El objetivo principal de este artículo es precisar los elementos que operan en el substrato de la filosofía de Thomas Hobbes y que permiten dar coherencia lógica a su sistema. Para ello se apela a la idea de concepto operatorio desarrollada por Eugen Fink, a la vez que se llama la atención sobre la importancia de considerar la filosofía hobbesiana en su totalidad. Esto implica que cualquier análisis de la teoría política de Hobbes debe considerar las relaciones existentes con (...) las concepciones científicas hobbesianas. (shrink)

Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, (...) as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations form a (...) bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L₃, the intuitionistic system Iⁱ and several para-consistent logics, as for instance Sette's system Pⁱ, the D'Ottaviano and da Costa system J₃ and da Costa's systems $\text{C}_{n}$ , 1 ≤ n ≤ ω. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to (...) consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. (shrink)

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

El problema de marco cuestiona cómo los procesos cognitivos determinan qué información, de entre toda la disponible, es relevante dada una tarea determinada. Aunque postulamos una definición posible, especificar de qué trata este problema es una tarea complicada. Una manera de obtener claridad sobre esta cuestión es explorar distintas interpretaciones del problema de marco, interpretación lógica y filosófica, para dilucidar luego la dificultad en común. Como resultado de nuestro análisis concluimos que, sea la interpretación del problema de marco que (...) se tenga en mente, la dificultad subyacente es la misma: examinar vasta cantidad de información. La dificultad de la vastedad de información, a modo de puente entre ambas interpretaciones, permitiría no solo comprender más claramente de qué trata el problema de marco sino también resolverlo. The frame problem questions how cognitive processes determine what information is relevant, among all available, given a particular task. Although we proposed a possible definition, it is a difficult task to specify what this problem is about. One way to get insight into this matter is to explore different interpretations of the frame problem, a logical interpretation and a philosophical interpretation, to then elucidate the difficulty in common. The conclusion drawn from our analysis is that, irrespective of the interpretation of the frame problem you may have in mind, the underlying difficulty is the same: to examine a vastness of information. The difficulty posed by the vastness of information, taken as a bridge between both interpretations, would not only allow a clearer insight into the frame problem but also to solve it. (shrink)

I discuss this collection of essays on Hegel and the sciences while stressing the interest of Hegel's philosophy of nature in the light of later non-mainstream developments in the life-sciences and medicine. I compare then the chapters dedicated to Hegel's logic with recent literature on para-consistent logic and re-interpretations of Hegel's own logic.

RESUMO Um problema de longa data na erudição aristotélica diz respeito à questão de como reconciliar a dupla descrição de Aristóteles da metafísica como ontologia (a ciência universal do ser enquanto ser) e teologia (a ciência da substância imutável e separada). Uma tentativa importante de responder a esta questão (iniciada primeiro por G. Patzig) consiste em dizer que a substância imutável e separada é focalmente anterior (ou o significado focal da) substância e, portanto, de ser em geral (uma vez que (...) a substância é focalmente anterior a ser no geral). Este artigo visa refutar esse tipo de abordagem para o problema da unidade da metafísica de Aristóteles, argumentando que (i) as relações de significado focal implicam a prioridade lógica (definição) dos itens anteriores sobre os itens dependentes situados em tais relações; (ii) a substância imutável e separada não é logicamente anterior aos outros tipos de substâncias distinguidas por Aristóteles; e, portanto, (iii) a substância imutável e separada não é focalmente anterior à (o significado focal de) substância. (shrink)

The aim of this study consists of postulating an ethical approach to Human Rights conceiving them as rights of the other, claiming that this approach represent a starting point for the development of a legal pluralism that contributes to the achievement of an effective and authentic peace. For this study is based on the premise that the law must be able to deal with the serious and still pending social and environmental problems that underlie the violations of human rights, especially (...) those of ecological nature. By way of conclusion, the need for a radical change of the paradigm is evidenced, for which a perspective of ethics as a linguistic praxis is proposed, since such praxis has the potential to inspire the reformulation of human rights, for the purpose of reversing the dynamics of violence imposed by the logic of identity underlying the current legal system. (shrink)

Resumen Se sostiene que la operación general de la silogística en la Lógica cumple con un programa trazado tempranamente por Hegel, consistente en concebir la unidad de la forma racional del silogismo y el contenido absoluto de la idea como resultado de una dialéctica inmanente a la propia forma del silogismo, mediante la cual éste abandona su carácter meramente formal y alcanza su propio concepto: ser forma objetiva o racional. Para esto se acude a la sección Subjetividad de la (...) Doctrina del Concepto, procurando reconstruir los antecedentes sistemáticos del capítulo sobre el silogismo, así como la operación específica que éste realiza en vista del tránsito a la sección Objetividad. El sentido de dicha operación consiste en la superación de un exceso que, al comienzo, la necesidad de la sustancia tiene por sobre la libertad del concepto, de lo cual resulta un exceso del concepto por sobre la sustancia. Para verificar esta operación se examinan los silogismos del estar y la necesidad, concluyendo con la determinación del tipo de objeto y de conocimiento que resulta del silogismo, y con la proyección del problema a estadios ulteriores de la Lógica, en los cuales el problema avistado adquiere un cariz marcadamente práctico.In this paper I argue that the general operation of syllogistic in the Logic fulfills a philosophical program that was early outlined by Hegel. This program conceives the unity of the rational form of the syllogism and the absolute content of the idea as a result of an immanent dialectic to the syllogism's form, by which it abandons its merely formal character and attains its own concept: being an objective or rational form. For the purposes of this argument, I analyze the Subjectivity section of the Doctrine of the Concept and reconstruct the systematic background of the chapter on syllogism, as well as the specific operation it performs in view of the transit to the Objectivity section. This operation consists in overcoming a certain excess that the necessity of the substance has, first, over the freedom of the concept, which results in an excess of the concept over the substance. To clarify this operation, I examine the syllogisms of qualitative existence and necessity. Finally, I show the determination of the type of object and knowledge that results from the syllogism and explain how this problem projects to later stages of Logic, in which the analyzed problem acquires a relevant practical aspect. (shrink)

In (Béziau 2001), Béziau provides a means by which Gentzen’s sequent calculus can be combined with the general semantic theory of bivaluations. In doing so, according to Béziau, it is possible to construe the abstract "core" of logics in general, where logical syntax and semantics are "two sides of the same coin". Thecentral suggestion there is that, by way of a modification of the notion of maximal consistency, it is possible to prove the soundness and completeness for any normal (...) class='Hi'>logic (without invoking the role of classical negation in the completeness proof). However, the reduction to bivaluation may be a side effect of the architecture of ordinary sequents, which is both overly restrictive, and entails certain expressive restrictions over the language. This paper provides an expansion of Béziau’s completeness results for logics, by showing that there is a natural extension of that line of thinking to n-sided sequent constructions. Through analogical techniques to Béziau’s construction, it is possible, in this setting, to construct abstract soundness and completeness results for n-valued logics.En (Béziau 2001), Béziau ofrece un recurso para combinar el cálculo de secuentes de Gentzen con la teoría semántica general de bivaluaciones. Al hacer esto, según Béziau, es posible construir el “núcleo” abstracto de la lógica en general, donde sintaxis y semántica son las dos caras de una misma moneda. La sugerencia clave es que, mediante una modificación de la noción de consistencia máxima, es posible probar la corrección y completud de cualquier lógica normal (sin invocar la función de la negación clásica en la prueba de completud). Sin embargo, la reducción a bivaluaciones puede ser un efecto colateral de la arquitectura de los secuentes ordinarios, que es abiertamente restrictiva y entraña determinadas restricciones expresivas sobre el lenguaje. Este artículo ofrece una expansión de los resultados de completud de Béziau para la lógica, mostrando que existe una extensión natural de esta línea de pensamiento a construcciones de secuentes de n lados. Mediante técnicas análogas a la construcción de Béziau, en este marco es posible construir resultados abstractos decorrección y completud para la lógica n-valuada. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to (...) consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to (...) consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major logical (...) theory and offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

A fundamental question asked in modal logic is whether a given theory is consistent. But consistent with what? A typical way to address this question identifies a choice of background knowledge axioms (say, S4, D, etc.) and then shows the assumptions codified by the theory in question to be consistent with those background axioms. But determining the specific choice and division of background axioms is, at least sometimes, little more than tradition. This paper introduces generic theories (...) for propositional modal logic to address consistency results in a more robust way. As building blocks for background knowledge, generic theories provide a standard for categorical determinations of consistency. We argue that the results and methods of this paper help to elucidate problems in epistemology and enjoy sufficient scope and power to have purchase on problems bearing on modalities in judgement, inference, and decision making. (shrink)

Although logical consistency is desirable in scientific research, standard statistical hypothesis tests are typically logically inconsistent. To address this issue, previous work introduced agnostic hypothesis tests and proved that they can be logically consistent while retaining statistical optimality properties. This article characterizes the credal modalities in agnostic hypothesis tests and uses the hexagon of oppositions to explain the logical relations between these modalities. Geometric solids that are composed of hexagons of oppositions illustrate the conditions for these modalities to be (...) logically consistent. Prisms composed of hexagons of oppositions show how the credal modalities obtained from two agnostic tests vary according to their threshold values. Nested hexagons of oppositions summarize logical relations between the credal modalities in these tests and prove new relations. (shrink)

The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this in (...) a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective. (shrink)

G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R. Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to Gödel incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility, viz., that the logic of the mind can simultaneously be Gödel incomplete, consistent, mechanical, and recursion complete (capable of all means of recursion). A representational approach is pursued, which owes its origin to works by, among others, J. Myhill (1964), P. Benacerraf (...) (1967), J. Webb (1980, 1983) and M. Arbib (1987). It is shown that the fallacy shared by the two arguments under discussion lies in misidentifying two systems, the one for which the Gödel sentence is constructable and to be proved, and the other in which the Gödel sentence in question is indeed provable. It follows that the logic of the mind can surpass its own Gödelian limitation not by being inconsistent or non-mechanistic, but by being capable of representing stronger systems in itself; and so can a proper machine. The concepts of representational provability, representational maximality, formal system capacity, etc., are discussed. (shrink)

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and (...) T prove the existence of infinitely many non-logical objects (T deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of T toCantor's theorem which is somewhat surprising. (shrink)

Simultaneous hypothesis tests can fail to provide results that meet logical requirements. For example, if A and B are two statements such that A implies B, there exist tests that, based on the same data, reject B but not A. Such outcomes are generally inconvenient to statisticians (who want to communicate the results to practitioners in a simple fashion) and non-statisticians (confused by conflicting pieces of information). Based on this inconvenience, one might want to use tests that satisfy logical requirements. (...) However, Izbicki and Esteves shows that the only tests that are in accordance with three logical requirements (monotonicity, invertibility and consonance) are trivial tests based on point estimation, which generally lack statistical optimality. As a possible solution to this dilemma, this paper adapts the above logical requirements to agnostic tests, in which one can accept, reject or remain agnostic with respect to a given hypothesis. Each of the logical requirements is characterized in terms of a Bayesian decision theoretic perspective. Contrary to the results obtained for regular hypothesis tests, there exist agnostic tests that satisfy all logical requirements and also perform well statistically. In particular, agnostic tests that fulfill all logical requirements are characterized as region estimator-based tests. Examples of such tests are provided. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. We define the logics BK+ and BK'+ by adding to B+ the K rule and to BK+ the characteristic S4 axiom, respectively. These logics are endowed with a relatively strong non-constructive negation. We prove that all the logics defined lack the K axiom and the standard paradoxes of consistency.

In a paper published in 1975, Robert Jeroslow introduced the concept of an experimental logic as a generalization of ordinary formal systems such that theoremhood is a (or in practice ) rather than . These systems can be viewed as (rather crude) representations of axiomatic theories evolving stepwise over time. Similar ideas can be found in papers by Putnam (1965) and McCarthy and Shapiro (1987). The topic of the present article is a discussion of a suggestion by Allen Hazen, (...) that these experimental logics might provide an illuminating way of representing “the human mathematical mind”. This is done in the context of the well-known Lucas-Penrose thesis. Though we agree that Jeroslow's model has some merit in this context, and that the Lucas-Penrose arguments certainly are less than persuasive, some semi-technical doubts are raised concerning the alleged impact of experimental logics on the question of knowable self-consistency. (shrink)

Arif Ahmed has recently argued that causal decision theory is dynamically inconsistent and that we should therefore prefer evidential decision theory. However, the principal formulation of the evidential theory, Richard Jeffrey’s Logic of Decision, has a mixed record of its own when it comes to evaluating plans consistently across time. This note probes that neglected record, establishing the dynamic consistency of evidential decision theory within a restricted class of problems but then illustrating how evidentialists can fall into sequential incoherence (...) outside of this class. Uncovering the evidentialist’s own dynamic inconsistencies reveals, contra Ahmed, that sequential choice considerations do not significantly favor the evidentialist’s theory over the causalist’s. (shrink)

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A (A B)) (A B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions (...) and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional. (shrink)

The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal (...) logic that has persisted for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields. (shrink)

In “Truth and Probability” Ramsey claimed that the logic of consistency for probability is not a logic of truth. After supporting this claim, he proceeded to explore the possibilities for a logic of truth for probability. An examination of Ramsey's intent reveals that Ramsey was far from being an orthodox Bayesian when it comes to statistical reasoning. The relations between Ramsey's thought and the ideas of Keynes and Peirce are discussed.

We prove Robinson consistency theorem as well as Craig, Lyndon and Herbrand interpolation theorems in linear continuous logic.

It has long been recognized that the concept of inconsistency is a central part of commonsense reasoning. In this issue, a number of authors have explored the idea of reasoning with maximal consistent subsets of an inconsistent stratified knowledge base. This paradigm, often called “coherent-based reasoning", has resulted in some interesting proposals for para-consistent reasoning, non-monotonic reasoning, and argumentation systems. Unfortunately, coherent-based reasoning is computationally very expensive. This paper harnesses the approach of approximate entailment by Schaerf and (...) Cadoli [SCH 95] to develop the concept of “approximate coherent-based reasoning". To this end, we begin to present a multi-modal propositional logic that incorporates two dual families of modalities: □S and?S defined for each subset S of the set of atomic propositions. The resource parameter S indicates what atoms are taken into account when evaluating formulas. Next, we define resource-bounded consolidation operations that limit and control the generation of maximal consistent subsets of a stratified knowledge base. Then, we present counterparts to existential, universal, and argumentative inference that are prominent in coherence-based approaches. By virtue of modalities □S and?S, these inferences are approximated from below and from above, in an incremental fashion. Based on these features, we show that an anytime view of coherent-based reasoning is tenable. (shrink)

The logic B+ is Routley and Meyer’s basic positive logic. Wedefine the logics BK+ and BK′+ by adding to B+ the K rule and to BK+the characteristic S4 axiom, respectively. These logics are endowed witha relatively strong non-constructive negation. We prove that all the logicsdefined lack the K axiom and the standard paradoxes of consistency.

The book Paraconsistent Logic: Consistency, Contradiction and Negation by Walter Carnielli and Marcelo Coniglio is the most thorough study of Logics of Formal Inconsistency...

The logics of formal inconsistency are logics tolerant to some amount of contradiction, but in which some versions of explosion still hold. The main result of this paper is a reconstruction of two such logics in the dialogical framework. By doing so, we achieve two things. On the one hand, we provide a formal approach to argumentative situations where some contradictions may occur while keeping the idea that there may still be situations in which some propositions are ‘safe’ in the (...) sense of immunity to the contradictions. On the other hand, we open a new line of study on these logics, in the context of the game-theoretical approach to semantics born in the 1960s, with various interesting perspectives, some of which are discussed at the end of this article. (shrink)

In “Truth and Probability” Ramsey claimed that the logic of consistency for probability is not a logic of truth. After supporting this claim, he proceeded to explore the possibilities for a logic of truth for probability. An examination of Ramsey's intent reveals that Ramsey was far from being an orthodox Bayesian when it comes to statistical reasoning. The relations between Ramsey's thought and the ideas of Keynes and Peirce are discussed.

In this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of (...) whether we have a Dutch Book Argument for Conditionalization is left open. (shrink)

A typical theorem of conaexive logics is Aristotle''s Thesis(A), (AA).A cannot be added to classical logic without producing a trivial (Post-inconsistent) logic, so connexive logics typically give up one or more of the classical properties of conjunction, e.g.(A & B)A, and are thereby able to achieve not only nontriviality, but also (negation) consistency. To date, semantical modellings forA have been unintuitive. One task of this paper is to give a more intuitive modelling forA in consistent logics. In (...) addition, while inconsistent but nontrivial theories, and inconsistent nontrivial logics employing prepositional constants (for which the rule of uniform substitution US fails), have both been studied extensively within the paraconsistent programme, inconsistent nontrivial logics (closed under US) do not seem to have been. This paper gives sufficient conditions for a logic containingA to be inconsistent, and then shows that there is a class of inconsistent nontrivial logics all containingA. A second semantical modelling forA in such logics is given. Finally, some informal remarks about the kind of modellingA seems to require are made. (shrink)

The claim that nature is self-consistent has recently been contested by a number of paraconsistent logicians. In this paper I will survey the arguments which paraconsistent logicians have presented for the thesis that nature is actually inconsistent. My conclusion is that these arguments all fail.The paraconsistency programme has to date been concerned primarily with outlining the philosophical inadequacy of classical logic, and detailed discussions of issues bearing upon the philosophical adequacy of the paraconsistency position itself are not to (...) be found as yet in the literature. This inadequacy will be illustrated here with respect to the question of the self-consistency of nature. (shrink)

Moral conflicts are real, and while a deontic logic containing a modified "agglomeration rule" may be able to accommodate this fact, even the most sophisticated logic will still overlook much of what everyday normative reasoning involves.

On an expressivist view, ethical claims are understood as expressions of our attitudes, desires, and feelings. A famous puzzle for this view concerns the use of logic in ethical reasoning, and two standard treatments try to solve the puzzle by explaining logical inconsistency in terms of conflicting attitudes. I argue, however, that this general strategy fails: because we can reason effectively even in the presence of conflicting moral attitudes – in cases of moral dilemmas – avoiding these conflicts cannot (...) be a ground for correct moral reasoning. The result is a dilemma for expressivists: if they take all kinds of attitudes to be under consideration, then conflict cannot play the required role, since attitudes can fail to be compatible in cases of moral conflict. If they restrict attention to ‘all-in attitudes’ or to intentions or plans, then there is an important notion of obligation, used in standard arguments – one for which conflicts are allowed – that they fail to capture. I explain why expressivists should be especially tolerant of conflicting attitudes, and I conclude that they should pursue a different strategy for grounding logical normativity. (shrink)