OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Dung's (1995) argumentation framework takes as input two abstract entities: a set of arguments and a binary relation encoding attacks between these arguments. It returns acceptable sets of arguments, called extensions, w.r.t. a given semantics. While the abstract nature of this setting is seen as a great advantage, it induces a big gap with the application that it is used to. This raises some questions about the compatibility of the setting with a logical formalism (i.e., whether it is possible to instantiate it properly from a logical knowledge base), and about the significance of the various semantics in the application context. In this paper we tackle the above questions. We first propose to fill in the previous gap by extending Dung's (1995) framework. The idea is to consider all the ingredients involved in an argumentation process. We start with the notion of an abstract monotonic logic which consists of a language (defining the formulas) and a consequence operator. We show how to build, in a systematic way, arguments from a knowledge base formalised in such a logic. We then recall some basic postulates that any instantiation should satisfy. We study how to choose an attack relation so that the instantiation satisfies the postulates. We show that symmetric attack relations are generally not suitable. However, we identify at least one 'appropriate' attack relation. Next, we investigate under stable, semi-stable, preferred, grounded and ideal semantics the outputs of logic-based instantiations that satisfy the postulates. For each semantics, we delimit the number of extensions an argumentation system may have, characterise the extensions in terms of subsets of the knowledge base, and finally characterise the set of conclusions that are drawn from the knowledge base. The study reveals that stable, semi-stable and preferred semantics either lead to counter-intuitive results or provide no added value w.r.t. naive semantics. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970) . Ideal and grounded semantics either coincide and generalise the free consequence relation developed by Benferhat, Dubois, and Prade (1997), or return arbitrary results. Consequently, Dung's (1995) framework seems problematic when applied over deductive logical formalisms. instantiation is built, and fully characterise its set of plausible inferences that may be drawn from the knowledge base. The results show that unlike the abstract framework, its good instantiations have a finite number of extensions under stable, semi-stable and preferred semantics. This is particularly the case when the knowledge base is finite. We show that the set of all formulas used in the supports of the arguments of a stable extension is a maximal (for set inclusion) consistent subset of the knowledge base. However, not every maximal consistent subset of the knowledge base necessarily has a corresponding stable extension. This leads to arbitrary plausible inferences. In the case of a full correspondence, stable semantics does not play any role since the same result is already ensured by the naive semantics. This means that stable semantics either returns arbitrary results or provides no added value. Besides, naive semantics either leads to arbitrary results or generalises the coherence-based approach initially developed by Rescher and Manor (1970) . The situation is worse for preferred semantics. The set of formulas that are used in support of the arguments of a preferred extension is a consistent (but not necessarily maximal for set inclusion) subset of the knowledge base. Thus, arbitrary inferences may be drawn from the knowledge base. Semi-stable extensions are shown to always coincide with stable ones. Thus, semi-stable semantics provides no added value w.r.t. stable semantics. Regarding the ideal and grounded semantics, there are two cases as well. In the first case, both semantics coincide, i.e., the ideal extension of any argumentation framework satisfying the postulates coincides with the grounded extension, which itself coincides with the set of arguments that are built from the free part of a knowledge base, i.e., using the subset of formulas which are not involved in the inconsistency of the knowledge base. Consequently, under these semantics, the set of plausible conclusions is the so-called free consequences in the coherence-based approach for reasoning about inconsistent information (Benferhat, Dubois, & Prade, 1997) . In the second case, both semantics return arbitrary conclusions. The overall study reveals that Dung's (1995) framework can be properly instantiated; i.e., there are instantiations that satisfy some basic rationality postulates. However, stable, semistable, preferred, ideal and grounded semantics are not suitable. Thus, Dung's framework is problematic when applied over a logical formalism, specifically a deductive one. The paper is structured as follows: Section 2 recalls the abstract argumentation framework of Dung (1995) . Section 3 details our instantiation of Dung's framework. Section 4 defines rationality postulates that such instantiation should satisfy, and investigates when those postulates are satisfied/violated. Section 5 analyses the different acceptability semantics introduced by Dung. Section 6 compares our contribution with existing works. Finally, Section 7 concludes the paper with some remarks and perspectives. In Dung (1995), an argumentation framework consists of a set of arguments and a binary relation expressing attacks among the arguments. An argumentation framework is a pair (A, R) where A is a set of arguments and R ⊆ A × A is an attack relation. A pair (a, b) ∈ R means that a attacks b. A set E ⊆ A attacks an argument b iff ∃a ∈ E such that (a, b) ∈ R. We sometimes use the infix notation aRb to denote (a, b) ∈ R. An argumentation framework (A, R) is a graph, the nodes of which are the arguments of A and the edges of which are the attacks in R. The arguments are evaluated using a Property 5 Let X, X ′ , X ′′ ⊆ L. Almost all well-known monotonic logics (classical logics, intuitionistic logics, modal logics, etc.) can be viewed as special cases of Tarski's notion of an abstract logic. AI introduced non-monotonic logics, which do not satisfy monotonicity (Bobrow, 1980) . Once (L, CN) is fixed, a notion of consistency arises as follows: Definition 6 (Consistency) Let X ⊆ L. X is consistent w.r.t. the logic (L, CN) iff CN(X ) = L. It is inconsistent otherwise. Property 11 Let (L, CN) be adjunctive, C ⊆ L be a minimal conflict. For all X ⊂ C, if X = ∅, then: (1) ∃x ∈ L such that CN({x}) = CN(X ). (2) ∃x 1 ∈ CN(X ) and ∃x 2 ∈ CN(C \ X ) such that the set {x 1 , x 2 } is inconsistent. Let (L, CN) be a fixed abstract logic. From now on, we will consider a knowledge base , which is a subset of the logical language L (in symbols, ⊆ L). This base may be infinite, however, with no loss of generality and for the sake of simplicity, it is assumed to be free of tautologies: Assumption 12 Let be a knowledge base. For all x ∈ , x / ∈ CN(∅). The first parameter of an argumentation framework is the set of arguments. In Dung's (1995) framework, an argument is an abstract entity. In what follows, it is built from a knowledge base . It gives a reason for believing a conclusion. Formally, an argument satisfies three main requirements: (i) the reason is a subset of the knowledge base, thus restricting the origin of the arguments; (ii) the reason should be consistent, thus avoiding absurd reasons; and (iii) the reason is minimal. The third requirement means that only relevant information w.r.t. the conclusion is considered.
Interestingly, the study of reasoning with maximal consistency by deductive argumentation frameworks in relation to the question of the compatibility of the Dung-style setting with logical formalisms has led the authors of REF to the following conclusions:
Logical limits of abstract argumentation frameworks
Journal of Applied Non-Classical Logics
Journal of Applied Non-Classical Logics
Computer Science