# Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic

Knowledge-Based System Group, Institut für Logic and Computation, Technische Universität Wien, Favoritenstraße 9-11, 1040 Vienna, Austria

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This paper is an extended version of our paper published in the proceedings of the 15th International Conference on Logic Programming and Non-monotonic Reasoning (LPNMR 2019) as well as of an abstract published in the proceedings of the conference “Kurt Gödel’s Legacy: Does Future lie in the Past?” held 2019 in Vienna. This paper is dedicated to the memory of Khimuri Rukhaia, logician, professor, and a kind man, who was the teacher of the first author during her Bachelor studies and who sadly passed away during the preparation of this work.

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These authors contributed equally to this work.

Received: 9 June 2020 / Revised: 14 July 2020 / Accepted: 15 July 2020 / Published: 21 July 2020

(This article belongs to the Special Issue Deductive Systems)

Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults.