First-degree Entailment Research Articles

Variable Domain First-Order First-Degree Entailment and Some of Its Children

Variable Domain First-Order First-Degree Entailment and Some of Its Children

A note on negation inconsistent variants of FDE-negation

H. Omori and H. Wansing introduced in a recent paper possible alternatives for the negation of the logic of first-degree entailment. One of their observations with regard to these alternative negations is that some of them turn out to induce negation inconsistency, meaning that some contradictions become provable (under an arbitrary premise) when used in place of the original negation. Omori and Wansing also considered a non-deterministic generalisation of such operators, but it was left open whether the generalised negation similarly induces negation inconsistency. In this paper, we provide an answer to this question in the positive, and moreover look into further generalisation and characterisation of non-deterministic operators which satisfy the formal criteria of negation inconsistency and its pair notion of negation incompleteness in the setting of Omori and Wansing.

First-Degree Entailment and Truthmaker Functions

First-Degree Entailment and Truthmaker Functions

On Garfield and Priest’s interpretation of the use of the catuskoti in Mūlamadhyamakakārikā

ABSTRACT According to Garfield and Priest’s interpretation, the positive use of the catuskoti by Nāgārjuna in Mūlamadhyamakakārikā (MMK) shows that he endorses a four-valued semantics similar to that of Belnap’s First-Degree Entailment (FDE), while the negative use of the catuskoti by Nāgārjuna in MMK indicates that what he really has in mind is a plurivalent five-valued semantics. This paper argues that their interpretation suffers from a number of problems: adequate logic, collapse of kotis, lack of literature support, and a suitable explanation of the third koti. The final part of the paper combines insights from Westerhoff, Cotnoir, Garfield and Priest together and describes the right interpretation of Nāgārjuna’s use of the catuskoti in MMK with the right semantics and the right logic of MMK. The authors argue that Nāgārjuna’s ‘rebuttal’ of all four kotis in a catuskoti should be understood as a mere denial or a mere rejection of these kotis.

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 $.

Two 4-valued implicative expansions of first-degree entailment logic: The relevant logic BN4VSP and the (relevant) entailment logic BN4AP

AbstractA logic L has the ‘variable-sharing property’ (VSP) if in all L-theorems of the form $A\rightarrow B$, $A$ and $B$ share at least a propositional variable. A logic L has the ‘Ackermann property’ (AP) if in all L-theorems of the form $A\rightarrow (B\rightarrow C)$, $A$ contains at least a conditional connective ($\rightarrow $). Anderson and Belnap consider the VSP a necessary property of any relevant logic, and both the VSP and the AP necessary properties of any (relevant) entailment logic. Now, among relevant logicians, Brady’s logic BN4 is widely viewed as the adequate 4-valued implicative logic. But BN4 lacks the VSP and the AP. The aim of this paper is to define the logics BN4$^{\text {VSP}}$ and BN4$^{\text {AP}}$. The former one has the VSP, whereas the latter one has the VSP and the AP. Moreover, both logics have some properties that do not support their consideration as mere artificial constructs.

Non-contingency in a paraconsistent setting

Abstract We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator $\blacktriangle \phi $ which is construed as ‘$\phi $ has the same value in all accessible states’ or ‘all sources give the same information on the truth value of $\phi $’. We equip this logic dubbed $\textbf {K}^\blacktriangle _{\textbf {FDE}}$ with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the $\blacktriangle $ operator modelling search for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that $\blacktriangle $ is not definable via the necessity modality $\Box $ of $\textbf {K}_{\textbf{FDE}}$. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, $\textbf {S4}$ and $\textbf {S5}$ (among others) frames are definable.

Negation, Truth and Falsity

The present article includes thematically connected questions on negation in the areas of the philosophy of logic, philosophical logic and metaphysics. It highlights a central problem negation poses to a certain area of philosophy. The purpose of the article is also to analyze negation in the areas of the philosophy of logic, philosophical logic and metaphysics. It highlights a central problem negation poses to a certain area of philosophy. It includes historical references, puts the problems discussed into context, and ties together a number of the main themes and problems. The investigation discusses the problem of logical revisionism and whether there is any room for genuine disagreement, and hence shared meaning, between the classicist and deviant’s respective uses of ‘not’. If there is not, revision is impossible. We argue that revision is indeed possible and provide an account of negation as contradictoriness according to which a number of alleged negations are declared genuine. Among them are the negations of FDE (First-Degree Entailment) and a wide family of other relevant logics, LP (Priest’s dialetheic “Logic of Paradox”), Kleene weak and strong 3-valued logics with either “exclusion” or “choice” negation, and intuitionistic logic. We also discuss the problem of furnishing intuitionistic logic with an empirical negation for adequately expressing claims of the form ‘A is undecided at present’ or ‘A may never be decided’ the latter of which has been argued to be intuitionistically inconsistent. We reveal the importance of various notions of consequence-as-s-preservation where s may be falsity (versus untruth), indeterminacy or some other semantic (or “algebraic”) value, in formulating rationality constraints on speech acts and propositional attitudes such as rejection, denial and dubitability. The article provides an account of the nature of truth values regarded as objects. It is argued that only truth exists as the maximal truthmaker. The consequences this has for semantics representationally construed are considered and it is argued that every logic, from classical to non-classical, is gappy. Moreover, a truthmaker theory is developed whereby only positive truths, an account of which is also developed therein, have truthmakers. We investigate the definability of negation as “absolute” impossibility, i.e. where the notion of necessity or possibility in question corresponds to the global modality.

Minimally Nonstandard K3 and FDE

Graham Priest has formulated the minimally inconsistent logic of paradox (MiLP), which is paraconsistent like Priest’s logic of paradox (LP), while staying closer to classical logic. We present logics that stand to (the propositional fragments of) strong Kleene logic (K3) and the logic of first-degree entailment (FDE) as MiLP stands to LP. That is, our logics share the paracomplete and the paraconsistent-cum-paracomplete nature of K3 and FDE, respectively, while keeping these features to a minimum in order to stay closer to classical logic. We give semantic and sequent-calculus formulations of these logics, and we highlight some reasons why these logics may be interesting in their own right.

Two-sided Sequent Calculi for FDE-like Four-valued Logics

We present a method that generates two-sided sequent calculi for four-valued logics like first degree entailment (FDE). (We say that a logic is FDE-like if it has finitely many operators of finite arity, including negation, and if all of its operators are truth-functional over the four truth-values ‘none’, ‘false’, ‘true’, and ‘both’, where ‘true’ and ‘both’ are designated.) First, we show that for every n-ary operator ⋆ every truth table entry f⋆(x1,…,xn) = y can be characterized in terms of a pair of sequent rules. Secondly, we use these sequent rules to build sequent calculi and prove their completeness. With the help of two simplification procedures we then show that the 2 ⋅ 4n sequent rules that characterize an n-ary operator can be systematically reduced to at most four sequent rules. Thirdly, we use our method to investigate the proof-theoretical consequences of including intuitive truth-functional implications in FDE-like logics.

Hybrid Deduction-Refutation Systems for FDE-Based Logics

Hybrid deduction-refuation systems are presented for four first-degree entailment based logics. The hybrid systems are shown to be deductively and refutationally sound with respect to their logics. The proofs of completeness are presented in a uniform way. The paper builds on work by Goranko, who presented a deductively and refutationally sound and complete hybrid system for classical logic.

Hilbert-style axiomatization of first-degree entailment and a family of its extensions

This paper presents a Hilbert-style system for the logic of first-degree entailment defined in a Fmla-Fmla framework. The effective use of this formulation as a basis for a whole family of systems extending the logic of first-degree entailment in various directions is shown. By systematizing this family, some new systems are uncovered, and some other well-established logics (such as the first-degree entailment fragment of Priest's Logic of Paradox) obtain new axiomatization. Semantics for the key systems from the family is formulated.

The (Greatest) Fragment of Classical Logic that Respects the Variable-Sharing Principle (in the FMLA-FMLA Framework)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics is defined in terms of a \(p\)-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints.

NONCLASSICAL TRUTH WITH CLASSICAL STRENGTH. A PROOF-THEORETIC ANALYSIS OF COMPOSITIONAL TRUTH OVER HYPE

AbstractQuestions concerning the proof-theoretic strength of classical versus nonclassical theories of truth have received some attention recently. A particularly convenient case study concerns classical and nonclassical axiomatizations of fixed-point semantics. It is known that nonclassical axiomatizations in four- or three-valued logics are substantially weaker than their classical counterparts. In this paper we consider the addition of a suitable conditional to First-Degree Entailment—a logic recently studied by Hannes Leitgeb under the label HYPE. We show in particular that, by formulating the theory PKF over HYPE, one obtains a theory that is sound with respect to fixed-point models, while being proof-theoretically on a par with its classical counterpart KF. Moreover, we establish that also its schematic extension—in the sense of Feferman—is as strong as the schematic extension of KF, thus matching the strength of predicative analysis.

God, Gluts and Gaps: Examining an Islamic Traditionalist Case for a Contradictory Theology

In this paper, I examine the deep theological faultline generated by divergent understandings of the divine attributes (ṣifāt) among two early antagonistic Muslim groups – the traditionalists (mainly Ḥanbalites) and the scholastic rationalists (mutakallimūn). I attempt to defend the traditionalist stance of literally accepting on the one hand physical qualities and actions described about God and on the other transcendent descriptions about Him by framing it as an exceptional contradiction to be embraced. I then argue using the paraconsistent logical system of First Degree Entailment (FDE) as expounded by philosopher and logician Jc Beall that accepting such a contradiction regarding any number of predicates about God have no logical impediments. Hence, the traditionalist stance (at least) cannot be accused of being incoherent. I will use as a case-study the discussion over the divine attribute of speech (kalām). Finally, I respond to some criticisms against such a proposal and offer some concluding remarks on the wider implications this proposal of a contradictory or glut-theoretic model would have on Muslim theology in general.

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.

A Note on FDE “All the Way Up”

A very natural and philosophically important subclassical logic is FDE (for first-degree entailment). This account of logical consequence can be seen as going beyond the standard two-valued account (of “just true” and “just false”) to a four-valued account (adding the additional values of “both true and false” and “neither true nor false”). A natural question arises: What account of logical consequence arises from considering further (positive) combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different (more algebraic) context. In this note we generalize Priest’s (and indirectly Shramko and Wansing’s) result to show that even if one considers ordinal-many (positive) combinations of the previous values, for any ordinal, the resulting consequence relation (the resulting logic) remains FDE.

Exactly true and non-falsity logics meeting infectious ones

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding ‘exactly true’ and ‘non-falsity’ versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125–135; Shramko (2019). Dual-Belnap logic and anything but falsehood. Journal of Logics and their Applications, 6, 413–433; Shramko et al. (2017). First-degree entailment and its relatives. Studia Logica, 105(6), 1291–1317] to the case of infectious logics, namely to the case of Deutsch's logic introduced in Deutsch [Relevant analytic entailment. The Relevance Logic Newsletter, 2(1), 26–44; The completeness of S. Studia Logica, 38(2), 137–147]. The particular systems obtained in this way are and . We present them in the form of sequent calculi and prove corresponding soundness and completeness theorems. We illuminate the connection between , and two well-known systems, strong Kleene three-valued logic and Priest's Logic of Paradox . This connection allows us to investigate the characterisation of the entailment relations associated with and as well as to introduce the notion of ‘infectious analogue’ of a certain logic. We also study implicative extensions of and and prove soundness and completeness theorems for them as well.

Correspondence analysis and automated proof-searching for first degree entailment

In this paper, we present correspondence analysis for the well-known four-valued logic First Degree Entailment (FDE). Correspondence analysis is Kooi and Tamminga’s technique for finding adequate natural deduction systems for all the truth-functional unary and binary extensions of an arbitrary functionally incomplete many-valued logic. In particular, Kooi and Tamminga initially presented correspondence analysis for Asenjo–Priest’s three-valued the Logic of Paradox. Generally speaking, a tabular functionally incomplete many-valued logic is added with an arbitrary unary or binary connective $$ \circ $$ following the truth-table definition of $$ \circ $$ . As a result, a tremendous amount of logics obtains a sound and complete natural deduction system in one go. In this paper, we generalize its application proposed by Kooi and Tamminga for the unary extensions of FDE and obtain correspondence analysis for the binary extensions of the logic in question. On the other hand, we use a proof-searching algorithm to have been successfully applied to classical and a variety of non-classical logics and present a finite, sound, and complete proof-searching algorithm for natural deduction systems for the binary extensions of FDE obtained via correspondence analysis. In the end, a comparative study of this method and the method presented by Avron and his collaborators is given.

On Definability of Connectives and Modal Logics over FDE

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK □ ×BK □ , which underlines the central role played by BK among FDE-based modal logics.