Abstract
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.
Highlights
Most formal logics studied in the literature are monotonic in the sense that an increased set of premisses never yields a reduced set of conclusions
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
Prominent among these are the sequent-type calculi for default logic and autoepistemic logic introduced by Bonatti [7] and those for default logic, autoepistemic logic, and circumscription by Bonatti and Olivetti [8]
Summary
Most formal logics studied in the literature are monotonic in the sense that an increased set of premisses never yields a reduced set of conclusions. Given the large body of works devoted to nonmonotonic reasoning, only few investigations exist dealing with concrete proof systems for it Prominent among these are the sequent-type calculi for default logic and autoepistemic logic introduced by Bonatti [7] and those for default logic, autoepistemic logic, and circumscription by Bonatti and Olivetti [8]. The first variant of default logic employs Łukasiewicz’s three-valued logic [11] as the underlying base logic and the second variant generalises default rules by allowing a selection of consequents in defaults, closely related to the answer-set semantics of disjunctive logic programs [4] Both versions have been introduced to address certain representational shortcomings of standard default logic.
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