Search Results (5)

We introduce De Morgan Heyting logic for Heyting algebras with De Morgan negation (DH-algebras). The variety $\mathcal{DH}$ of all DH-algebras is congruence distributive. The lattice of all subvarieties of $\mathcal{DH}$ is distributive. We show the discrete dualities between De Morgan frames and DH-algebras. The Kripke completeness and finite approximability of some DH-logics are proven. Some conservativity of DH expansion of a Kripke complete superintuitionistic logic is shown by the construction of frame expansion. Finally, a cut-free terminating Gentzen sequent calculus for the DH-logic of De Morgan Boolean algebras is developed. Full article

Residuated basic logic ($\mathsf{RBL}$) is the logic of residuated basic algebras, which constitutes a conservative extension of basic propositional logic ($\mathsf{BPL}$). The basic implication is a residual of a non-associative binary operator in $\mathsf{RBL}$. The conservativity is shown by relational semantics. A Gentzen-style sequent calculus $\mathsf{GRBL}$, which is an extension of the distributive full non-associative Lambek calculus, is established for residuated basic logic. The calculus $\mathsf{GRBL}$ admits the mix-elimination, subformula, and disjunction properties. Moreover, the class of all residuated basic algebras has the finite embeddability property. The consequence relation of $\mathsf{GRBL}$ is decidable. Full article

The cut-free single-succedent Gentzen sequent calculus ${\mathsf{GK}}_t$ for the minimal tense logic ${\mathsf{K}}_t$ is introduced. This sequent calculus satisfies the displaying property. The proof proceeds in terms of a Kolmogorov translation and three intermediate sequent systems. Finally, we show that tense logics axiomatized by strictly positive implication have cut-free Gentzen sequent calculi uniformly. Full article

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. Full article

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic $\mathsf{SCI}$ . The language of $\mathsf{SCI}$ is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, $\mathsf{SCI}$ is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for $\mathsf{SCI}$ known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for $\mathsf{SCI}$ and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper. Full article