Paraconsistency

In this paper, I examine anti-realist remarks made by Wittgenstein in the begin- ning of the 1930s on the nature of contradictions. An anti-realist view of logic rejects the paradigm of correspondence to facts and an independent reality to deal with important logical notions such as logical connectives and contradic- tions. In the 1930s,1 the so-called Middle Wittgenstein discussed the nature of language, logic, and mathematics based on the notion of rules with mem- bers of the Vienna Circle at the close of his influential, but highly problematic, Tractarianproject.2 Inspiredbythesediscussions,Isuggestaphilosophicalac- count using the notion of rules and normativity to understand the philosoph- ical meaning of contradictions in paraconsistent logic. I defend, pace (middle) Wittgenstein and Frege, that contradictions should be taken as conflicting rules in practices rather than as dialetheia, that is, as corresponding to some peculiar but real contradictory entity or state of affairs in the world.

Eine Analyse der Unendlichkeit Gottes Obwohl sich die Idee der Unendlichkeit Gottes wie ein Tiefenstrom durch die religionsphilosophische Analyse des Wesens Gottes zieht, ist es nicht klar, was genau es bedeutet, zu behaupten, dass Gott unendlich ist. Um diese Aussage zu klären, werde ich zunächst einen synkategorematischen und einen kategorematischen Unendlichkeitsbegriff voneinander unter-scheiden. Während eine synkategorematische Analyse der Unendlichkeit Gottes auf die Unendlichkeit einzelner göttlicher Attribute beschränkt ist, besteht der Kern der kategorematischen Analyse darin, dass Unendlichkeit als ein eigenständiges kategoriales Attribut Gottes aufgefasst wird. Nach einer kurzen Kritik der Leistungsfähigkeit einer synkategorematischen Analyse der Unendlichkeit Gottes wird die Analyse des kategorematischen Unendlichkeitsverständnisses zeigen, dass die Rede von der Unendlichkeit Gottes als eigenständigem göttlichen Attribut eine Chiffre dafür ist, dass Gott als erstes Prinzip der Schöpfung alle in der Schöpfung zu findenden Unterschiede in der Einheit seines Wesens vereint. Ich ende mit einigen kurzen Bemerkungen über die Voraussetzungen und Konsequenzen dieser Interpretation der Unendlichkeit Gottes.

The present work is devoted to the exploration of some formal possibilities suggesting, since some years, the possibility to elaborate a new, whole geometry, relative to the concept of “opposition”. The latter concept is very important and vast (as for its possible applications), both for philosophy and science and it admits since more than two thousand years a standard logical theory, Aristotle’s “opposition theory”, whose culminating formal point is the so called “square of opposition”. In some sense, the whole present enterprise consists in discovering and ordering geometrically an infinite amount of “avatars” of this traditional square structure (also called “logical square” or “Aristotle’s square”). The results obtained here go even beyond the most optimistic previous expectations, for it turns out that such a geometry exists indeed and offers to science many new conceptual insights and formal tools. Its main algorithms are the notion of “logical bi-simplex of dimension m” (which allows “opposition” to become “n-opposition”) and, beyond it, the notions of “Aristotelian pq-semantics” and “Aristotelian pq-lattice” (which allow opposition to become p-valued and, more generally, much more fine-grained): the former is a game-theoretical device for generating “opposition kinds”, the latter gives the structure of the “opposition frameworks” containing and ordering the opposition kinds. With these formal means, the notion of opposition reaches a conceptual clarity never possible before. The naturalness of the theory seems to be maximal with respect to the object it deals with, making this geometry the new standard for dealing scientifically with opposition phenomena. One question, however, philosophical and epistemological, may seem embarrassing with it: this new, successful theory exhibits fundamental logical structures which are shown to be intrinsically geometrical: the theory, in fact, relies on notions like those of “simplex”, of “n-dimensional central symmetry” and the like. Now, despite some appearances (that is, the existence, from time to time, of logics using some minor spatial or geometrical features), this fact is rather revolutionary. It joins an ancient and still unresolved debate over the essence of mathematics and rationality, opposing, for instance, Plato’s foundation of philosophy and science through Euclidean geometry and Aristotle’s alternative foundation of philosophy and science through logic. The geometry of opposition shows, shockingly, that the logical square, the heart of Aristotle’s transcendental, anti-Platonic strategy is in fact a Platonic formal jungle, containing geometrical-logical hyper-polyhedra going into infinite. Moreover, this fact of discovering a lot of geometry inside the very heart of logic, is also linked to a contemporary, raging, important debate between the partisans of “logic-inspired philosophy” (for short, the analytic philosophers and the cognitive scientists) and those, mathematics-inspired, who begin to claim more and more that logic is intrinsically unable to formalise, alone, the concept of “concept” (the key ingredient of philosophy), which in fact requires rather geometry, for displaying its natural “conceptual spaces” (Gärdenfors). So, we put forward some philosophical reflections over the aforementioned debate and its deep relations with questions about the nature of concepts. As a general epistemological result, we claim that the geometrical theory of oppositions reveals, by contrast, the danger implicit in equating “formal structures” to “symbolic calculi” (i.e. non-geometrical logic), as does the paradigm of analytic philosophy. We propose instead to take newly in consideration, inspired by the geometry of logic, the alternative paradigm of “structuralism”, for in it the notion of “structure” is much more general (being not reduced to logic alone) and leaves room to formalisations systematically missed by the “pure partisans” of “pure logic”.

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question.

This article is focused on answering the question to what extent one can be a sceptic. Sextus Empiricus’s Outlines of Scepticism serves as a guide. In section 1, it is investigated whether three logical laws have a certain foundation or are subject to doubt. In section 2, Sextus’s way to deal with these laws is examined; the question arises how dogmatic his approach is. After that, a possible ‘reply’ by Sextus to the criticism receives attention. Section 3 is concentrated on a possible alternative to Sextus’s approach. Besides logical laws, some important methods are concerned.

Esta investigación ontológica aborda lo relativo al existir: qué es el existir; qué es aquello de lo que cabe decir que es o existe; si hay sentidos irreducibles de ese verbo `existir'; qué sea el no-existir -lo cual conduce al
problema de si el existir se da por grados; si cabría la posibilidad de que no existiera nada; cómo se relaciona la existencia con la referencialidad lingüística; cuáles sean las leyes generales que rigen todo lo existente. Este último problema se desglosa en el estudio de varios principios ontológicos: los de inteligibilidad, razón suficiente, identidad de los indiscernibles, interdependencia, gradualidad y otros. Esos siete problemas se estudian con las técnicas de la filosofía analítica y desde un punto de vista dialéctico, o sea el de la afirmación de que hay verdades mutuamente contradictorias que estriban en la parcial coexistencia de propiedades opuestas en el mismo sujeto, cuando se dan por grados.

PALABRAS-CLAVE:
    ontología, existencia, ser, no-ser, inexistencia, grados de ser, grados de realidad, nada, referencialidad lingüística, principios ontológicos, inteligibilidad, razón suficiente, identidad de los indiscernibles, interdependencia, gradualidad, filosofía analítica, dialéctica, verdades contradictorias,  verdad parcial, coexistencia de opuestos, grados de verdad.


Abstract:
    Dialectical ontology foundations.
This ontological research takes up the question of existence: what is to exist; what does exist; whether there are irreducible senses of `existence'; what is non-existence, which raises the problem of whether existence comes in degrees; if there could be nothing; how existence relates with linguistic referentiality; what are the most general laws governing whatever exists. The latter problem breaks down into several ontological principles: those of intelligibility, sufficient reason, identity of indiscernibles, interdependence, graduality and others.
Those seven problems are gone into through the techniques belonging to analytical philosophy and from a dialectical view-point, which means the assertion of mutually contradictory truths, resulting from the partial coexistence of opposite properties in the same subject, when they come in degrees.

Keywords:
ontological research, existence, non-existence, degrees of reality, nothingness, linguistic referentiality, ontological principles, intelligibility of being,
sufficient reason, identity of indiscernibles, interdependence, graduality, analytical philosophy, dialectics, contradictory truths, partial truth, coexistence of opposites, degrees of truth.

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for (the admissible) negations. The approach unifies in a philosophically motivated picture the following results: nothing can be called a negation properly if it does not satisfy (Minimal) Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies; and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility, but on the interpretation of worlds.

In this paper, I examine the possibility that recent developments in non-classical and paraconsistent logics might offer us a way to better understand the 'paradoxical', yet strikingly coherent nature of Cusanus' philosophical method for deriving 'names for God', particularly the 'three-step' form he proposes in De Docta Ignorantia - up to and including the ultimate 'paradoxical' element in his systematic account, the 'infinite-finite' Christ.
Building on previous research by Marko Ursic and Lorenzo Pena, I propose a four-valued logic NC which can serve as a model for illustrating Cusanus' method, as it is able to generate strikingly similar arguments to those Cusanus himself presents.
This approach provides some striking proof of the internal coherency of Cusanus' method and of the philosophical interest it should still present to the contemporary reader.

Logic has traditionally been construed as a normative discipline; it sets forth standards of correct reasoning. Explosion is a valid principle of classical logic. It states that an inconsistent set of propositions entails any proposition whatsoever. However, ordinary agents presumably do---occasionally, at least---have inconsistent belief sets. Yet it is false that such agents may, let alone ought, to believe any proposition they please. Therefore, our logic should not recognize explosion as a logical law. Call this the `normative argument against explosion'. Arguments of this type play---implicitly or explicitly---a central role in motivating paraconsistent logics.  Branden Fitelson (2008), in a throwaway remark, has conjectured that there is no plausible 'bridge principle' articulating the normative link between logic and reasoning capable of supporting such arguments. This paper offers a critical evaluation of Fitelson's conjecture and hence of normative arguments for paraconsistency and the conceptions of logic's normative status on which they repose. It is argued that Fitelson's conjecture turns out to be correct: normative arguments for paraconsistency probably fail.

Classical propositional logic can be characterized, indirectly , by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit "in the negative". More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.

In recent editions of this journal, Jean-Yves Beziau [8] and Marcin Tkaczyk [42] have criticised a prominent dialetheic logic and common arguments for dialetheism, respectively. While Beziau argues that Priest's logic LP commits the dialetheist to trivialism, the thesis that all propositions are true, Tkaczyk maintains that the arguments traditionally proposed for dialetheism are faulty and ultimately that dialetheism should be rejected as self-refuting. This paper shows that both are mistaken in their contentions. Beziau's argument conflates truth-in-an-interpretation with truth simpliciter and Tkaczyk misconstrues the substance of dialetheic arguments. In the process of identifying these weaknesses of both arguments , the paper clarifies elements of both dialetheic logics and dialethe-ism which these discussions demonstrate are still misunderstood within the literature.

En esta tesis se explora la noción de negación estricta entendida según el paradigma de la negación clásica y la posibilidad de extender sus características esenciales a un contexto no clásico de lógicas no bivalentes. Para ello se comparan diferentes operadores de negación no clásicos, enfocándose particularmente en operadores de negación de la familia de lógicas paraconsistentes, sus características técnicas y sus respectivas interpretaciones filosóficas. Finalmente se brinda un sistema cuatrivaluado basado en el conjunto FOUR con un modelo semántico basado en un orden total que induce nuevas valuaciones solucionando algunos problemas señalados a la lógica relevante FDE. Se introduce en el mismo una negación paraconsistente que cumple con los criterios mínimos de negación estricta no clásica y puede funcionar como conmutador en un sistema de cuatro valores de verdad.

We single out a type of set-theoretical paradoxes, following the famous
Bertrand Russell scheme which was recently spotlit by Graham Priest. It is the
formula of “prohibited words” from Dino Buzzati’s 1958 short story bearing
the same name. We show up that the not less famous Palamite distinction
between the Divine essence and energy satisfi es the formula conditions, the
same being true for its correspondence with the Russell scheme. Seen from
the set-theoretical viewpoint, the Palamite prohibition to name the Divine
essence is structurally isomorphic to the “prohibited words” depicted by the
great Italian postmodernist writer. Hereby the set of Divine names stemming
from His energies corresponds in Palamism to the domain of “normal” words
in Buzzati, i.e., of those words, expressions, and idioms which are made use
of in the ordinary language.

The epsilon calculus contains terms of the form 'εxFx' for every predicate in the language.  This means that it includes what I shall call 'empty' terms (when there are no Fs) and also what I shall call 'indexical' terms (when there is more then one F).  These particular terms give the epsilon calculus a very distinctive character in comparison with the standard predicate calculus.  For instance, an 'empty' term is central to understanding how we can do things that Russell and Whitehead's Principia Mathematica cannot do, in connection with Gödel's Theorems.  And attention to 'indexical' terms is crucial to solving the major Logical Paradoxes that have been a puzzle for over a century.  Most particularly they are crucial to the solution of what has been called 'Gödel's Paradox', which has been claimed to show that natural language is
paraconsistent.

Bunge's widely known so-called refutation of dialectics is shown to fail. Once unreasonable logical constraints are waived, while some non-classical logical approaches are taken seriously, Bunge's allegations emerge as groundless. In particular, the paper argues that there are interesting versions of the dialectical principle of the unity of opposites which can be implemented and formalized through some paraconsistent logics, esp. those belonging to the family of fuzzy logics.

RESUMEN.-
En su libro Materialismo y ciencia dedica Bunge el cap. 4º a una crítica de la dialéctica. En esta comunicación someto a discusión esa crítica. Mi conclusión es que la crítica falla; porque algunos de los argumentos de Bunge, siendo certeros, no consiguen dar en el blanco, ya que sus referencias son globales y no aporta evidencia textual que avale la atribución de las tesis que él refuta como formando parte del bagaje de las ideas dialécticas; y, en cambio, otros de sus argumentos son de hecho inválidos o contienen premisas erróneas.
El meollo de mi discusión con Bunge estriba en que él sobreentiende que la lógica, la única lógica desde la cual quepa enjuiciar a las teorías, es y ha de ser la lógica clásica. Pues bien, aunque muchos de sus argumentos tienen fuerza desde esa presuposición, algunos de ellos la pierden cuando se plantean los problemas desde otro ángulo, que la moderna investigación lógico-matemática ha revelado perfectamente viable, cual es el de una lógica paraconsistente.


PALABRAS CLAVE.- Mario Bunge, dialéctica, contradicción, unidad de los opuestos, lógica paraconsistente, lógica clásica

KEYWORDS.- Mario Bunge, dialectics, contradiction, unity of oposites, paraconsistent logic, classical logic

Philosophical dialetheism, whose main exponent is Graham Priest, claims that some contradictions hold, are true, and it is rational to accept and assert them. Such a position is naturally portrayed as a challenge to the Law of Non-Contradiction (LNC). But all the classic formulations of the LNC are, in a sense, not questioned by a typical dialetheist, since she is (cheerfully) required to accept them by her own theory. The goal of this paper is to develop a formulation of the Law which appears to be unquestionable, in the sense that the Priestian dialetheist is committed to accept it without also accepting something inconsistent with it, on pain of trivialism—that is to say, on pain of lapsing into the position according to which everything is the case. This will be achieved via (a) a discussion of Priest's dialetheic treatment of the notions of rejection and denial; and (b) the characterization of a negation via the primitive intuition of content exclusion. Such a result will not constitute a cheap victory for the friends of consistency. We may just learn that different things have been historically conflated under the label of 'Law of Non-Contradiction'; that dialetheists rightly attack some formulations of the Law, and orthodox logicians and philosophers have been mistaken in assimilating them to the indisputable one.

Ambivalence (as in practical conflicts, moral dilemmas, conflicting beliefs, and mixed feelings) is a central phenomenon of human life. Yet ambivalence is incompatible with entrenched philosophical conceptions of personhood, judgement, and action, and is denied or marginalised by thinkers of diverse concerns. This book takes a radical new stance, bringing the study of core philosophical issues together with that of ambivalence. The book proposes new accounts in several areas – including subjectivity, consciousness, rationality, and value – while elucidating a wide range of phenomena expressive of ambivalence, from emotional ambivalence to self-deception. The book rejects the view that ambivalence makes a person divided, showing that our tension-fraught attitudes are profoundly unitary. Ambivalence is not tantamount to confusion or to paralysis: it is always basically rational, and often creative, active, and perceptive as well. The book develops themes from Wittgenstein, Davidson, Sartre, and Freud. It engages with contemporary debates in Analytic Philosophy in addition to work ranging from Aristotle to Cultural Studies and Empirical Psychology, and considers a rich set of examples from daily life and literature.

Since its first appearance in 1966, the notion of a supervaluation has been regarded by many as a powerful tool for dealing with semantic gaps. Only recently, however, applications to semantic gluts have also been considered. In previous work [4:1991, 4:1994a] I proposed a general framework exploiting the intrinsic gap/glut duality. Here I also examine an alternative account where gaps and gluts are treated on a par: although they reflect opposite situations, the semantic upshot is the same in both cases—the value of some expressions is not uniquely defined. Other strategies for generalizing supervaluations are considered and some comparative facts are discussed.

Aristotle grants that two mutually contradictory properties can be simultaneously present in the same subject to some extent. However in order to keep clear of recognizing true contradictions, he falls back on the dubious manoeuver of regarding intermediaries as as entirely different in kind from the extremes, rather than as lesser degrees of the properties fully realized at the extremes.

We single out a type of set-theoretical paradoxes, following the famous Bertrand Russell scheme which was recently spotlit by Graham Priest. It is the formula of “prohibited words” from Dino Buzzati’s 1958 short story bearing the same name. We show up that the not less famous Palamite distinction between the Divine essence and energy satisfi es the formula conditions, the same being true for its correspondence with the Russell scheme. Seen from the set-theoretical viewpoint, the Palamite prohibition to name the Divine essence is structurally isomorphic to the “prohibited words” depicted by the great Italian postmodernist writer. Hereby the set of Divine names stemming from His energies corresponds in Palamism to the domain of “normal” words in Buzzati, i.e., of those words, expressions, and idioms which are made use of in the ordinary language.

Inconsistency toleration is the phenomenon of working with inconsistent information without threatening one's rationality. Here I address the role that ignorance plays for the tolerance of contradictions in the empirical sciences. In particular, I contend that there are two types of ignorance that, when present, can make epistemic agents to be rationally inclined to tolerate a contradiction. The first is factual ignorance, understood as temporary undecidability of the truth values of the conflicting propositions. The second is what I call "ignorance of theoretical structure", which is lack of knowledge of relevant inference patterns within a specific theory. I argue that these two types of ignorance can be explanatory of the sci-entists' rational disposition to be tolerant towards contradictions, and I illustrate this with a case study from neutrino physics.

Paraconsistent Belief Revision based on a formal consistency operator (in portuguese)

Belief Revision studies how rational agents change their beliefs when they receive new information. The AGM system, most influential work in this area of study investigated by Alchourrón, Gärdenfors and Makinson, postulates rationality criteria for different types of belief change and provides explicit constructions for them -- the equivalence between the postulates and operations is called representation theorem. Recent studies show how the AGM paradigm can be compliant with different non-classical logics, which is called the AGM-compliance -- this is the case of the paraconsistent logics family we analyze in this thesis, the Logics of Formal Inconsistency (LFIs).

Despite the AGM-compliance, when a new logic is taken into account its underlying rationality must be understood and its language should be used. In that way new constructions are proposed, which actually captures the intuition of LFIs -- what we call the AGMo system. Thus, we provide a new interpretation for these logics, more in line with formal epistemology. In an alternative approach, by considering the AGM-compliance, we show how the AGM results can be directly applied to LFIs -- resulting the AGMp system. In both approaches, we prove the corresponding representation theorems where needed.

Key-words: belief revision, non-classical logics, formal epistemology, paraconsistent logic, theory dynamics, logical consistency, logical contradicition.


Resumo

A Revisão de Crenças estuda como agentes racionais mudam suas crenças ao receberem novas informações. O sistema AGM, trabalho mais influente desta área apresentado por Alchourrón, Gärdenfors e Makinson, postula critérios de racionalidade para os diferentes tipos de mudança de crenças e oferece construções explícitas para tais -- a equivalência entre os postulados e operações é chamado de teroema da representação. Trabalhos recentes mostram como o paradigma AGM pode ser compatível com diferentes lógicas não-clássicas, o que é chamado de AGM-compatibilidade -- este é o caso da família de lógicas paraconsistentes que analisamos, as Lógicas da Inconsistência Formal (LFIs, da sigla em inglês).

A despeito da AGM-compatibilidade, ao se partir de uma nova lógica sua racionalidade subjacente deve ser entendida e sua linguagem deve ser efetivamente usada. Propomos assim novas construções que de fato capturam a intuição presente na LFIs -- é o que chamamos de sistema AGMo. Com isso, possibilitamos a estas lógicas uma nova interpretação, na esteira da epistemologia formal. Em uma abordagem alternativa, ao se partir da AGM-compatibilidade os resultados AGM podem ser diretamente aplicados às LFIs -- o que chamamos de sistema AGMp. Em ambas abordagens, provamos os respectivos teoremas da representação sempre que necessário.

Palavras-chave: revisão de crenças, lógicas não-clássicas, epistemologia formal, lógica paraconsistente, dinâmica de teorias, consistência lógica, contradição lógica.

A non-monotonic logic, the Logic of Plausible Reasoning (LPR), capable of coping with the demands of what we call complex reasoning, is introduced. It is argued that creative complex reasoning is the way of reasoning required in many instances of scientific thought, professional practice and common life decision taking. For managing the simultaneous consideration of multiple scenarios inherent in these activities, two new modalities, weak and strong plausibility, are introduced as part of the Logic of Plausible Deduction (LPD), a deductive logic specially designed to serve as the monotonic support for LPR. Axiomatics and semantics for LPD, together with a completeness proof, are provided. Once LPD has been given, LPR may be defined via a concept of extension over LPD. Although the construction of LPR extensions is first presented in standard style, for the sake of comparison with existing non-monotonic formalisms, alternative more elegant and intuitive ways for constructing non-monotonic LPR extensions are also given and proofs of their equivalence are presented.

Paraconsistent logics are logics which allow solid deductive reasoning under contradictions by offering a mathematical and philosophical support to contradictory yet non-trivial theories.

Due to its role in models of scientific reasoning and to its philosophical implications, as well as to its connections to topics such as abduction, automated reasoning, logic programming, and belief revision, paraconsistency has becoming a fast growing area.

The present volume, edited by Jean-Yves Beziau, Walter Carnielli and Dov Gabbay, expert logicians versed on heterodosx logics, originated around the III World Congress on Paraconsistency (WCP3) held in Toulouse, France, in July, 2003. It contains the most recent results on several aspects of paraconsistent logic, including philosophical debates on paraconsistency and its connections to philosophy of language, argumentation theory, computer science, information theory, and artificial intelligence.

The book is a basic tool for those who want to know ore about paraconsistent logic, its history and philosophy, the various systems of paraconsistent logic and their applications.

The present volume is edited by Jean-Yves Beziau, Walter Carnielli and Dov Gabbay, expert logicians versed in a wide variety of topics.

ISBN 978-1-904987-73-4

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational efficiency. A prototype based on this second system was effectively implemented.

In his early Reportatio, Walter Chatton (1290--1343) makes a proposal that appears to be a procedure based on paraconsistent logic games. Walter's procedure (as I interpret it) is able to determine whether one of a pair of opposite propositions is more compatible with another proposition. In particular, Walter believes his procedure can determine which of a pair of counter-possible propositions is more compatible with a selected factual proposition. Shortly after, Adam Wodeham (1298--1358) finds a crucial flaw in Walter's proposal. Apparently persuaded by Adam's critique, Walter weakens his view in his later Lectura. On the weakened view, Walter's procedure cannot handle counter-possible propositions. But it can determine whether one of a pair of opposite factual or counter-factual propositions is more compatible with a selected factual proposition.

In this paper two systems of AGM-like Paraconsistent Belief Revision
are overviewed, both defined over Logics of Formal Inconsistency
(LFIs) due to the possibility of defining a formal consistency
operator within these logics.

The AGM° system is strongly based on this operator and internalize the
notion of formal consistency in the explicit constructions and
postulates. Alternatively, the AGMp system uses the AGM-compliance of
LFIs and thus assumes a wider notion of paraconsistency – not
necessarily related to the notion of formal consistency.

El sistema LBVF se construye como una extensión de la lógica clásica positiva, al incluir operadores de afirmación alterna, negación alterna y completez, además, se definen a partir de ellos los operadores de negación clásica y buena fundamentación. El sistema es caracterizado por una semántica de valuaciones tradicionales con la cual se prueba que, respecto a los operadores de afirmación y de negación alterna el sistema es para completo. En el sistema se caracterizan las definiciones de verdad y falsedad presentadas por Aristóteles, representando la falsedad aristotélica con el operador de negación alterna y la verdad aristotélica con el operador de afirmación alterna, lográndose con esta interpretación dar solución a una variante de la paradoja del mentiroso.

MSC: 03BXX, 03B53, 03B42

The aim of this paper is to apply synthetic tableaux method (STM) to the paraconsistent logic CLuN, developed by Diderik Batens. Soundness and completeness of STM with respect to CLuN semantics are proved. It is also shown how to interpret CLuN negation in terms of relations represented by the square of oppositions of traditional syllogistic.

El libro constituye un ejercicio de lógica aplicada a los problemas de la teología filosófica, empleando una lógica contradictorial o paraconsistente, que admite la posibilidad de contradicciones verdaderas. La primera sección se dedica a la fenomenología de la religión. La segunda examina  las determinaciones contradictorias del concepto de Dios, definiendo `Dios' como un ser al que no falta ninguna perfección o realidad. A la luz de las discusiones de la filosofía analítica, se examinan las dos tesis, teísmo y ateísmo. Se propone abordar ese concepto desde una lógica gradualista y una teoría no-estándar de conjuntos difusos en la cual ciertos seres desbordan el ámbito de aplicabilidad del principio de comprensión.

PALABRAS-CLAVE:
filósofos analíticos, lógica, ontología modal, posible, necesario, contingente, David Lewis, S. Kripke, teología filosófica, lógica paraconsistente, contradicciones verdaderas, concepto de Dios, perfección, lógica contradictorial, lógica gradualista, contradicciones parciales, principio de comprensión, teoría axiomática de conjuntos, inteligibilidad racional, teísmo, ateísmo

My paper is a brief presentation of conjunctive paraconsistency, that is the idea that there might be true contradictions without trivialism, but two contradictory terms cannot be (and cannot be considered) both true. Such kind of paraconsistency implies the rejection of any dialetheia, intended as a true proposition whose negation is true, and hence the acceptance of the Law of Non-Contradiction in its alethic-distributed form: 'there cannot be any true-and-false proposition', which I call the Law of Non-Dialetheia.
The conjunctivist approach is symmetrically opposed to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true, and its negation is true too, but the conjunction is not true. While non-adjunctive accounts are quite widespread nowadays, and dialetheism is a very famous paraconsistent logic, there is no clear and definite conjunctivist position in recent literature, though such kind of paraconsistency was not unusual in the tradition: Hegel is alleged to be one of the upholders, and possibly Aristotle himself endorsed a similar position.

The AGM model is the most remarkable framework for modeling belief revision. However, it is not perfect in all aspects. Paraconsistent belief revision, multi-agent belief revision and non-prioritized belief revision are three different extensions to AGM to address three important criticisms applied to it. In this article, we propose a framework based on AGM that takes a position in each of these categories. Also, we discuss some features of our framework and study the satisfiability of AGM postulates in this new context. KEYWORDS belief revision, the AGM model, multi-source belief revision, non-prioritized belief revision, para-consistent belief revision, the logic PAC