In this paper, the positive fragment of the logic math formula of contraction-less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four-valued logic math formula. This extended relevant logic is called math formula, and it has the property of constructible falsity which is known to be a characteristic property of math formula. A Gentzen-type sequent calculus math formula for math formula is introduced, and the cut-elimination and decidability (...) theorems for math formula are proved. Two extended Routley-Meyer semantics are introduced for math formula, and the completeness theorems with respect to these semantics are proved. (shrink)

In this article we firstly present a Kripke semantics for the description logic ALC which is directly inspired by the semantics for Intuitionistic logic. Moreover, we discuss why a direct translation of this kind of semantics is not adequate in the description logic context and propose a constructive semantics that differs from the previous one by the fact that we impose a condition on the partial order. We also present a tableau calculus which is sound and complete with respect to (...) our semantics. As an application of the calculus we prove that this semantics meets the disjunction property, a key criterion used in assessing whether a logic is constructive. (shrink)

The paper discusses properties of 4QL, a DATALOG⌉⌉-like query language, originally outlined by Maluszyński and Szalas (Maluszyński & Szalas, 2011). 4QL allows one to use rules with negation in heads and bodies of rules. It is based on a simple and intuitive semantics and provides uniform tools for “lightweight” versions of known forms of nonmonotonic reasoning. Negated literals in heads of rules may naturally lead to inconsistencies. On the other hand, rules do not have to attach meaning to some literals. (...) Therefore 4QL is founded on a four-valued semantics, employing the logic introduced in (Maluszyński et al., 2008; Vitória et al., 2009) with truth values: ‘true’, ‘false’, ‘inconsistent’ and ‘unknown’. In addition, 4QL is tractable w.r.t. data complexity and captures PTIME queries. Even though DATALOG⌉⌉ is known as a concept for the last 30 years, to our best knowledge no existing approach enjoys these properties.In the current paper we:investigate properties of well-supported models of 4QLprove the correctness of the algorithm for computing well-supported modelsshow that 4QL has PTIME data complexity and captures PTIME. (shrink)