ABSTRACT A bilattice is said to be regular provided its truth conjunction and disjunction are monotonic with respect to its knowledge ordering. The principal result of this paper is that the following properties of a bilattice B are equivalent: 1. B is regular; 2. the truth conjunction and disjunction of B are definable through the rest of the operations and constants of B; 3. B is isomorphic to a bilattice of the form L 1 · L 2 (...) where L 1 and L 2 are bounded lattices. We also derive from this metaequivalence a number of corollaries concerning distributive bilattices, degenerated bilattices and bilattices with negation. (shrink)

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 presents an information-based logic that is applied to the analysis of entailment, implicature and presupposition in natural language. The logic is very fine-grained and is able to make distinctions that are outside the scope of classical logic. It is independently motivated by certain properties of natural human reasoning, namely partiality, paraconsistency, relevance, and defeasibility: once these are accounted for, the data on implicature and presupposition comes quite naturally.The logic is based on the family of semantic spaces known as (...) bilattices, originally proposed by Ginsberg (1988), and used extensively by Fitting (1989, 1992). Specifically, the logic is based on a subset of bilattices that I call evidential bilattices, constructed as the Cartesian product of certain algebras with themselves. The specific details of the epistemic agent approach of the logical system is derived from the work of Belnap (1975, 1977), augmented by the use of evidential links for inferencing. An important property of the system is that it has been implemented using an extension of Fitting's work on bilattice logic programming (1989, 1991) to build a model-based inference engine for the augmented Belnap logic. This theorem prover is very efficient for a reasonably wide range of inferences. (shrink)

While Kripke's original paper on the theory of truth used a three-valued logic, we believe a four-valued version is more natural. Its use allows for possible inconsistencies in information about the world, yet contains Kripke's development within it. Moreover, using a four-valued logic makes it possible to work with complete lattices rather than complete semi-lattices, and thus the mathematics is somewhat simplified. But more strikingly, the four-valued version has a wide, natural generalization to the family of interlaced bilattices. Thus, with (...) little more work, the theory is extended to a broad class of settings. Indeed, a result like Theorem 6.2 would not even be possible to state without the interlaced bilattice machinery. We hope the notion of interlaced bilattice will make apparent further such connections. (shrink)

Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiple-valued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the paper is relatively self-contained.

This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory for an adequate definition of the deduction apparatus for multi-valued logic. We argue that bilattice theory enables us to obtain a nice extension of the graded approach to fuzzy logic. To give an example, a completeness theorem for a logic based on Boolean algebras is proved.

In a previous work we studied, from the perspective ofAlgebraic Logic, the implicationless fragment of a logic introduced by O. Arieli and A. Avron using a class of bilattice-based logical matrices called logical bilattices. Here we complete this study by considering the Arieli-Avron logic in the full language, obtained by adding two implication connectives to the standard bilattice language. We prove that this logic is algebraizable and investigate its algebraic models, which turn out to be distributive bilattices with (...) additional implication operations. We axiomatize and state several results on these new classes of algebras, in particular representation theorems analogue to the well-known one for interlaced bilattices. (shrink)

Baltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further (...) extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective. (shrink)

This paper discusses a dualization of Fitting's notion of a "cut-down" operation on a bilattice, rendering a "track-down" operation, later used to represent the idea that a consistent opinion cannot arise from a set including an inconsistent opinion. The logic of track-down operations on bilattices is proved equivalent to the logic d_Sfde, dual to Deutsch's system S_fde. Furthermore, track-down operations are employed to provide an epistemic interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combinations of cut-and (...) track-down operations allow settling positively the question of whether bilattice-based semantics are available for subsystems of S_fde. (shrink)

One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while (...) making the mathematical machinery simpler and more perspicuous. In addition, work such as that of Yablo fits naturally into the bilattice setting. What I do here is present the general background of bilattices, discuss why they are natural, and show how fixpoint approaches to truth in languages that allow self-reference can be applied. This is not new work, but rather is a summary of research I have done over many years. (shrink)

The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our (...) work. The outcome are paraconsistent logics with a lot of desirable properties.1. (shrink)

We introduce a new product bilattice con- struction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lat- tice factors. Finally, we (...) show how to employ our product construction to define first-order definable classes of bi- lattices corresponding to any first-order definable subclass of residuated lattices. (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)

In this paper we reexamine the question of whether questions areinherently intensional entities. We do so by proposing a novelextensional theory of questions, based on a re-interpretation of thedomain of t as a bilattice rather than the usual booleaninterpretation. We discuss the adequacy of our theory with respect tothe adequacy criteria imposed on the semantics of questionsby (Groenendijk and Stokhof 1997). We show that the theory is able to account in astraightforward manner for some complex issues in the semantics (...) ofquestions including coordinated questions, combined indicative andinterrogative sentences, questions with quantifiers, and theimpossibility of negating questions. (shrink)

Lattice logic, bilattice logic, and paraconsistent quantum logic are investigated based on monosequent systems. Paraconsistent quantum logic is an extension of lattice logic, and bilattice logic is an extension of paraconsistent quantum logic. Monosequent system is a sequent calculus based on the restricted sequent that contains exactly one formula in both the antecedent and succedent. It is known that a completeness theorem with respect to a lattice-valued semantics holds for a monosequent system for lattice logic. A completeness theorem (...) with respect to a lattice-valued semantics is proved for paraconsistent quantum logic, and a completeness theorem with respect to a bilattice-valued semantics is proved for bilattice logic. Some syntactical properties, including cut-elimination and duality, are also investigated for the monosequent systems for these logics. (shrink)

This paper sheds light on the relationship between the logic of generalized truth values and the logic of bilattices. It suggests a definite solution to the problem of axiomatizing the truth and falsity consequence relations, \ and \ , considered in a language without implication and determined via the truth and falsity orderings on the trilattice SIXTEEN 3 . The solution is based on the fact that a certain algebra isomorphic to SIXTEEN 3 generates the variety of commutative and distributive (...) bilattices with conflation. (shrink)

We develop a Priestley-style duality theory for different classes of algebras having a bilattice reduct. A similar investigation has already been realized by B. Mobasher, D. Pigozzi, G. Slutzki and G. Voutsadakis, but only from an abstract category-theoretic point of view. In the present work we are instead interested in a concrete study of the topological spaces that correspond to bilattices and some related algebras that are obtained through expansions of the algebraic language.

Since Popper and Lakatos, the demarcation line between science and non-science has been considered one of the fundamental issues of the philosophy of science. According to Lakatos, pseudoscience is a non-science, which appears as science, using science's public authority. Since then, mountains of texts have been published on how non-sciences, such as astrology, are not sciences. -/- But the enemy is not on the other side of the border. The enemy is in our midst. Science has been institutionalized. The best (...) way to do the pseudoscience is to fuss inside the big enterprise recognized as a science. (shrink)

A general Gentzen-style framework for handling both bilattice (or strong) negation and usual negation is introduced based on the characterization of negation by a modal-like operator. This framework is regarded as an extension, generalization or re- finement of not only bilattice logics and logics with strong negation, but also traditional logics including classical logic LK, classical modal logic S4 and classical linear logic CL. Cut-elimination theorems are proved for a variety of proposed sequent calculi including CLS (a conservative (...) extension of CL) and CLScw (a conservative extension of some bilattice logics, LK and S4). Completeness theorems are given for these calculi with respect to phase semantics, for SLK (a conservative extension and fragment of LK and CLScw, respectively) with respect to a classical-like semantics, and for SS4 (a conservative extension and fragment of S4 and CLScw, respectively) with respect to a Kripke-type semantics. The proposed framework allows for an embedding of the proposed calculi into LK, S4 and CL. (shrink)

:Advocates of cosmopolitan ideals, to the extent that they engage with questions of institutional design, typically imagine replicating or refining existing, nation-state models of governance but on an international scale. This essay argues that cosmopolitan ethics need not go hand in hand with international government, and may be better served by a different approach. I explore the concept of degeneracy as a principle of institutional evaluation and design in international politics. Degeneracy is a characteristic of complex systems in which multiple (...) components of the system offer overlapping functions, and is a key component in the robustness of such systems. Non-degenerate systems, by contrast, exhibit fragility in the face of adverse conditions. When applied to systems of governance, degeneracy commends polycentricity and allows for some evaluation of the robustness of different mechanisms and forms of polycentric governance. Cosmopolitan ideals are better served by providing alternatives to existing forms of governance than by building on them. I consider some concrete policy applications of this idea, focusing on immigration and intellectual property. (shrink)

The chapter discusses the following quotation from Charles Peirce: "One of these days, perhaps, there will come a writer of opinions less humdrum than those of Dr. (Alfred Russel) Wallace, and less in awe of the learned and official world...who will argue, like a new Bernard Mandeville, that man is but a degenerate monkey, with a paranoic talent for self-satisfaction, no matter what scrapes he may get himself into, calling them 'civilization,' and who, in place of the unerring instincts of (...) other races, has an unhappy faculty for occupying himself with words and abstractions, and for going wrong in a hundred ways before he is driven, willy-nilly, into the right one. Dr. Wallace would condemn such an extravagant paradoxer." Charles Peirce, 1901 -/- . (shrink)

Focusing on Kentucky’s immense and world-famous Mammoth Cave, this essay considers contexts from across the nineteenth century in which subterranean darkness was envisaged as a driving force in the transformation of living things. In fact, the cave was the stage for several allied discourses of “dark degeneracy” that conjured images of both generative and destructive mutability, from the generation of animals without eyes to the apparent disintegration of some kinds of human bodies and minds through exposure to the darkness. In (...) elucidating several parallel discourses of dark degeneracy, the essay offers a distinct contribution to several bodies of scholarship by highlighting an important and hitherto neglected manifestation of an intensifying nineteenth-century focus on the how of natural processes and a growing sense of the transformative significance of the apparent cohesion of malleable bodies with their wider environments. At the same time, it evidences the influence of Mammoth Cave in forging and contributing to late nineteenth-century debates about degeneration and shows how dominant sensory hierarchies framed both popular and scientific understandings of life in the deep darkness. (shrink)

Summary Theories of familial, racial, and national degeneration in the late nineteenth and early twentieth centuries have been explored by historians in the context of social and moral pathology. At the same time nerve degeneration was studied in the post mortem room and in the laboratory but links to the broader ideology of degeneration have not been investigated by scholars. This paper joins these domains by examining the concept of Wallerian degeneration. It argues that various discourses—including those of the laboratory (...) scientist, the clinician, and the social theorist—employed the term degeneration, and these discourses frequently overlapped demonstrating that degeneration was a ubiquitous fact of Victorian and Edwardian nature. (shrink)

The essay concerns the negative end of the political argument of the Republic, that injustice—the rule of unreason—is both widespread and undesirable, and that whatever shadows of virtue or order might be found in its midst are corrupt and unstable. This claim is explained in detail in Republic 8 and 9. These passages explain recognizable faults in recognizable regimes in terms of the failure of the rule of reason and the corresponding success of the rule of non-rational forms of motivation. (...) I will first look at degenerate regimes as they appear in a less systematic way in the Ship of State passage in Republic 6 and in the discussion with Thrasymachus in book 1. I then give a general overview of the system of degenerate regimes in book 8 to examine what exactly goes wrong with them and why, and explain how the process of degeneration ought be understood as the progressive decay of the rule of reason. Finally, I argue that a close look at this decay reveals something surprising: that degenerate regimes and characters feature weak versions of virtue, shadow-virtues that are based on appearances and held in place by force. Thus in the end the whole process of degeneration ought be understood as an extended conflict between reason and appetite. (shrink)

The paper contains some algebraic results on several varieties of algebras having an (interlaced) bilattice reduct. Some of these algebras have already been studied in the literature (for instance bilattices with conflation, introduced by M. Fit- ting), while others arose from the algebraic study of O. Arieli and A. Avron’s bilattice logics developed in the third author’s PhD dissertation. We extend the representation theorem for bounded interlaced bilattices (proved, among others, by A. Avron) to un- bounded bilattices and (...) prove analogous representation theorems for the other classes of bilattices considered. We use these results to establish categorical equivalences between these structures and well-known varieties of lattices. (shrink)

This paper focuses on natural dualities for varieties of bilattice-based algebras. Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to (...) build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation. Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras. (shrink)

A striking feature of John Dewey’s philosophical attitude in his later period is that for self-description, he did not prefer the term “pragmatism.” Instead, he employed such isms as “experimentalism” and “naturalism.” In the period in which he moved towards developing his own original philosophy, he even stated that “I reject root and branch to the term ‘pragmatism.’”1 As he was at the time drawn to naturalism, it might be revealing indeed that he rejects “root and branch” to “pragmatism.” Also (...) in the newly released notes for his lectures on social philosophy in China 1919, the word “pragmatism” goes without a single mention.2 The reader does not encounter the creed for a “pragmatist social... (shrink)

The Y chromosome in the XY sex-determination system is often shorter than its X counterpart, a condition attributed to degeneration after Y recombination ceases. Contrary to the traditional view of continuous, gradual degeneration, our study reveals stabilization within large mating populations. In these populations, we demonstrate that both mutant and active alleles on the Y chromosome can reach equilibrium through a mutation-selection balance. However, the emergence of a new species, particularly through the founder effect, can disrupt this equilibrium. Specifically, if (...) the male founders of a new species carry only a mutant allele for a particular Y-linked gene, this allele becomes fixed, leading to the loss of the corresponding active gene on the Y chromosome. Our findings suggest that the rate of Y-chromosome degeneration may be linked to the frequency of speciation events associated with single-male founder events. (shrink)

The Hungarian-born philosopher of science Imre Lakatos introduces the methodology of scientific research programs, and also makes a famous distinction between “progressive” and “degenerating” programs. Although Lakatos does not give extensive guidance as to whether philosophical rather than scientific theories could also be judged in this way, he does give some intriguing hints in his discussion of a debate on induction between Rudolf Carnap and Karl Popper. After considering two extant but misguided attempts to use “degenerating” as a polemical term (...) in philosophy, this article gathers the aforementioned hints from Lakatos and a few others from Alfred North Whitehead in an effort to identify what a “progressive philosophy” might look like. (shrink)

Background: Primary open-angle glaucoma patients exhibit widespread white matter degeneration throughout their visual pathways. Whether this degeneration starts at the pre- or post-geniculate pathways remains unclear. In this longitudinal study, we assess the progression of WM degeneration exhibited by the pre-geniculate optic tracts and the post-geniculate optic radiations of POAG patients over time, aiming to determine the source and pattern of spread of this degeneration.Methods: Diffusion-weighted MRI scans were acquired for 12 POAG patients and 14 controls at two time-points 5.4 (...) ± 2.1 years apart. Fiber density, an estimate of WM axonal density, was computed for the OTs and ORs of all participants in an unbiased longitudinal population template space. First, FD was compared between POAG patients and the controls at time-point 1 and time-point 2 independently. Secondly, repeated measures analysis was performed for FD change in POAG patients between the two time-points. Finally, we compared the rate of FD change over time between the two groups.Results: Compared to the controls, POAG patients exhibited significantly lower FD in the left OT at TP1 and in both OTs and the left OR at TP2. POAG patients showed a significant loss of FD between the time-points in the right OT and both ORs, while the left OR showed a significantly higher rate of FD loss in POAG patients compared to the controls.Conclusions: We find longitudinal progression of neurodegenerative WM changes in both the pre- and post-geniculate visual pathways of POAG patients. The pattern of changes suggests that glaucomatous WM degeneration starts at the pre-geniculate pathways and then spreads to the post-geniculate pathways. Furthermore, we find evidence that the trans-synaptic spread of glaucomatous degeneration to the post-geniculate pathways is a prolonged process which continues in the absence of detectable pre-geniculate degenerative progression. This suggests the presence of a time window for salvaging intact post-geniculate pathways, which could prove to be a viable therapeutic target in the future. (shrink)

Strict/tolerant logic is a formally defined logic that has the same consequence relation as classical logic, though it differs from classical logic at the metaconsequence level. Specifically, it does not satisfy a cut rule. It has been proposed for use in work on theories of truth because it avoids some objectionable features arising from the use of classical logic. Here we are not interested in applications, but in the formal details themselves. We show that a wide range of logics have (...) strict/tolerant counterparts, with the same consequence relations but differing at the metaconsequence level. Among these logics are Kleene’s K3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {K_3}$$\end{document}, Priest’s LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {LP}}$$\end{document}, and first-degree entailment, FDE. The primary tool we use is the bilattice. But it is more than a tool, it seems to be the natural home for this kind of investigation. (shrink)

Ken Gemes's “The Biology of Evil” makes significant advances over previous discussions in its recognition of the centrality of the Jews in Nietzsche's account of the rise of slave morality, and in its differentiation between Nietzsche's virulent opposition to the anti-Semitic movements of his era and his embrace of prejudice regarding Jews and Jewry. There are three areas in which his claims are deficient, however. He does not realize Nietzsche's lifelong interest in the contemporary Jewish Question in Germany. He disregards (...) Nietzsche's avid interest in eugenics and eugenic solutions in the discussion of degeneration; Nietzsche knew the works of Francis Galton and others, and he himself wrote explicitly about positive and negative eugenics. Finally, Gemes believes Nietzsche advocated a mixture of slave and master morality, while it is clear from his rhetoric and his substantive remarks that he envisions no harmony of these two antagonistic value systems. (shrink)

Over the years, nursing research and practice have been through remarkable transformations in response to evolving and emerging healthcare systems and practices. Regarding research, nurses moved beyond merely using the quantitative methodology to combining qualitative, quantitative and mixed methods. In practice, nurses have transitioned from the delivery of medical‐based care to nursing theory‐guided practice, evidence‐based practice, knowledge translation and transformative practice. Some domains of nursing research and practice became progressive, while others degenerated. This paper aims to examine how different (...) domains of nursing research and practice progress and degenerate using Imre Lakatos's methodology of scientific research programs. Lakatos differentiated scientific and pseudoscientific knowledge and coined the idea of scientific research programs. He believed that science in any discipline develops so that some programs are more progressive and others’ degenerative. The degeneration and progression of programs occur steadily and rationally and are determined based on the extent of development and programs’ potential to predict new solutions to old problems. Adopting this idea could enable nurses to critically analyse research programs in everyday knowledge development to use valid and legitimate programs for informing nursing practice. (shrink)

[Accepted for publication in Lakatos's Undone Work: The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, special issue of Kriterion: Journal of Philosophy. Edited by S. Nagler, H. Pilin, and D. Sarikaya.] Lakatos’s analysis of progress and degeneration in the Methodology of Scientific Research Programmes is well-known. Less known, however, are his thoughts on degeneration in Proofs and Refutations. I propose and motivate two new criteria for degeneration based on the discussion in Proofs and Refutations (...) – superfluity and authoritarianism. I show how these criteria augment the account in Methodology of Scientific Research Programmes, providing a generalized Lakatosian account of progress and degeneration. I then apply this generalized account to a key transition point in the history of entropy – the transition to an information-theoretic interpretation of entropy – by assessing Jaynes’s 1957 paper on information theory and statistical mechanics. (shrink)