W on (S,R) and such metastatements can attack and be attacked. For example we can attack an argument a by saying that a is a generic argument which attacks all other arguments {x | x 6= a} and therefore a should be out. What we are saying is W (a) where: W (a) = ∀x(x 6= a→ aRx) attacks a. We shall use Gödel’s intuitionistic logic G3, semantically defined by all intuitionistic Kripke models with just two linearly ordered worlds t < s (t the actual world and s a possible world), with < the intuitionistic accessibility relation. Appendix A describes the logic G3 in detail. It can be axiomatised. We present the complete extensions of A = (S,R), using the Caminada labelling approach, [2]. A Caminada labelling of S is a function λ : S 7→ {in, out, und} such that the following holds (C1) λ(x) = in iff for all y attacking x, λ(y) = out. (C2) λ(x) = out iff for some y attacking x, λ(y) = in. (C3) λ(x) = und iff for all y attacking x, λ(y) 6= in, and for some y0 attacking x, λ(y0) = und. 1 (C4) If x is not attacked at all, then λ(x) = in. Let us use the following notation for G3. For a proposition e, write e = (>,>) to mean t e and s e. Write e = (⊥,>) to mean t 6 e and s e. Write e = (⊥,⊥) to mean t 6 e and s 6 e. Note that since {t < s} is an intuitionistic model, the option t e and s 6 e is not allowed. We denote assignments h of truth values to atoms in the model by h(e) = (>,>) or h(e) = (⊥,>) or h(e) = (⊥,⊥). We also write e = (>,>), (⊥,>), (⊥,⊥) respectively, using abuse of notation, when the assignment is known. So t e means t h e and s e means s h e. 1That is, λ(x) = und iff neither λ(x) = in nor λ(x) = out. 2This condition follows from (C1), since the empty conjunction is considered >.