Logic N4 Research Articles

Abstract Argumentation and Answer Set Programming: Two Faces of Nelson’s Logic

AbstractIn this work, we show that both logic programming and abstract argumentation frameworks can be interpreted in terms of Nelson’s constructive logic N4. We do so by formalising, in this logic, two principles that we call noncontradictory inference and strengthened closed world assumption: the first states that no belief can be held based on contradictory evidence while the latter forces both unknown and contradictory evidence to be regarded as false. Using these principles, both logic programming and abstract argumentation frameworks are translated into constructive logic in a modular way and using the object language. Logic programming implication and abstract argumentation supports become, in the translation, a new implication connective following the noncontradictory inference principle. Attacks are then represented by combining this new implication with strong negation. Under consideration in Theory and Practice of Logic Programming (TPLP).

Logical Pluralism and Interpretations of Logical Systems

Logical pluralism is a general idea that there is more than one correct logic. Carnielli and Rodrigues [2019a] defend an epistemic interpretation of the paraconsistent logic N4, according to which an argument is valid in this logic just in case it necessarily preserves evidence. The authors appeal to this epistemic interpretation to briefly motivate a kind of logical pluralism: “different accounts of logical consequence may preserve different properties of propositions”. The aim of this paper is to study the prospect of a logical pluralism based on different interpretations of logical systems. First, we give our analysis of what it means to interpret a logic – and make some hopefully useful distinctions along the way. Second, we present what we call an interpretational logical pluralism: there is more than one correct logic and a logic is correct only if it has some adequate interpretation. We consider four variants of this idea, bring up some possible objections, and try to find plausible solutions on behalf of the pluralist. We will argue that interpretations of logical systems provide a promising – albeit not unproblematic – route to logical pluralism.

Normalisation for Some Quite Interesting Many-Valued Logics

In this paper, we consider a set of quite interesting three- and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a detailed version of Prawitz’s proof of Nelson’s logic N4 and its extension by intuitionist negation.

Kripke-Style Models for Logics of Evidence and Truth

In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment (FDE) with a classicality operator ∘ that recovers classical logic for formulas in its scope. According to the intended interpretation here proposed, these models represent a database that receives information as time passes, and such information can be positive, negative, non-reliable, or reliable, while a formula ∘A means that the information about A, either positive or negative, is reliable. This proposal is in line with the interpretation of N4 and FDE as information-based logics, but adds to the four scenarios expressed by them two new scenarios: reliable (or conclusive) information (i) for the truth and (ii) for the falsity of a given proposition.

Kripke-Completeness and Cut-elimination Theorems for Intuitionistic Paradefinite Logics With and Without Quasi-Explosion

Two intuitionistic paradefinite logics N4C and N4C+ are introduced as Gentzen-type sequent calculi. These logics are regarded as a combination of Nelson’s paraconsistent four-valued logic N4 and Wansing’s basic constructive connexive logic C. The proposed logics are also regarded as intuitionistic variants of Arieli, Avron, and Zamansky’s ideal paraconistent four-valued logic 4CC. The logic N4C has no quasi-explosion axiom that represents a relationship between conflation and paraconsistent negation, but the logic N4C+ has this axiom. The Kripke-completeness and cut-elimination theorems for N4C and N4C+ are proved.

Tarskian consequence relations bilaterally: some familiar notions

This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, of definitional equivalence, and of a conservative extension.

Paraconsistent Logic, Evidence, and Justification

In a forthcoming paper, Walter Carnielli and Abilio Rodrigues propose a Basic Logic of Evidence (BLE) whose natural deduction rules are thought of as preserving evidence instead of truth. BLE turns out to be equivalent to Nelson’s paraconsistent logic N4, resulting from adding strong negation to Intuitionistic logic without Intuitionistic negation. The Carnielli/Rodrigues understanding of evidence is informal. Here we provide a formal alternative, using justification logic. First we introduce a modal logic, KX4, in which \(\square X\) can be read as asserting there is implicit evidence for X, where we understand evidence to permit contradictions. We show BLE embeds into KX4 in the same way that Intuitionistic logic embeds into S4. Then we formulate a new justification logic, JX4, in which the implicit evidence motivating KX4 is made explicit. KX4 embeds into JX4 via a realization theorem. Thus BLE has both implicit and explicit possibly contradictory evidence interpretations in a formal sense.

Proof theory of Nelson’s paraconsistent logic: A uniform perspective

The aim of this paper is to obtain a theoretical foundation of inconsistency-tolerant (or paraconsistent) reasoning by presenting a comprehensive study of the structural proof-theory of David Nelson’s paraconsistent logic. Inconsistency handling has a growing importance in Computer Science since inconsistencies may frequently occur in knowledge-based and intelligent information systems. Paraconsistent, inconsistency-tolerant logics have been studied to cope with such inconsistencies. In this paper, proof systems for Nelson’s paraconsistent logic N4 are comprehensively studied. The logic N4 is a fundamental system and known to be a common basis for various extended and useful paraconsistent logics. Some basic theorems including cut-elimination, normalization and completeness are uniformly proved using various embedding theorems. A variety of sequent calculi and natural deduction systems for N4 and some closely related systems are presented and compared.

Modal logics with Belnapian truth values

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation.

The Class of Extensions of Nelson's Paraconsistent Logic

The article is devoted to the systematic study of the lattice eN4⊥ 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 eN4⊥ and the lattice of superintuitionistic logics. Distinguish in eN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.

Natural deduction systems for Nelson's paraconsistent logic and its neighbors

Firstly, a natural deduction system in standard style is introduced for Nelson's para-consistent logic N4, and a normalization theorem is shown for this system. Secondly, a natural deduction system in sequent calculus style is introduced for N4, and a normalization theorem is shown for this system. Thirdly, a comparison between various natural deduction systems for N4 is given. Fourthly, a strong normalization theorem is shown for a natural deduction system for a sublogic of N4. Fifthly, a strong normalization theorem is proved for a typed λ-calculus for a neighbor of N4. Finally, it is remarked that the natural deduction frameworks presented can also be adapted for Wansing's basic connexive logic C.

On the Representation of N4-Lattices

N4-lattices provide algebraic semantics for the logic N4, the paraconsistent variant of Nelson's logic with strong negation. We obtain the representation of N4-lattices showing that the structure of an arbitrary N4-lattice is completely determined by a suitable implicative lattice with distinguished filter and ideal. We introduce also special filters on N4-lattices and prove that special filters are exactly kernels of homomorphisms. Criteria of embeddability and to be a homomorphic image are obtained for N4-lattices in terms of the above mentioned representation. Finally, subdirectly irreducible N4-lattices are described.

Algebraic Semantics for Paraconsistent Nelson's Logic

In the present article, different types of semantics for the logic N4, the paraconsistent variant of Nelson's constructive logic with strong negation, will be considered. N4 will be characterized in terms of so-called Fidel-structures. It will be stated that Fidel-structures are equivalent to twist-structures. Further, the N4-lattices, generalization of N-lattices, will be defined and it will be proved that they can be represented as twist-structures. It will be proved that N4-lattices form a variety νN4 and there is a natural dual isomorphism between the lattice of subvarieties of νN4 and the lattice of N4-extensions.

Displaying the modal logic of consistency

AbstractIt is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.