Belnap-Dunn Logic Research Articles

Belnap-Dunn Logic and Query Answering in Inconsistent Databases with Null Values

This paper concerns an expansion of first-order Belnap-Dunn logic, called BD^{\supset,F}_{\bot}, and an application of this logic in the area of relational database theory. The notion of a relational database, the notion of a query applicable to a relational database, and several notions of an answer to a query with respect to a relational database are considered from the perspective of this logic, taking into account that a database may be an inconsistent database and/or a database with null values. The chosen perspective enables among other things the definition of a notion of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. For each of the notions of an answer considered, being an answer to a query with respect to a database of the kind considered is decidable.

Knowledge and ignorance in Belnap–Dunn logic

Abstract In this paper, we argue that the usual approach to modelling knowledge and belief with the necessity modality $\Box $ does not produce intuitive outcomes in the framework of the Belnap–Dunn logic ($\textsf{BD}$, alias $\textbf{FDE}$—first-degree entailment). We then motivate and introduce a nonstandard modality $\blacksquare $ that formalizes knowledge and belief in $\textsf{BD}$ and use $\blacksquare $ to define $\bullet $ and $\blacktriangledown $ that formalize the unknown truth and ignorance as not knowing whether, respectively. Moreover, we introduce another modality $\textbf{I}$ that stands for factive ignorance and show its connection with $\blacksquare $. We equip these modalities with Kripke-frame-based semantics and construct a sound and complete analytic cut system for $\textsf{BD}^{\blacksquare }$ and $\textsf{BD}^{\textbf{I}}$—the expansions of $\textsf{BD}$ with $\blacksquare $ and $\textbf{I}$. In addition, we show that $\Box $ as it is customarily defined in $\textsf{BD}$ cannot define any of the introduced modalities, nor, conversely, neither $\blacksquare $ nor $\textbf{I}$ can define $\Box $. We also demonstrate that $\blacksquare $ and $\textbf{I}$ are not interdefinable and establish the definability of several important classes of frames using $\blacksquare $.

Reasoning with belief functions over Belnap–Dunn logic

We design an expansion of Belnap–Dunn logic with belief and plausibility functions that allows non-trivial reasoning with contradictory and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in [23]. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of Łukasiewicz logic. The second approach is inspired by [3]. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.

The lattice of all 4-valued implicative expansions of Belnap–Dunn logic containing Routley and Meyer’s basic logic Bd

Abstract The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with $\wedge $, $\vee $ and $\sim $ as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer’s basic logic B and its useful disjunctive extension B$^{\textrm {d}}$. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class.

A Class of Implicative Expansions of Belnap-Dunn Logic in which Boolean Negation is Definable

Belnap and Dunn’s well-known 4-valued logic FDE is an interesting and useful non-classical logic. FDE is defined by using conjunction, disjunction and negation as the sole propositional connectives. Then the question of expanding FDE with an implication connective is of course of great interest. In this sense, some implicative expansions of FDE have been proposed in the literature, among which Brady’s logic BN4 seems to be the preferred option of relevant logicians. The aim of this paper is to define a class of implicative expansions of FDE in whose elements Boolean negation is definable, whence strong logics such as the paraconsistent and paracomplete logic PŁ4 and BN4 itself are definable, in addition to classical propositional logic.

Logics of involutive Stone algebras

An involutive Stone algebra (IS-algebra) is simultaneously a De Morgan algebra and a Stone algebra (i.e., a pseudo-complemented distributive lattice satisfying the Stone identity \(\mathop {\sim }x \vee \mathop {\sim }\mathop {\sim }x \approx 1\)). IS-algebras have been studied algebraically and topologically since the 1980s, but a corresponding logic (here denoted \(\mathcal {IS}_{\le }\)) has been introduced only very recently. This logic is the departing point of the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that \(\mathcal {IS}_{\le }\) is a conservative expansion of the Belnap-Dunn four-valued logic (i.e., the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic (i.e., every strengthening of the Belnap-Dunn logic) so as to obtain an extension of \(\mathcal {IS}_{\le }\). We show that every logic thus defined can be axiomatized by adding a fixed finite set of multiple-conclusion rule schemata to the corresponding super-Belnap base logic. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of \(\mathcal {IS}_{\le }\). In fact, as in the super-Belnap case, we establish that the finitary extensions of \(\mathcal {IS}_{\le }\) are already uncountably many. When the base super-Belnap logic possesses a disjunction, we show that we can reduce the multiple-conclusion calculus to a traditional one; some of the multiple-conclusion axiomatizations so introduced are analytic and are thus of independent interest from a proof-theoretic standpoint. We also consider a few extensions of \(\mathcal {IS}_{\le }\) that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods.

Probabilities with Gaps and Gluts

Belnap-Dunn logic (BD), sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes’ and Jeffrey updating and also suggest mechanisms for aggregating information from different sources.

THE LATTICE OF SUPER-BELNAP LOGICS

AbstractWe study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.

De Morgan clones and four-valued logics

We study clones on a four-element set related to the clone \({\mathsf {DMA}}\) of all term functions of the subdirectly irreducible four-element De Morgan algebra \({\mathbf {DM}}_{{{\mathbf {4}}}}\). We find generating sets for the clones of all functions preserving the subalgebras of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the automorphisms of \({\mathbf {DM}}_{{{\mathbf {4}}}}\), the truth order and the information order on \({\mathbf {DM}}_{{{\mathbf {4}}}}\), as well as clones defined by conjunctions of these conditions. We identify the covers of \({\mathsf {DMA}}\) in the lattice of four-valued clones and describe the lattice of clones above \({\mathsf {DMA}}\) which contain the discriminator function. Finally, observing that each clone above \({\mathsf {DMA}}\) defines an expansion of the four-valued Belnap–Dunn logic, we classify these clones by their metalogical properties, specifically by their position within the Leibniz and Frege hierarchies of abstract algebraic logic.

Modal and Intuitionistic Variants of Extended Belnap–Dunn Logic with Classical Negation

In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm.

Four-Valued Logics of Truth, Nonfalsity, Exact Truth, and Material Equivalence

The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.

Measuring evidence: a probabilistic approach to an extension of Belnap–Dunn logic

This paper introduces the logic of evidence and truth \({ LET}_{F}\) as an extension of the Belnap–Dunn four-valued logic \({ FDE}\). \({ LET}_{F}\) is a slightly modified version of the logic \({ LET}_{J}\), presented in Carnielli and Rodrigues (Synthese 196:3789–3813, 2017). While \({ LET}_{J}\) is equipped only with a classicality operator \({\circ }\), \({ LET}_{F}\) is equipped with a non-classicality operator \({\bullet }\) as well, dual to \({\circ }\). Both \({ LET}_{F}\) and \({ LET}_{J}\) are logics of formal inconsistency and undeterminedness in which the operator \( \circ \) recovers classical logic for propositions in its scope. Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition \( \alpha \) even if \( \alpha \) is not true. As well as \({ LET}_{J}\), \({ LET}_{F}\) is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for \({ LET}_{F}\) where statements \(P(\alpha )\) and \(P({\circ }\alpha )\) express, respectively, the amount of evidence available for \(\alpha \) and the degree to which the evidence for \( \alpha \) is expected to behave classically—or non-classically for \( P({\bullet }\alpha ) \). A probabilistic scenario is paracomplete when \(P(\alpha ) + P(\lnot \alpha ) < 1\), and paraconsistent when \(P(\alpha ) + P(\lnot \alpha ) > 1\), and in both cases, \(P(\circ \alpha ) < 1\). If \(P(\circ \alpha ) = 1\), or \(P({\bullet }\alpha ) = 0\), classical probability is recovered for \( \alpha \). The proposition \( {\circ }\alpha \vee {\bullet }\alpha \), a theorem of \({ LET}_{F}\), partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory.

Polarity Semantics for Negation as a Modal Operator

The minimal weakening $${{\textsf {N}}}_0$$ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of $${{\textsf {N}}}_0$$ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of $${{\textsf {N}}}_0$$ . These logics have the finite model property and they are decidable.

A capacity-based framework encompassing Belnap–Dunn logic for reasoning about multisource information

Belnap–Dunn four-valued logic is one of the best known logics for handling elementary information items coming from several sources. More recently, a conceptually simple framework, namely a two-tiered propositional logic augmented with classical modal axioms (here called BC logic), was suggested by the second author and colleagues, for the handling of multisource information. It is a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set functions called Boolean capacities. We show BC logic is more expressive than Belnap–Dunn logic by proposing a consequence-preserving translation of Belnap–Dunn logic in this setting. As special cases, we can recover already studied translations of three-valued logics such as Kleene and Priest logics. Moreover, BC logic is compared with the source-processor logic of Avron, Ben Naim and Konikowska. Our translation bridges the gap between Belnap–Dunn logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various methods for handling inconsistency due to several conflicting sources of information.

Gentzen-Type Sequent Calculi for Extended Belnap–Dunn Logics with Classical Negation: A General Framework

Gentzen-type sequent calculi GBD+, GBDe, GBD1, and GBD2 are respectively introduced for De and Omori’s axiomatic extensions BD+, BDe, BD1, and BD2 of Belnap–Dunn logic by adding classical negation. These calculi are constructed based on a small modification of the original characteristic axiom scheme for negated implication. Theorems for syntactically and semantically embedding these calculi into a Gentzen-type sequent calculus LK for classical logic are proved. The cut-elimination, decidability, and completeness theorems for these calculi are obtained using these embedding theorems. Similar results excluding cut-elimination results are also obtained for alternative Gentzen-type sequent calculi gBD+, gBDe, gBD1, and gBD2 for BD+, BDe, BD1, and BD2, respectively. These alternative calculi are constructed based on the original characteristic axiom scheme for negated implication.

Countably Many Weakenings of Belnap–Dunn Logic

Every Berman’s variety $$\mathbb {K}_p^q$$ which is the subvariety of Ockham algebras defined by the equation $${\sim ^{2p+q}}a = {\sim ^q}a$$ ($$p\ge 1$$ and $$q\ge 0$$) determines a finitary substitution invariant consequence relation $$\vdash _p^q$$. A sequent system $$\mathsf {S}_p^q$$ is introduced as an axiomatization of the consequence relation $$\vdash _p^q$$. The system $$\mathsf {S}_p^q$$ is characterized by a single finite frame $$\mathfrak {F}_p^q$$ under the frame semantics given for the formal language. By the duality between frames and algebras, $$\mathsf {S}_p^q$$ can be viewed as a $$4^{2p+q}$$-valued logic as it is characterized by a distributive lattice of $$4^{2p+q}$$ elements with a unary operator. Moreover, a structural-rule-free, cut-free and terminating sequent system $$\mathsf {G}_p^q$$ is established for $$\vdash _p^q$$. The Craig interpolation property of $$\vdash _p^q$$ is shown proof-theoretically utilizing $$\mathsf {G}_p^q$$.

An Algebraic View of Super-Belnap Logics

The Belnap–Dunn logic (also known as First Degree Entailment, or FDE) is a well-known and well-studied four-valued logic, but until recently little has been known about its extensions, i.e. stronger logics in the same language, called super-Belnap logics here. We give an overview of several results on these logics which have been proved in recent works by Přenosil and Rivieccio. We present Hilbert-style axiomatizations, describe reduced matrix models, and give a description of the lattice of super-Belnap logics and its connections with graph theory. We adopt the point of view of Abstract Algebraic Logic, exploring applications of the general theory of algebraization of logics to the super-Belnap family. In this respect we establish a number of new results, including a description of the algebraic counterparts, Leibniz filters, and strong versions of super-Belnap logics, as well as the classification of these logics within the Leibniz and Frege hierarchies.

Rivals to Belnap–Dunn Logic on Interlaced Trilattices

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic \(\mathsf {E}_{\mathtt {fde}}\). This correspondence has been interpreted as evidence that \(\mathsf {E}_{\mathtt {fde}}\) is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which \(\mathsf {E}_{\mathtt {fde}}\) is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting’s “cut-down” connectives—propositional connectives that “cut down” available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch’s four-valued \(\mathsf {S}_{\mathtt {fde}}\) and Richard Angell’s \(\mathsf {AC}\)—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic.

A Gentzen Calculus for Nothing but the Truth

In their paper Nothing but the Truth Andreas Pietz and Umberto Rivieccio present Exactly True Logic (ETL), an interesting variation upon the four-valued logic for first-degree entailment FDE that was given by Belnap and Dunn in the 1970s. Pietz & Rivieccio provide this logic with a Hilbert-style axiomatisation and write that finding a nice sequent calculus for the logic will presumably not be easy. But a sequent calculus can be given and in this paper we will show that a calculus for the Belnap-Dunn logic we have defined earlier can in fact be reused for the purpose of characterising ETL, provided a small alteration is made—initial assignments of signs to the sentences of a sequent to be proved must be different from those used for characterising FDE. While Pietz & Rivieccio define ETL on the language of classical propositional logic we also study its consequence relation on an extension of this language that is functionally complete for the underlying four truth values. On this extension the calculus gets a multiple-tree character—two proof trees may be needed to establish one proof.

The B&lt;sub&gt;4&lt;/sub&gt;-Modalized Propositional Logic

A modalized propositional Belnap-Dunn logic will be proposed in this paper which thereare four modalities [t]; [T]; [⊥]; [f] to represent the four values t;T;⊥; f; respectively, and a Gentzentypeddeduction system will be given so that the the system is sound and complete with the four-valuedsemantics of the Belnap-Dunn logic.