A logic of defeasible argumentation: Constructing arguments in justification logic

Abstract

In the 1980s, Pollock’s work on default reasons started the quest in the AI community for a formal system of defeasible argumentation. The main goal of this paper is to provide a logic of structured defeasible arguments using the language of justification logic. In this logic, we introduce defeasible justification assertions of the type t : F that read as “t is a defeasible reason that justifies F”. Such formulas are then interpreted as arguments and their acceptance semantics is given in analogy to Dung’s abstract argumentation framework semantics. We show that a large subclass of Dung’s frameworks that we call “warranted” frameworks is a special case of our logic in the sense that (1) Dung’s frameworks can be obtained from justification logic-based theories by focusing on a single aspect of attacks among justification logic arguments and (2) Dung’s warranted frameworks always have multiple justification logic instantiations called “realizations”. We first define a new justification logic that relies on operational semantics for default logic. One of the key features that is absent in standard justification logics is the possibility to weigh different epistemic reasons or pieces of evidence that might conflict with one another. To amend this, we develop a semantics for “defeaters”: conflicting reasons forming a basis to doubt the original conclusion or to believe an opposite statement. This enables us to formalize non-monotonic justifications that prompt extension revision already for normal default theories. Then we present our logic as a system for abstract argumentation with structured arguments. The format of conflicting reasons overlaps with the idea of attacks between arguments to the extent that it is possible to define all the standard notions of argumentation framework extensions. Using the definitions of extensions, we establish formal correspondence between Dung’s original argumentation semantics and our operational semantics for default theories. One of the results shows that the notorious attack cycles from abstract argumentation cannot always be realized as justification logic default theories.

Structured argumentation dynamics

“…The logic of default justifications was first defined in [50] and our definitions in this section follow those given in [54]. We start by outlining the underlying justification logic on which we base default theories.…”

“…In the logic of default reasons from Section 2.2, the main idea of the semantics is to give a procedure that determines the admissibility of reasons. This justifies our choice to fully focus on the admissibility condition by taking the single-world models perspective, as done in [54]. Definition 2 (JT CS model) We define a function reason assignment based on CS, * (•) : T m → 2 F m , a function mapping each term to a set of formulas from F m. We assume that it satisfies the following conditions:…”

“…Moreover, Dung's frameworks, modulo some attack cycles, always have multiple justification logic instantiations called "realizations". These results are shown in [54]. Throughout the rest of the paper, we only rely on the procedure to obtain an abstract argumentation framework from a default theory T called Forgetful projection [54, p. 26].…”

“…The basic format of our default theories has been developed in [50,51,54], where we interpret justification formulas of the form t : F as arguments that can defeat other formulas by means of undercut or rebuttal. Informally, for some defeasible argument understood as a premise-claim pair, a rebutting defeater is a reason for the opposite claim.…”

“…The paper is organized as follows. The logic introduced in [50,51,54] already models non-monotonicity with the use of undercut and rebuttal. We use this logic to exemplify Toulmin's [67] argumentation model in our default theories.…”