In this paper we study 12 four-valued logics arisen from Belnap's truth and/or knowledge four-valued lattices, with or without constants, by adding one or both or none of two new non-regular operations—classical negation and natural implication. We prove that the secondary connectives of the bilattice four-valued logic with bilattice constants are exactly the regular four-valued operations. Moreover, we prove that its expansion by any non-regular connective (such as, e.g., classical negation or (...) natural implication) is strictly functionally complete. Further, finding axiomatizations of the quasi varieties generated by the 12 logics involved (that prove to be varieties), we find naturell equational axiomatizations of these logics. Finally, applying Pynko's general theory of algebraizable sequential consequence operations, we also find equivalent natural sequentiell axiomatizations of the logics under consideration that expand either of two Pynko's sequential calculi for the constant-free truth-lattice four-valued logic. (shrink)

The paper discusses functional properties of some four-valued logics which are the expansions of four-valued Belnap’s logic DM4. At first, we consider the logics with two designated values, and then logics defined by matrices having the same underlying algebra, but with a different choice of designated values, i.e. with one designated value. In the preceding literature both approaches were developed independently. Moreover, we present the lattices of the functional expansions of DM4.

Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is de_ned by a unique conjunctive matrix ℳ4 with exactly two distinguished values over an expansion.

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De (...) Morgan's laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

We investigate some non-normal variants of well-studied paraconsistent and paracomplete modal logics that are based on N. Belnap’s and M. Dunn’s four-valued logic. Our basic non-normal modal logics are characterized by a weak extensionality rule, which reflects the four-valued nature of underlying logics. Aside from introducing our basic framework of bi-neighbourhood semantics, we develop a correspondence theory in order to prove completeness results with respect to our neighbourhood semantics for non-normal variants of \, \ (...) and \. (shrink)

Kleene’s strong three-valued logic extends naturally to a four-valued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it four-valued analogs of Kleene’s weak three-valued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for (...) bilinear bilattices the extensions do not produce any new equivalences. (shrink)

The main aim of this paper is to introduce the logics of evidence and truth $$LET_{K}^+$$ and $$LET_{F}^+$$ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics $$LET_{K}$$ and $$LET_{F}^-$$ with rules of propagation of classicality, which are inferences that express how the classicality operator $${\circ }$$ is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with (...) 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for $$LET_{K}$$ is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for $$LET_{K}^+$$ is then obtained by imposing restrictions on the semantics of $$LET_{K}$$. These restrictions correspond exactly to the rules of propagation of classicality that extend $$LET_{K}$$. The logic $$LET_{F}^+$$ is obtained as the implication-free fragment of $$LET_{K}^+$$. We also show that the 6 values of $$LET_{K}^+$$ and $$LET_{F}^+$$ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that $$LET_{K}^+$$ is Blok-Pigozzi algebraizable and that its implication-free fragment $$LET_{F}^+$$ coincides with the degree-preserving logic of the involutive Stone algebras. (shrink)

In this paper we introduce a Gentzen calculus for (a functionally complete variant of) Belnap's logic in which establishing the provability of a sequent in general requires \emph{two} proof trees, one establishing that whenever all premises are true some conclusion is true and one that guarantees the falsity of at least one premise if all conclusions are false. The calculus can also be put to use in proving that one statement \emph{necessarily approximates} another, where necessary approximation is a natural (...) dual of entailment. The calculus, and its tableau variant, not only capture the classical connectives, but also the `information' connectives of four-valued Belnap logics. This answers a question by Avron. (shrink)

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to special (...) kinds of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification. (shrink)

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of (...) Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. (shrink)

In Belnap's useful 4-valued logic, the set 2 = {T, F} of classical truth values is generalized to the set 4 = ������(2) = {Ø, {T}, {F}, {T, F}}. In the present paper, we argue in favor of extending this process to the set 16 = ᵍ (4) (and beyond). It turns out that this generalization is well-motivated and leads from the bilattice FOUR₂ with an information and a truth-and-falsity ordering to another algebraic structure, namely the trilattice (...) SIXTEEN₃ with an information ordering together with a truth ordering and a (distinct) falsity ordering. Interestingly, the logics generated separately by the algebraic operations under the truth order and under the falsity order in SIXTEEN₃ coincide with the logic of FOUR₂, namely first degree entailment. This observation may be taken as a further indication of the significance of first degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. We semantically define the logics of the two orderings in the extended language and in both cases axiomatize a certain fragment comprising three unary operations: a negation, an involution, and their combination. We also suggest two other definitions of logics in the full language, including a bi-consequence system. In other words, in addition to presenting first degree entailment as a useful 16-valued logic, we define further useful 16-valued logics for reasoning about truth and (non-)falsity. We expect these logics to be an interesting and useful instrument in information processing, especially when we deal with a net of hierarchically interconnected computers. We also briefly discuss Arieli's and Avron's notion of a logical bilattice and state a number of open problems for future research. (shrink)

ukasiewicz''s four-valued modal logic is surveyed and analyzed, together with ukasiewicz''s motivations to develop it. A faithful interpretation of it in classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed in the light of the presented results, ukasiewicz''s own texts, and related literature.

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN 3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are (...) a — incoming data is asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT 4 . Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a –order (truth order) and d –order (falsity order) coincide with first-degree entailment. Finally logic with two kinds of operations ( a –connectives and d –connectives) and consequence relation defined via a –ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

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. 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. (shrink)

In their useful logic for a computer network Shramko and Wansing generalize initial values of Belnap’s 4-valued logic to the set 16 to be the power-set of Belnap’s 4. This generalization results in a very specific algebraic structure — the trilattice SIXTEEN3 with three orderings: information, truth and falsity. In this paper, a slightly different way of generalization is presented. As a base for further generalization a set 3 is chosen, where initial values are a (...) — incoming data is asserted, d — incoming data is denied, and u — incoming data is neither asserted nor denied, that corresponds to the answer “don’t know”. In so doing, the power-set of 3, that is the set 8 is considered. It turns out that there are not three but four orderings naturally defined on the set 8 that form the tetralattice EIGHT4. Besides three ordering relations mentioned above it is an extra uncertainty ordering. Quite predictably, the logics generated by a–order and d–order coincide with first-degree entailment. Finally logic with two kinds of operations and consequence relation defined via a–ordering is considered. An adequate axiomatization for this logic is proposed. (shrink)

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.

Formal aspects of various ways of description of Jan Łukasiewicz’s four-valued modal logic £ are discussed. The original Łukasiewicz’s description by means of the accepted and rejected theorems, together with the four-valued matrix, is presented. Then the improved E.J. Lemmon’s description based upon three specific axioms, together with the relational semantics, is presented as well. It is proved that Lemmon’s axiomatic is not independent: one axiom is derivable on the base of the remanent two. Several (...) axiomatizations, based on three, two or one single axiom are provided and discussed, including S. Kripke’s axiomatics. It is claimed that (a) all substitutions of classical theorems, (b) the rule of modus ponens, (c) the definition of “◊” and (d) the single specific axiom schema: ⬜A ∧ B → A ∧ ⬜B, called the jumping necessity axiom, constitute an elegant axiomatics of the system £. (shrink)

A deterministic weakening \ of the Belnap–Dunn four-valued logic \ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \ is formalized as a sequent calculus. The completeness and decidability of \ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \ is described, and found to be (...) a subvariety of Berman’s variety \. Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \ is the logic of a nine-element distributive lattice with a negation. Moreover, \ is embedded into \ by Glivenko’s double-negation translation. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by (...) a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior (...) of the Truth and the False, which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki. (shrink)

A survey is given of three-valued paraconsistent propositionallogics connected with Jaśkowski’s criterion for constructing paraconsistentlogics. Several problems are raised and four new matrix three-valued paraconsistent logics are suggested.

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 and the other for falsification. 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. (shrink)

ABSTRACT In this paper we introduce a set of six logical values, arising in the application of three-valued logics to time intervals, find its algebraic structure, and use it to define a six-valued logic. We then prove, by using algebraic properties of the class of De Morgan algebras, that this semantically defined logic can be axiomatized as Belnap's ?useful? four-valued logic. Other directions of research suggested by the construction of this set of six (...) logical values are described. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new log- ics and prove a completeness result for the first and last element of the chain stating that both logics are determined (...) by a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

The logic BN4 was defined by R.T. Brady as a four-valued extension of Routley and Meyer’s basic logic B. The system EF4 is defined as a companion to BN4 to represent the four-valued system of implication. The system Ł was defined by J. Łukasiewicz and it is a four-valued modal logic that validates what is known as strong Łukasiewicz-type modal paradoxes. The systems EF4-M and EF4-Ł are defined as alternatives to Ł (...) without modal paradoxes. This paper aims to define a Belnap-Dunn semantics for EF4, EF4-M and EF4-Ł. It is shown that EF4, EF4-M and EF4-Ł are strongly sound and complete w.r.t. their respective semantics and that EF4-M and EF4-Ł are free from strong Łukasiewicz-type modal paradoxes. (shrink)

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic \. This correspondence has been interpreted as evidence that \ is “the” logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which \ 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 \ and Richard Angell’s \—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic. (shrink)

This study aims to introduce a modal extension M4CC of Arieli, Avron, and Zamansky's ideal paraconsistent four-valued logic 4CC as a Gentzen-type sequent calculus and prove the Kripke-completeness and cut-elimination theorems for M4CC. The logic M4CC is also shown to be decidable and embeddable into the normal modal logic S4. Furthermore, a subsystem of M4CC, which has some characteristic properties that do not hold for M4CC, is introduced and the Kripke-completeness and cut-elimination theorems for this (...) subsystem are proved. This subsystem is also shown to be decidable and embeddable into S4. (shrink)

The classical logician's principal dictum, «A proposition is either true or false, not neither, and not both,» still leaves considerable room for multi-valued logic.

A curious feature of Belnap’s “useful four-valued logic”, also known as first-degree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic.

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets (...) of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox. (shrink)

Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be developed in this framework.

The trilattice SIXTEEN₃ is a natural generalization of the wellknown bilattice FOUR₂. Cut-free, sound and complete sequent calculi for truth entailment and falsity entailment in SIXTEEN₃, are presented.

This paper sets out two semantics for the relevant logic R based on Dunn's four-valued semantics for first-degree entailments. Unlike Routley's semantics for weak relevant logics, they do not use two ternary accessibility relations. Unlike Restall's semantics, they capture all of R. But there is a catch. Both of the present semantics are neighbourhood semantics, that is, they include sets of propositions in the specification of their frames.

Here, I first prove that certain families of k-valued clones have the Gupta-Belnap fixed-point property. This essentially means that all propositional languages that are interpreted with operators belonging to those clones are such that any net of self-referential sentences in the language can be consistently evaluated. I then focus on two four-valued generalisations of the Kleene propositional operators that generalise the strong and weak Kleene operators: Belnap’s clone and Fitting’s clone, respectively. I apply the theorems from (...) the initial part of the paper to analyse the fixed-point property of Belnap’s and Fitting’s clones when some special operators that reflect the semantics are added. The conclusion of the paper is that Fitting’s clone is better suited than Belnap’s to provide self-referential languages with highly expressive resources. (shrink)

As the subtitle indicates, this book intends to discuss Descartes’ attempt of laying a new foundation of knowledge. In a lively and critical interpretation of Descartes’ writings, especially of his Discours de la Méthode and of his Meditationes, and a competent use of the corresponding philosophical literature the success of this attempt of enlightenment and its shortcomings, identified with the Cartesian re-introduction of the traditional metaphysics, are explained in order to allow the author in a concluding discussion to present his (...) ideas of the necessary revision of Descartes philosophy. Presenting the I as the subject not only for the first time but in an unique way, and concentrating upon the problem of the self-originating human knowledge in an exemplary manner, Descartes is found to have become the founder of subjective philosophy. As knowledge of knowledge, philosophy is not concerned primarily with objects, but with knowledge and has self-foundation as its main task. So it has to start with human knowledge and to place the self as the fountain from which all knowledge proceeds. The demands of subjective philosophy dictate its method described in the Discours. The foundation of knowledge, which as such can only be true and certain knowledge, leads to enlightenment, i.e., to the rebellion against and to the great confrontation with the culture of the time, that is, with the guiding principles of understanding and evaluating the world of his society, and to the introduction of self-experience as the primary source of knowledge. Appointing human reason as the only leader of his intellectual life, Descartes determines the program and future order of his thoughts; the method, in the form of the four rules, has "correctly to direct his reason and to seek the truth in the sciences." It brings Descartes over the methodical doubt to the unquestionable and unsurpassed foundation of human knowledge, the absolute certitude of the self in the Cogito. His analysis of the poor, unprotected I, related to itself knowingly, is unique and of priceless philosophical value. However, the further stations of the philosophical way of Descartes who, unable to stand the ontological loneliness of this philosophically determined self-awareness, sacrificed the idea of scientific progress to a rational theology, cannot be followed. The claim of the metaphysician to be able to discover or, at least, to test every humanly possible knowledge in terms of the absolute knowledge, disintegrates in view of the progress of the new sciences. Not eternal, necessary truth, but the finite human I is the origin of subjective philosophy. The stability of rational relations does not lie in these relations themselves as if they were existing for themselves, but in the strict consequence of logical reason. "'God’ is... the title for the transcendental-philosophical background of consciousness, for the conditions to which the I-think is always already subjected." All contents which allegedly are already given in the infinite knowledge and can always be deduced from it, have to be produced and developed out of the first human knowledge by way of scientific research. Science which received its new foundation in subjective philosophy and is sought in the efforts of communication of the scientists, is not characterized by truth, but by certitude. "'Truth’ is the concern of metaphysics or better of religion. It is of the essence of such truth, not to be capable of demonstration. The absolutely true has no need to prove itself; it is entitled to faith." Differences between metaphysics and religion, and the nature and possible justification of such metaphysical and religious faith are regrettably not discussed.—M. S. (shrink)

In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic known as 4CC. These theorems are proved using Schütte’s method, which can simultaneously prove completeness and cut-elimination.

Bilattices, due to M. Ginsberg, are a family of truth value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnap’s four-valued logic, based on classical two-valued logic. Among other examples are those based on finite many-valued logics, and on probabilistic valued logic. A fixed point semantics is developed for logic programming, allowing any bilattice as the space of truth values. The mathematics is little more complex (...) than in the classical two-valued setting, but the result provides a natural semantics for distributed logic programs, including those involving confidence factors. The classical two-valued and the Kripke/Kleene three-valued semantics become special cases, since the logics involved are natural sublogics of Belnap’s logic, the logic given by the simplest bilattice. (shrink)

This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic (...) examples are the local and the global consequences associated with a normal modal logic, and the logics preserving degrees of truth and preserving truth associated with certain substructural and many-valued logics. For protoalgebraic logics the results in the paper coincide with those obtained by two of the authors in 2001, so the main novelty of the approach is its suitability for all kinds of logics. The paper also studies three kinds of definability of the Leibniz filters, and their consequences for the determination of the strong version. In a second part of the paper several case studies are developed, comprising positive modal logic, Dunn–Belnap’s four-valued logic, the large family of substructural logics, and some relevance logics. (shrink)

The logic E4 is related to Brady’s BN4 in a similar way to which Anderson and Belnap’s logic of entailment E is related to their logic of the relevant implication R. In ‘A companion to Brady’s 4-valued relevant logic: the 4-valued logic of entailment E4’, quoted in this paper, three alternatives to BN4 and another three to E4 are summarily introduced in a couple of pages as the only alternatives containing Routley and (...) Meyer’s basic logic B, provided some conditions are fulfilled. The aim of this note is to prove BN4 and its three alternatives are the same logic up to some point, since they are functionally equivalent to each other, as it is the case with E4 and its three alternatives, which are also functionally equivalent to each other; to some extent, E4 is superior to BN4, since the latter is functionally included in the former, but not conversely. (shrink)

Bilattices, introduced by Ginsberg as a uniform framework for inference in artificial intelligence, are algebraic structures that proved useful in many fields. In recent years, Arieli and Avron developed a logical system based on a class of bilattice-based matrices, called logical bilattices, and provided a Gentzen-style calculus for it. This logic is essentially an expansion of the well-known Belnap–Dunn four-valued logic to the standard language of bilattices. Our aim is to study Arieli and Avron’s logic (...) from the perspective of abstract algebraic logic . We introduce a Hilbert-style axiomatization in order to investigate the properties of the algebraic models of this logic, proving that every formula can be reduced to an equivalent normal form and that our axiomatization is complete w.r.t. Arieli and Avron’s semantics. In this way, we are able to classify this logic according to the criteria of AAL. We show, for instance, that it is non-protoalgebraic and non-self-extensional. We also characterize its Tarski congruence and the class of algebraic reducts of its reduced generalized models, which in the general theory of AAL is usually taken to be the algebraic counterpart of a sentential logic. This class turns out to be the variety generated by the smallest non-trivial bilattice, which is strictly contained in the class of algebraic reducts of logical bilattices. On the other hand, we prove that the class of algebraic reducts of reduced models of our logic is strictly included in the class of algebraic reducts of its reduced generalized models. Another interesting result obtained is that, as happens with some implicationless fragments of well-known logics, we can associate with our logic a Gentzen calculus which is algebraizable in the sense of Rebagliato and Verdú . We also prove some purely algebraic results concerning bilattices, for instance that the variety of distributive bilattices is generated by the smallest non-trivial bilattice. This result is based on an improvement of a theorem by Avron stating that every bounded interlaced bilattice is isomorphic to a certain product of two bounded lattices. We generalize it to the case of unbounded interlaced bilattices. (shrink)

The four essays collected in this dossier are directed upon the contemporary understandings of blackness, as an ontology, a phenomenology, or a historicity. In the order of their presentation they encompass and situate what seems first to limit black being or overflow it, but which, when questioned, that is, disclosed, or unconcealed, does not fit into this logos, nor is ordered by it, even making what is most discernable about blackness in its past, future, or present, seem imaginary, moored (...) in a non-present,and presence, but in ways which make the non-being that marks blackness as sign, affect, or speaking being, unthinkable except as an appearing, a manifestation, an identity.As a dossier directed upon the present philosophical understanding of blackness, each paper seeks to unanchor blackness from the meanings and presuppositions that it normally calls forth. Beyond what is discernable, or ready-made, each paper addresses structures that make (and have made) blackness discernable as the always given to be seen, as meaningless speech, as pathology, as a concept beyond reason.The dossier opens with a question: what kind of disturbance does blackness present to being? Why does it seem to madden reason, or call forth frenzy? In Calvin Warren’s essay, “The Karen Call,” what comes into correlation with blackness are various kinds of emergency that structure white thought and presence but that belie witnessing and intelligibility, and in whose chain of significations can be discerned an ontological privilege that constitutes the world. To appear emergent, it seems, all blackness has to do is become one with the myth that bans or compromises it; a manifestation in which subject-object relations are already polarized by the dangers that deploy it in dramas of emergency. Once blackness becomes emergent, there is no escape, for it comes up against what Warren, reading Heidegger against the grain, suggests is a racial destiny of Being. To appear black is forthwith to resemble an es gibt that is indissociable from threat, falsehood, fantasy, and obliteration. To that extent, blackness attests to a kind of ontological emergency without surcease or hope, and one whose trace remains in every moment or glimpse; whose technical modality and structure always returns us to the call that is neither speech nor gaze, word nor symbol, but represents an extimacy that is undeconstructable, beyond utterance, meaning, or reparation.Such extimacy seems to induce a crisis in the philosophical notion of being itself. In Axelle Karera’s essay, “Paraontology,” the question asked is why blackness cannot be contained or fitted into ontology itself; or, if philosophy is ontology, why does blackness seem to signify an irreducible resistance to it?How does such resistance occur? If blackness can only be presented as “paraontology,” a word that seems to shelter ontology at the same time as it presents the search for a new order of being, what does the para teach us about blackness as disorder, resistance, or identity? What is its philosophical genealogy and affiliation, and why does the para itself seem to never get beyond its own conciliation, or filiation to philosophy, whose ordering and racist historicity thereby remains the seme and mark of the para as interval, trope, and index? Or, put more simply, can blackness ever resist its own resistance as trope, conjuncture, or representation? Retracing that genealogy through the work of Oskar Becker, Karera uncovers a surreptitious repetition in black uses of the word, in whose weave can be discerned a fascist interference that not only undermines the fugitivity of the resistance so claimed, but one that unwittingly repeats a fascist ontology of world, history, and being.1 It’s a startling, deeply disturbing claim. But the apparent echoing of the Heideggerian bell in black significations of ontology suggests that what is being reabsorbed here is not so much a radical difference but a concept that remains justified and in fee to, pardoned and annulled by, a white notion of being. What, then, can it mean to think blackness as para? Nothing more than a philosophical confusion that confuses l’être qui fuit, that is, the lack that Lacan says cannot be completed or resisted, for an appositional fantasy of displacement. Here everything that is deemed black has to be made meaningful in apposition to ontology, but everything that is black must also, it seems, always return apposition to the white para of ontology; to be black and appositional would, it seems, produce only the disavowal of the philosophical genealogy that always already determines it.One would be right to distrust such claims—the moralism of the refusal, and the scandal of the faux resistance by which the para hides behind its own whited-out screen.In the third contribution to the dossier, the question asked is what would it mean for blackness to repair itself, to let go or retrieve and so redeem its own historical conjuncture? How do we refer to an irreparable past, that is, a past that keeps on coming back, but one that has no sign or signified, no memory that can recapture or reveal it, or simply rejoin us to it without remainder? Rinaldo Walcott’s essay, “‘Retrospective Significance,’” revisits the reparations debate as a question of black being. Instead of repair, he finds desolation and emptiness, or what he calls a staggeringly incoherent rhetoric of coherency. It’s a paradox that addresses what it means to remember or to forget a nothing; if blackness is nothing, how does one reconnect with it, bond with it, compensate it in its historic devastation? What would it mean to receive meaning from such devastation, to express it if this nothingness is absolute, or absolutely valueless? If ontology is a structure, then blackness, Walcott suggests, expresses nothing less than a “slave-ness” that continues to confront and disturb us absolutely, irrevocably. As such slave-ness is the trace that remains as a sign of the irreparable. As a trace, and not a sign, slave-ness becomes the primordial signifier of being, nation, history, and diaspora. But it is what makes it irrecuperable that defines its value and difficulty for the very language of reparation. Walcott pursues this difficult, enigmatic thread through the work of black thinkers on diaspora and identity. He concludes: black being cannot be dissociated from the disturbance it offers value, conscience, and any thought of alterity.The final and fourth contribution returns to the question of blackness as the trace or thread (the monstration) of being. How does one follow the thread (montrer) of blackness if it is always already withdrawn from being? David Marriott’s essay, “Ontology and Lalangue,” evokes Lacan’s concept of lalangue in pursuit of this enigmatic equivocation between letter and signifier, philosophy and psychoanalysis. Showing how both discourses have sought to master this exorbitance as master and/or free it as slave, I show how blackness is extinguished by the very words used to precipitate it into being. The slavish audacity of this lalangue is then shown to disarticulate, or more exactly, withdraw from ontology, but only so that whiteness can be presented and proclaimed all the more as the un-representable truth and logic of Dasein. This way of excluding blackness while preserving whiteness is not only central to philosophical and psychoanalytic understandings of language, Marriott argues, but this way of manifesting itself without manifesting itself is also crucial to the racialization of the logos as such.What is essential here is the way a meaning that is beyond meaning or representation is inserted into a problematic of mastery and servitude that advances-conceals a racial genealogy but in ways that makes the being of blackness an enigma. As such, blackness becomes an exorbitance that is always effaced in its monstration. More, its revelation must necessarily be unrecognized as the very locus of knowledge and truth. In that sense, what would it mean to figure, or to speak, this n’est pas without the consolation of being, meaning, truth, or logic? What would it mean to represent it as situated or defined by the language of ontology? Is black speaking being always in a relationship to and simultaneously the manifestation of a lalangue of lalangue? (shrink)

This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic (...) examples are the local and the global consequences associated with a normal modal logic, and the logics preserving degrees of truth and preserving truth associated with certain substructural and many-valued logics. For protoalgebraic logics the results in the paper coincide with those obtained by two of the authors in 2001, so the main novelty of the approach is its suitability for all kinds of logics. The paper also studies three kinds of definability of the Leibniz filters, and their consequences for the determination of the strong version. In a second part of the paper several case studies are developed, comprising positive modal logic, Dunn–Belnap’s four-valued logic, the large family of substructural logics, and some relevance logics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Dynamic Encounters between Buddhism and the West ReportLaura Langone and Alexandra S. IlievaThe following is a summary of the 2021 Postgraduate Conference titled "Dynamic Encounters between Buddhism and the West," which took place online on June 28 and 29. The conference was conceptualized, organized, and run by three AHRC funded PhD students at the University of Cambridge: Laura Langone (Faculty of Modern and Medieval Languages); Alexandra S. Ilieva (Faculty (...) of Divinity); and Harry Harland (Faculty of Law). The conference took place over two days and featured eighteen presenters across six panels, and two keynote speakers. There were speakers from four continents (Europe, North America, Asia, and Oceania). A wide range of topics in multiple fields was discussed: religious studies, neuroscience, art history, politics, philosophy, literature, philology, mathematics, art, and music. Each day consisted of three panels, each with three presentations, one keynote presentation, and a networking session.The keynote speakers were Prof. Albrecht Classen (The University of Arizona) and Prof. Graham Parkes (University of Vienna).Prof. Classen opened the conference with his presentation. He taught us that we have been globalized and transcultural since at least the Middle Ages. To support this claim, he provided us with ample evidence of Buddhist-Western exchanges in Art and Literature dotted throughout the literature and architecture of Europe. Prof. Parkes observed that following Zen Master Dōgen's thought as well as Nietzsche's philosophy we can develop a saner approach to the environment, which provides us with resources to help us to tackle the challenges of the climate crisis.Our first panel was titled "Hellenistic-Buddhist Encounters," which followed on nicely from Prof. Classen's thesis that encounters between Buddhism and the West have been ubiquitous and have dated as far back as the Hellenistic period. Our second panel, "Encounters with the Contemporary World," were impressive examples of intercultural philosophy. Each participant demonstrated a unique ability to bring Buddhist ideas into dialogue with contemporary Western philosophy, specifically ethics, philosophy of mathematics, and epistemology. The third panel, titled "Therapeutic Encounters," taught us of the therapeutic value of Buddhist meditation and philosophical inquiry. As our presenters emphasized, this therapeutic value is not just an ancient artifact to be studied anthropologically, but a relevant and applicable approach to the modern idea of well-being that therefore deserves continued scholarship.On the second day, we started with our fourth panel called "Encounters through Western Conceptual Lenses," which among others gave us insight into the reception [End Page 393] of Japanese Buddhism by the German liberal protestants of the nineteenth century. Our fifth panel—"Encounters with German Philosophy"—was an excellent example of contemporary comparative work between Buddhist philosophy, ethics, and logic, and the German thinkers Hegel and Nietzsche.We learned that Nietzsche labeled Buddhism as nihilism––a position that can be challenged through Ambedkar's reading of Buddhism. On the other hand, despite Nietzsche's criticism of Buddhism, striking similarities between his approach to life and that of Mahāyāna Buddhism can be found, both elaborating a life-affirming philosophy. We also became familiar with similarities and differences between the logic of Nishida and Hegel. Finally, our last panel, "Literary and Artistic Encounters" made for a fitting conclusion, reminding us that Buddhist-Western encounters are ongoing, not just politically and intellectually but also in art, music, and literature. These encounters are dynamic in nature and fundamentally lacking in an essential core. Instead, they are entities that echo through the ages allowing for reinterpretation and reintegration across geographic and temporal scopes.Both during and after the conference we received extremely positive feedback from the participants. The level of participation and enthusiasm was highly promising, showing that the future of intercultural studies of this sort is bright. Graduate conferences like these are key for early career scholars to set up networks and find likeminded peers to engage in ongoing dynamic encounters.In conclusion, this conference served to highlight the sophisticated emerging scholarship taking place that seeks to investigate Buddhist-Western interactions and dialogue. But, perhaps more importantly, the conference itself makes a metapoint: not only has the world been transcultural for many centuries, but the conference itself and in particular the global spatial locations of... (shrink)

In this paper we propose to enrich the four-valued modal logic associated to Monteiro's Tetravalent modal algebras (TMAs) with a deductive implication, that is, such that the Deduction Meta-theorem holds in the resulting logic. All this lead us to establish some new connections between TMAs, symmetric (or involutive) Boolean algebras, and modal algebras for extensions of S5, as well as their logical counterparts.