In this paper, we define a first-order logic CFʹ with strong negation and bounded static quantifiers, which is a variant of Thomason's logic CF. For the logic CFʹ, the usual Kripke formal semantics is defined based on situations, and a sound and complete axiomatic system is established based on the axiomatic systems of constructive logics with strong negation and Thomason's completeness proof techniques. With the use of bounded quantifiers, CFʹ allows the domain of quantification to be empty and allows for (...) nondenoting constants. CFʹ is intended as a fragment of a logic for situation theory. Thus the connection between CFʹ and infon logic is discussed. (shrink)

The paper is devoted to the contributions of Helena Rasiowa to the theory of non-classical negation. The main results of Rasiowa in this area concerns–constructive logic with strong (Nelson) negation.

Constructive logic with Nelson negation is an extension of the intuitionistic logic with a special type of negation expressing some features of constructive falsity and refutation by counterexample. In this paper we generalize this logic weakening maximally the underlying intuitionistic negation. The resulting system, called subminimal logic with Nelson negation, is studied by means of a kind of algebras called generalized N-lattices. We show that generalized N-lattices admit representation formalizing the intuitive idea of refutation by means of counterexamples giving in (...) this way a counterexample semantics of the logic in question and some of its natural extensions. Among the extensions which are near to the intuitionistic logic are the minimal logic with Nelson negation which is an extension of the Johansson's minimal logic with Nelson negation and its in a sense dual version — the co-minimal logic with Nelson negation. Among the extensions near to the classical logic are the well known 3-valued logic of Lukasiewicz, two 12-valued logics and one 48-valued logic. Standard questions for all these logics — decidability, Kripke-style semantics, complete axiomatizability, conservativeness are studied. At the end of the paper extensions based on a new connective of self-dual conjunction and an analog of the Lukasiewicz middle value ½ have also been considered. (shrink)

In this work we define, in a general way, an algebraic semantics for the logics of the hierarchy InPk. This semantics is defined by means of an alternative construction, with respect to the usual algebraic semantics, and it is known in the literature as Twist-structures semantics. Besides that, we modify such construction, defining the so-called ω-Twist-structures here. This adaptation allows us to prove adequacy theorems for every logic of the hierarchy InPk.

The article is devoted to the systematic study of the lattice εN4⊥ consisting of logics extending N4⊥. The logic N4⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between εN4⊥ and the lattice of superintuitionistic logics. Distinguish in εN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.

Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions (...) and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK. (shrink)

Earlier algebraic semantics for Belnapian modal logics were defined in terms of twist-structures over modal algebras. In this paper we introduce the class of BK -lattices, show that this class coincides with the abstract closure of the class of twist-structures, and it forms a variety. We prove that the lattice of subvarieties of the variety of BK -lattices is dually isomorphic to the lattice of extensions of Belnapian modal logic BK . Finally, we describe invariants determining a twist-structure over a (...) modal algebra. (shrink)

It was proved by Odintsov and Pearce that the logic is a deductive base for paraconsistent answer set semantics of logic programs with two kinds of negation. Here we describe the lattice of logics extending, characterise these logics via classes of -models, and prove that none of the proper extensions of is a deductive base for PAS.

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine the (...) structure of the lattice of extensions of n(LC). (shrink)

We study the sequent system mentioned in the author's work as CyInFL with ‘intuitionistic’ sequents. We explore the connection between this system and symmetric constructive logic of Zaslavsky and develop an algebraic semantics for both of them. In contrast to the previous work, we prove the strong completeness theorem for CyInFL with ‘intuitionistic’ sequents and all of its basic variants, including variants with contraction. We also show how the defined classes of structures are related to cyclic involutive FL‐algebras and Nelson (...) FLew‐algebras. In particular, we prove the definitional equivalence of symmetric constructive FLewc‐algebras (algebraic models of symmetric constructive logic) and Nelson FLew‐algebras (algebras introduced by Spinks and Veroff, as the termwise equivalent definition of Nelson algebras). Because of the strong completeness theorem that covers all basic variants of CyInFL with ‘intuitionistic’ sequents, we rename this sequent system to symmetric constructive full Lambek calculus (). We verify the decidability of this system and its basic variants, as we did in the case of their distributive cousins. As a consequence we obtain that the corresponding theories of (distributive and nondistributive) symmetric constructive FL‐algebras are decidable. (shrink)

ABSTRACT In the paper we present a survey of some approaches to the semantics of many-valued propositional systems. These approaches are inspired on one hand by classical problems in the investigations of logical aspects of epistemic activity: knowledge and truth, contradictions, beliefs, reliability of data, etc. On the other hand they reflect contemporary concerns of researchers in Artificial Intelligence (and Cognitive Science in general) with inferences drawn from imperfect information, even from total ignorance. We treat the mathematical apparatus that has (...) emerged recently: algebraic structures related to the new logical systems in the same way Boolean algebras correspond to classical logic. (shrink)

Motivated by Kalman residuated lattices, Nelson residuated lattices and Nelson paraconsistent residuated lattices, we provide a natural common generalization of them. Nelson conucleus algebras unify these examples and further extend them to the non-commutative setting. We study their structure, establish a representation theorem for them in terms of twist structures and conuclei that results in a categorical adjunction, and explore situations where the representation is actually an isomorphism. In the latter case, the adjunction is elevated to a categorical equivalence. By (...) applying this representation to the original motivating special cases we bring to the surface their underlying similarities. (shrink)