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1 Introduction
Assumptions are an important concept in defeasible reasoning. Often, in both expert and everyday reasoning, the information provided is not complete or it is inconsistent. By assuming additional information or considering consistent subsets of information, a conclusion can be reached in such cases. A well-known formal method for modeling defeasible reasoning is abstract argumentation theory, introduced by Dung [Dung95]. In logical argumentation, the arguments have a specific structure on which the attacks depend [BesHun01, Pra10]. One such logical argumentation framework is sequent-based argumentation [ArStr15argcomp], in which arguments are represented by sequents, as introduced by Gentzen [Gen34] and well-known in proof theory. Attacks between arguments are formulated by sequent elimination rules, which are special inference rules. The resulting framework is generic and modular, in that any logic, with a corresponding sound and complete sequent calculus can be taken as the deductive base (the so-called core logic).
In this paper we extend sequent-based argumentation. To each sequent a component for assumptions is added. This way, a distinction can be made between strict and defeasible premises, to reach further conclusions. As an instance of the obtained framework, assumption-based argumentation (ABA) [BDKT97, DKT09, Toni14] is studied and the relation to reasoning with maximally consistent subsets [ReMa70] is investigated. The latter is a well-known method to maintain consistency, in view of inconsistent information. ABA is a structural argumentation framework which is also abstract, in that there are only limited assumptions on the underlying deductive system. It was introduced to determine a set of assumptions that can be accepted as a conclusion from the given information.
Arguments in ABA are constructed by applying modus ponens to simple clauses of an inferential database. Only recently logic-based instantiations of ABA have been studied, mostly with classical logic as the core logic. Sequent-based argumentation, and the here introduced assumptive generalization, are more general and modular, in that these are based on a Tarskian core logic and the arguments are constructed via the inference rules of the corresponding sequent calculus. Logics that can be equipped with defeasible assumptions by means of assumptive sequent-based argumentation include, in addition to classical logic, intuitionistic logic, many of the well-known modal logics and several relevance logics. Hence, the results of this paper generalize to many deductive core systems, as long as the Tarskian conditions are fulfilled.
Sequent calculi and sequent-based argumentation have some further advantages as well. For example, the latter comes equipped with a dynamic proof theory [ArStr15LFSA, ArStr17DD], introduced to study argumentation from a proof theoretical perspective. These dynamic derivations provide a mechanism for deriving arguments as well as attacks and hence to reach conclusions for a given argumentation framework in an automatic way. Sequent calculi themselves have been investigated for many logics and purposes, mainly in the context of proof theory. A significant advantage over other proof systems is, that the premises can be manipulated within a proof, see also [SchHei03].
The paper is organized as follows. In the next section, sequent-based argumentation is recalled. Then, in Section LABEL:sec:AssumptiveSeq, the general framework for assumptive sequent-based argumentation is introduced. This framework will be considered in Section LABEL:sec:ABA, in which ABA is taken as an example, to show how the assumptive sequent-based framework can be applied. We conclude in Section LABEL:sec:Conclusion.
2 Sequent-based argumentation
Throughout the paper only propositional languages are considered, denoted by . Atomic formulas are denoted by , formulas are denoted by , sets of formulas are denoted by , and finite sets of formulas are denoted by , later on we will denote sets of assumptions by and finite sets of assumptions by , all of which can be primed or indexed.
Definition 1.
A logic for a language is a pair , where is a (Tarskian) consequence relation for , having the following properties: reflexivity: if , then ; transitivity: if and , then ; and monotonicity: if and , then .
As usual in logical argumentation (see, e.g., [BesHun01, Pol92, Prak96, SiLou92]), arguments have a specific structure based on the underlying formal language, the core logic. In the current setting arguments are represented by the well-known proof theoretical notion of a sequent.
Definition 2.
Let be a logic and a set of -formulas.
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An -sequent (sequent for short) is an expression of the form , where and are finite sets of formulas in and is a symbol that does not appear in .
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An -argument (argument for short) is an -sequent ,111Set signs in arguments are omitted. where . is called the support set of the argument and its conclusion.
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An -argument based on is an -argument , where . We denote by the set of all the -arguments based on .
Given an argument , we denote and . We say that is a sub-argument of iff . The set of all the sub-arguments of is denoted by .
The formal systems used for the construction of sequents (and so of arguments) for a logic , are sequent calculi [Gen34], denoted here by . In what follows we shall assume that is sound and complete for , i.e., is provable in iff . One of the advantages of sequent-based argumentation is that any logic with a corresponding sound and complete sequent calculus can be used as the core logic.222See [ArStr15argcomp] for further advantages of this approach. The construction of arguments from simpler arguments is done by the inference rules of the sequent calculus [Gen34].
Argumentation systems contain also attacks between arguments. In our case, attacks are represented by sequent elimination rules. Such a rule consists of an attacking argument (the first condition of the rule), an attacked argument (the last condition of the rule), conditions for the attack (the conditions in between) and a conclusion (the eliminated attacked sequent). The outcome of an application of such a rule is that the attacked sequent is ‘eliminated’. The elimination of a sequent is denoted by or .
Definition 3.
A sequent elimination rule (or attack rule) is a rule of the form:
| (1) |
It is said that -attacks .
Example 1.
Suppose contains a -negation (where and for every atom ) and a -conjunction (where iff and ). We refer to [ArStr15argcomp, StrAr15logcom] for a definition of a variety of attack rules. Assuming that , two such rules are:
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