Paraconsistent logic

Hay sufrimiento en el mundo, hay mal en el mundo. Existe evidencia de ambos. No todo sufrimiento parece ser con vistas a un bien mayor. Es decir, parece haber casos de mal gratuito o injustificado. El pasado 2 de febrero del 2024, observamos en Viña del Mar -Chile las trágicas consecuencias de un devastador incendio forestal, probablemente provocado intencionalmente por el ser humano. Ante escenarios de este tipo, el desafío epistémico para quienes creemos en Dios es preguntarnos si este es un caso de mal innecesario y si, de ser así, tiene alguna implicancia para la creencia en Dios. En este escrito plantearé los riesgos de racionalizar el sufrimiento, sin que ello implique que la búsqueda de razones sobre el mismo (su explicación) derive en una justificación indirecta del mismo. En última instancia, se trata de distinguir entre evaluar las causas de un evento, la justificación del evento y la evaluación ética del mismo.

This paper investigates the proof-theoretic foundations of double negation introduction (DNI: A ⊢ ¬¬A) and double negation elimination (DNE: ¬¬A ⊢ A) in classical logic. 1 By examining both sequent calculus and natural deduction, it is shown that these rules originate in reductio ad absurdum (RAA): DNI results from deriving ¬¬A via ¬I by discharging [¬A], while DNE arises from deriving A through a reductio on [¬A]. 2 Their significance extends beyond semantic equivalence, for DNI and DNE embody the identity relation A ⟷ ¬¬A as a structural principle of classical logic. The paper demonstrates that both rules possess harmony, ensuring balance between introduction and elimination, and normalisation, which guarantees that derivations reduce to canonical form without detours. These features reveal double negation not as a redundancy, but as a mechanism of proof-theoretic stability, securing the disciplined integration of RAA into classical logic.

The article explores the shift from classical binary logic (true and false) to a more complex landscape of logical frameworks. It introduces logical pluralism, which suggests that no single, universally accepted system of logic exists. The paper connects this idea to neutrosophy, a framework that extends fuzzy logic by incorporating a third value: indeterminacy. This triadic approach, where every proposition has a degree of truth (T), falsity (F), and indeterminacy (I), is particularly useful for managing incomplete or contradictory information. The author argues that different logical systems, like different tools, are suited for different problems, and that the search for a single "one true logic" is less important than recognizing the diverse tools available for navigating the complexities of reasoning.

This paper introduces a formal framework for Conversational Game Theory (CGT), a multi-agent paradigm integrating computational, cognitive, and psychological dimensions to achieve dynamic equilibrium in complex dialogues and any consensus type exercise.

As a content management system for community consensus publication, this means that CGT publishes content through an interface type of game that turns conflict into a “win-win” editing sport.

CGT defines equilibrium as a state where no participant can unilaterally improve their conversational utility—balancing contradiction resolution, logical coherence, and emotional satisfaction—without incurring trade-offs.
By leveraging a structured computational backbone, a paraconsistent cognitive framework grounded in a ternary logic (0, 1, 2), psychological engagement modeling, and a reward mechanism that incentivizes collaborative contributions, CGT provides a robust and scalable approach for guiding multi-agent interactions toward stable, constructive, and evolving outcomes.

Conversations culminate in “compositions,” published under “Contextual Completeness,” contributing to the Global Resolution, Alignment, and Inquiry Library (GRAIL).

This study presents a new Model Predictive Controller (MPC), built with algorithms based on Paraconsistent Annotated Logic (PAL), with application examples in the excitation control of a synchronous generator. PAL is a non-classical evidential and propositional logic that is associated with a Hasse lattice, and which presents the property of accepting the contradiction in its foundations. In this research, the algorithm was constructed with a version of the PAL that works with two information signals in the degrees of evidence format and, therefore, is called Paraconsistent Annotated Logic with annotation of two values (PAL2v). For the validation of the algorithmic structure, the computational tool MATLAB® Release 2012b, The MathWorks, Inc., Natick, MA, United States was used. Simulations were performed which compared the results obtained with PPC-PAL2v to those obtained in essays with the AVR (Automatic Voltage Regulator) controls in conjunction with the PSS (Power System Stabilize...

In the end of the analysis it is found a consensus value. In this work we presented numeric example and one example of application of the ParaExtr ctr in Load Profile Forecast used in support to decision of the operation in an Electric Power System, but his application potentiality is demonstrated in several fields of the Artificial Intelligence.

The article undertakes the issue of whether logic is empirical, that is, whether it is possible to modify logic due to empirical reasons. We will call this issue the problem of the empiricality of logic. Hilary Putnam's famous argument based on discoveries in quantum mechanics is discussed. Putnam claimed that classical logic should be discarded and replaced by quantum logic. This view is called the "Putnam's claim." It was met with harsh criticism from Dummett and Kripke, among others. While their criticism is largely valid, it does not dismiss the idea that logic itself could be empirical. The aim of this paper is to show that it is possible to present the discovery of the nondistributivity of quantum propositions as an empirical fact with implications for the revision of the adequacy of certain laws of logic, and thus, that empirical facts can have an impact on logic.

В издании представлены статьи по материалам докладов второй международной научной конференции «Византия, Европа, Россия: социальные практики и взаимосвязь духовных традиций», прошедшей 22-24 сентября 2022 г. в Социологическом институте РАН -филиале ФНИСЦ РАН. Конференция была посвящена интеллектуальной истории трех основных комплексов практик цивилизационного развития в европейском, средиземноморском и евро-азиатском регионах в исторических истоках их возникновения и существования -Византийской, Западноевропейской и Русской цивилизаций, а также цивилизаций, входящих в ареалы их влияния и взаимодействия. Статьи адресованы специалистам по философии, социологии, теологии, истории, филологии, политическим наукам, искусствоведению.

The liar and kindred paradoxes show that we can derive contradictions if our language possesses sentences lending themselves to paradox and we reason classically from schema (T) about truth: S is true iff p, where the letter p is to be replaced with a sentence and the letter S with a name of that sentence. This paper presents a theory of truth that keeps (T) at the expense of classical logic. The theory is couched in a language that possesses paradoxical sentences. It incorporates all the instances of the analogue of (T) for that language and also includes other platitudes about truth. The theory avoids contradiction because its logical framework is an appropriately constructed nonclassical propositional logic. The logic and the theory are different from others that have been proposed for keeping (T), and the methods used in the main proofs are novel.

To achieve a deeper clarification of the ontology of consciousness, perception, and logic, both classic and modern paradoxes are subjected to a systematic analysis. It is demonstrated that all genuine paradoxes can be traced back to a single fundamental schema-the paradox of the boundary. This is not an isolated special case but the structural core of all paradoxes: the irresolvable simultaneity of unity and difference. From this, far-reaching consequences arise for the theory of consciousness, for the episteme, and for the nature of the "thing-in-itself" in the sense of Immanuel Kant.

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of using the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed.

According to exceptionalism about logic, logic is special, it enjoys a privileged position among the sciences, its method is not the method of the natural sciences, and the knowledge it provides is different in kind from the knowledge provided by the natural sciences. According to anti-exceptionalism about logic, instead, logic isn't special, its method is continuous with the method of the natural sciences, logical theories are revisable, and when they are revised, they are revised on the same grounds on which scientific theories are revised. The heuristic view of logic can be regarded as an anti-exceptionalist stance on logic since it too conceives of logic as continuous with the natural sciences. Nevertheless, there are at least two main differences between the heuristic view of logic and standard formulations of anti-exceptionalism about logic: 1) anti-exceptionalists share with exceptionalists the idea that logic is the study of logical consequence, while according to the heuristic view of logic, logic should be understood as a logic of discovery; 2) antiexceptionalists usually claim that the method of logic is abductive, while according to the heuristic view of logic the method of logic is not abductive. It will be argued that the heuristic view of logic provides a more adequate anti-exceptionalist view on logic than standard formulations of anti-exceptionalism about logic.

It has been shown that the rules of logic for the principle 'Ex Contradictione quod libet' (ECQ) do not cause U8 to explode. This is because the antecedent is non-designated and modus ponens blocks detachment for the conclusion. Therefore, ECQ is not a theorem of U8, however, ECQ is acknowledged to be a hypothetical theorem. This leaves the axioms disjunctive introduction and the disjunctive syllogism intact for U8, unlike other paraconsistent logics. Using traffic light signals, an example showing paraconsistent behaviour is given for the U8 AND operation. It can be concluded U8 is a paraconsistent null logic system.

This paper introduces the Pre-Creation Memory Hypothesis (PCMH), a framework for understanding reports of human recollections that seemingly precede the creation of the current cosmos. Unlike conventional memory, which is temporally bound to biographical experience, PCMH proposes that consciousness can generate or access paradoxical states experienced as memories of primordial events, including the destruction and recreation of worlds. These experiences are phenomenologically characterized by absolute conviction, ineffability, and paradoxical simultaneity of clarity and obscurity. Drawing on sources from philosophy (Plato's anamnesis), theology (Genesis 1:2; Qur'an 7:172; Kabbalistic covenantal memory), and cognitive science (false memory, déjà vu, neardeath experiences), the paper situates PCMH within a cross-disciplinary dialogue. Cognitively, such experiences may reflect the brain's constructive simulation capacities operating at archetypal extremes. Theologically, they function as testimony (martyria) in which individuals bear witness to creation itself. Philosophically, they destabilize binary categories of time, truth, and memory, revealing paradox as structural rather than accidental. While unverifiable in empirical terms, PCMH highlights the need to expand current models of memory, reconsider theological categories of revelation, and re-evaluate paradox as a legitimate form of knowledge. The study concludes that pre-creation memory should be treated as a unique class of human experience, illuminating the intersection of cognition, theology, and metaphysics.

Contradiction has long been treated as an error by classical logic and modern artificial intelligence, yet paradox persists across philosophy, theology, science and daily cognition. Here we introduce Quantum-Inspired Cognitive Logic Systems (QCLS), a framework that formalizes paradox as structural essence rather than noise. QCLS draws from quantum logic, paraconsistent reasoning and cognitive neuroscience to model utterances as superpositions in semantic space, quantify contradiction through a Contradiction Index, and sustain contradictory states without collapse. We propose empirical validation via paradox-specific neural signatures—including theta–gamma oscillatory coupling, hippocampus–prefrontal connectivity, and a Neural Contradiction Index (nCI)—linking mathematical structure with brain dynamics. Beyond theory, QCLS suggests architectures for paradox-preserving AI, expanding current coherence-driven models toward theology, therapy and creativity. By positioning paradox as a measurable and generative invariant, QCLS reframes human cognition and points to a new epistemic paradigm where truth appears not in binary resolution but in superposed contradiction.

A pesquisa abordou o processo de tomada de decisão de gestores utilizando-se de algoritmo para-analisador baseado em lógicas não-clássicas: a lógica paraconsistente anotada (LPA). O método é apropriado ao tratamento de dados incertos, contraditórios ou paracompletos; consiste em estabelecer proposições e parametrizá-las para isolar os fatores de maior influência nas decisões. Especialistas são utilizados para obtenção de anotações sobre esses fatores, atribuindo-lhes graus de crença e descrença. No caso analisado, utilizou-se da LPA como instrumento de apoio ao processo de avaliação e seleção de profissional a ser promovido a gerente, dentre cinco possíveis, em pequena empresa familiar. Os três sócios-gerentes da empresa constituíram o grupo de especialistas, na medida em que conhecem o valor profissional dos cinco funcionários. A interpretação das avaliações realizadas pelos especialistas deu-se por intermédio dos baricentros no quadrado unitário do plano cartesiano (QUPC), que ind...

In this chapter we explore the topic of logical disagreement. Though disagreement in general has attracted widespread philosophical interest, both in epistemology and philosophy of language, the general issues surrounding disagreement have only rarely been applied to logical disagreement in particular. Here, we develop some of the fascinating semantic and epistemological puzzles to which logical disagreement gives rise. In particular, after distinguishing between different types of logical disagreement, we explore some connections between logical disagreements and deep disagreements over fundamental epistemic principles; we discuss several semantic puzzles that arise on various accounts of the meanings of logical terms; we investigate how such disagreements relate to Kripke’s so-called “Adoption Problem”; and we probe epistemological puzzles that arise from disagreements about logic in the light of central principles from the peer disagreement literature.

No dia 8 de agosto de 1900, Hilbert apresentou ao 2º Congresso Internacional de Matemáticos, reunido em Paris, uma lista de dez dentre 23 grandes problemas matemáticos ainda sem solução. Hilbert não duvidava que essas soluções existiam, bastando para encontrá-las a dose exata de esforço e engenho. O segundo problema da lista pedia que se demonstrasse a "compatibilidade dos axiomas aritméticos". Assim posto o problema carece da necessária clareza matemática, pois: 1) A que aritmética se referia Hilbert?; 2) Por que seriam necessárias demonstrações de consistência de teorias verdadeiras, como são, supõese, as aritméticas usuais dos números naturais, racionais e reais? (Haveria alguma razão para se duvidar que elas fossem, de fato, verdadeiras?) 3) Que ferramentas matemáticas seriam admissíveis nas demonstrações de consistência; ou seja, essas demonstrações deveriam ser levadas a cabo em que contexto matemático? Eu começo este artigo discutindo essas questões. Ainda que Hilbert tivesse em mente a aritmética dos números reais, nós aqui nos contentaremos em analisar exclusivamente o problema da consistência da aritmética dos inteiros não-negativos, e isso dá conta da primeira questão. Já a segunda requer uma distinção entre teorias propriamente ditas e teorias puramente formais 1 . A aritmética contentual, isto é, a teoria axiomática dos números inteiros não negativos (ou números naturais) é evidentemente uma teoria consistente, simplesmente por ser a teoria de um domínio dado de objetos, os números naturais. A consistência dessa aritmética está, assim, na dependência de uma intuição capaz de nos fornecer os objetos da teoria dos números. Ou, dito de outra forma, a consistência da aritmética contentual é garantida por uma intuição que, pressupõe-se, tem a capacidade de nos oferecer uma teoria verdadeira. Sendo verdadeira, a aritmética contentual é, a fortiori, consistente. A aritmética formal, entretanto, não é uma teoria de nenhum domínio pré-dado de objetos; logo, não é em nenhum sentido próprio nem verdadeira, nem falsa 2 . Cabe-lhe apenas descrever uma estrutura formal, cuja realidade está sub judice. Porisso o problema de sua consistência é tão importante. Trata-se de demonstrar que a estrutura formal que a teoria descreve é uma estrutura possível, ou seja, é a estrutura de um domínio possível de objetos. 1 Frege teve dificuldade em perceber essa distinção. Porisso nunca chegou a entender realmente por que Hilbert insistia na demonstração da consistência da aritmética. 2 Cabe aqui a "definição" de matemática dada por Russell: o discurso em que não sabemos do que falamos, nem se o que falamos é verdadeiro.

Logical matrices are widely accepted as the semantic structures that most naturally fit the traditional approach to algebraic logic. The behavioral approach to the algebraization of logics extends the applicability of the traditional methods of algebraic logic to a wider range of logical systems, possibly encompassing many-sorted languages and non-truth-functional phenomena. However, as one needs to work with behavioral congruences, matrix semantics are unsuited to the behavioral setting. In , a promising version of algebraic valuation semantics was proposed in order to fill in this gap. Herein, we define the class of valuations that should be canonically associated to a logic, and we show, by means of new meaningful bridge results, how it is related to the behaviorally equivalent algebraic semantics of a behaviorally algebraizable logic.

The theory of abstract algebraic logic aims at drawing a strong bridge between logic and universal algebra, namely by generalizing the well known connection between classical propositional logic and Boolean algebras. Despite of its successfulness, the current scope of application of the theory is rather limited. Namely, logics with a many-sorted language simply fall out from its scope. Herein, we propose a way to extend the existing theory in order to deal also with many-sorted logics, by capitalizing on the theory of many-sorted equational logic. Besides showing that a number of relevant concepts and results extend to this generalized setting, we also analyze in detail the examples of first-order logic and the paraconsistent logic C1 of da Costa.

Traditionally, a logic program is built up to reason about atomic first-order formulas. The key idea of parametrized logic programming is that, instead of atomic first-order formulas, a parametrized logic program reasons about formulas of a given parameter logic. Of course, the main challenge is to define the semantics of such general programs. In this work we introduce the novel definitions along with some motivating examples. This approach allows us to prove general results that can be instantiated for every particular choice of the parameter logic. Important general results we can prove include the existence of semantics and the alternating fix-point theorem of well-founded semantics. To reenforce the soundness of our general framework we show that some known approaches in the literature of logic programming, such as paraconsistent answer-sets and the MKNF semantics for hybrid knowledge bases, are obtained as particular choices of the parameter logic.

The class of so-called non-truth-functional logics constitutes a challenge to the usual algebraic based semantic tools, such as matrix semantics [ LS58]. The problem with these logics is the existence of non-congruent connectives that are not always interpreted ...

Belief Revision is a central area of research in Formal Philosophy that employs logic to model how human and artificial agents revise their beliefs in response to new information, examining the conditions under which such revisions may be considered rational. In this talk, I present a novel epistemic interpretation of Paraconsistency, grounded in Paraconsistent Belief Revision systems, in which the notions of (in)consistency are framed in terms of the epistemic attitudes of rational agents.
Among its contributions, this approach challenges the traditional assumption that accepting a new belief-representing sentence necessarily entails the retraction of its negation from the current epistemic state. Such automatic retraction conflicts with the rationality principle of informational economy, which plays a key role in epistemic agency. From this perspective, Paraconsistency is reexamined as a legitimate and rational response to conflicting information - a natural feature in the dynamics of belief. This epistemic standpoint also offers new insights into the structure and motivations behind the so-called Logics of Formal Inconsistency (LFIs), which provide the underlying framework for the present modeling.
Recent developments are also surveyed, including the proposal of a specific LFI tailored to better capture these intuitions: the Cbr system, along with its self-extensional extension, the RCbr system. These systems allow for a refined treatment of epistemic entrenchment in paraconsistent contexts, wherein strongly accepted beliefs - those both accepted and consistent - are aligned with the most entrenched beliefs of the agent. Furthermore, this framework makes it possible to interpret the attitude of accepting the consistency of a belief-representing sentence (without accepting the sentence itself) as a form of revision in the epistemic strength of that belief.

The historical development of the doctrine of the Trinity evidences an apparent contradiction. On the one hand, the doctrine says that there are three divine beings, which apparently implies that there are three gods. On the other hand, it declares that there is only one God. This problem has also been interpreted in terms of identity: f=g, s=g, and h=g, but f¹s, f¹h, and s¹h, which means that they are identical and not identical. Contradictions are problematic for many people. So, what to do with this contradiction? Some people have tried to solve it affirming one side of the contradiction and denying the other. Others have tried to dissolve it, i.e., they have attempted to show that there is no contradiction at all. In this paper, I propose a third alternative. To accept the contradiction. Taking this as my starting point, I develop two dialetheist models of the Trinity. These models are based on Priest's gluon theory and are inspired by Jeffry Pelletier's book review of Priest's book One. Through it, I explain how the trinitarian contradiction is possible by showing either how many things can form a unity or how three things that are identical to another are different from each other. At the end of the paper, I present some objections to these models and suggest, first, that metaphysical models are not as unnecessary as Beall makes them appear, and second, that if the gluon models that I present do not work, we probably must accept mysterianism. Unless another freetrouble, non-transitive-based model is offered.

Scientific inquiry is typically focused on particular questions about particular objects and properties.  This leads to a multiplicity of models which, even when they draw on a single, consistent body of concepts and principles, often employ different methods and assumptions to model different systems.  Pluralists have remarked on how scientists draw on different assumptions to model different systems, different aspects of systems and systems under different conditions and defended the value of distinct, incompatible models within science at any given time. (Cartwright, 1999; Chang, 2012) Paraconsistentists have proposed logical strategies to avoid trivialization when inconsistencies arise by a variety of means.(Batens, 2001; Brown, 1990; Brown, 2002)  Here we examine how chunk and permeate, a simple approach to paraconsistent reasoning which avoids heterodox logic by confining commitments to separate contexts in which reasoning with them is taken to be reliable while allowing ‘perm...

In "The Rule of Adjunction and Reasonable Inference" ~Kyburg, 1997! Henry Kyburg returns to a topic that has been a concern of his for more than 35 years ~Kyburg 1961, 1970!. Like anyone who approaches epistemological concerns from a probabilistic perspective, Kyburg is worried about how the logically trivial process of forming larger and larger conjunctions erodes high probabilities. The lottery paradox is the locus classicus of this problem. It shows that any rule that makes having a high enough probability sufficient for acceptance will run into inconsistency. By constructing a fair lottery with enough tickets we can give each sentence, T i , saying of ticket i that it will lose, a high enough probability to make it acceptable. By taking sufficient care to ensure the lottery is fair, we can also give the sentence "One of the tickets will win." a high enough probability. But these sentences are jointly inconsistent. Kyburg's response, unlike most others, is to tolerate this inconsistency, retain such a rule of acceptance, and reject the disease he playfully calls "conjunctivitis," the closure of the set of accepted claims under conjunction. This response, while it frees him to be true to his probabilistic roots, has a cost. The set of accepted claims is usually assumed to be closed under logical consequence. This makes it ~in the logical sense! a theory, and ~more importantly! expresses clearly the content of all that has been accepted: Any such closed set includes all and only those sentences that must be true if all the explicitly accepted sentences are. So this assumption makes it a straightforward matter to identify the implicit commitments of someone who explicitly accepts some set of sentences. If someone isn't committed to the logical consequences of what they've accepted, what ~one wonders! are they committed to? What would it take to show them they were wrong about something? Still, the gain is substantial; in fact, I believe it is more substantial than the cost. His position allows Kyburg's account of acceptance to hew closer to the NOÛS 33:2 ~1999! 273-283

Slater has argued that there cannot be any truly paraconsistent logics, because it's always more plausible to suppose whatever "negation" symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slater's argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics -the preservationist logics -are not subject to this objection. In addition I show that if we identify logics by means of consequence relations, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic) terms. Thus the interest of paraconsistent consequence relations -even those that emerge from dialetheic approaches -does not depend on the tenability of dialetheism. Of course, if dialetheism is defensible, then paraconsistent logic will be required to cope with it. But the existence (and interest) of paraconsistent logics does not depend on a defense of dialetheism.

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind -specifically concerning the preservation of the consistency of each chunkand concludes with some other possible applications and technical questions.

Nicholas Rescher has argued we must tolerate inconsistency because of our cognitive limitations. He has also produced, together with R. Brandom, a serious attempt at exploring the logic of inconsistency. Inconsistency tolerance calls for a systematic rewriting of our logical doctrines: it requires a paraconsistent logic. However, having given up all aggregation of premises, Rescher's proposal for a paraconsistenl logic fails to account for the reductive reasoning Rescher appeals to in his account of inconsistency tolerance. A non-adjunctive logic developed by P.K. Schotch and RE. Jennings provides just what Rescher's logic is lacking: It allows a tolerant attitude toward inconsistency while giving an account of reductive reasoning.

This paper touches on a number of seemingly disparate topics-Artificial Intelligence, Fuzzy Logic, String Theory, the search for extra-terrestrial intelligence, the Cantorian concept of infinite sets-in order to support the thesis that for a large part of the educated public in the Western world, the very concept of reality has been changing over the last few generations, and that the change is being accelerated by our increasing acceptance of the Virtual as a substitute for the traditional Real. This, as I hope to convince you, is a momentous shift in the our world view, and like so many profound but gradual shifts, has gone largely unnoticed. Whether the shift is ultimately a good thing or a bad, it ought not to go unscrutinized; this paper aims to bring it to public attention. (The paradox whose loss is referred to in the title is discussed at the end of the paper.

Argues for a 3-valued logic of vagueness. In distinction to other 3-valued approaches this logic is an extension of classical logic, due to the use of both a Boolean and a predicate negation. The approch is justified by semantic considerations. Critical issues like Sorites reasoning, higher order vagueness, and 'penumbral truth' are discussed from this perspective.

Here we initiate an investigation of the equational classes of msymmetric algebras endowed with two tense operators. These varieties is a generalization of tense algebras. Our main interest is the duality theory for these classes of algebras. In order to do this, we require Urquart's duality for Ockham algebras and Goldblatt's duality for bounded distributive lattice with operations. The dualities enable us to describe the lattices of congruences on tense m-symmetric algebras.

Alem da mais conhecida logica de Nelson (𝒩3) e da logica paraconsistente de Nelson (𝒩4), David Nelson introduziu no artigo de 1959 "Negation and separation of concepts in constructive systems", com motivacoes de aritmetica e construtividade, a logica que ele chamou de "𝒮". Naquele trabalho, a logica e definida por meio de um calculo (que carece crucialmente da regra de contracao) tendo infinitos esquemas de regras, e nenhuma semântica e fornecida. Neste trabalho nos tomamos o fragmento proposicional de 𝒮, mostrando que ele e algebrizavel (de fato, implicativo) no sentido de Blok & Pigozzi com respeito a uma classe de reticulados residuados involutivos. Assim, fornecemos a primeira semântica para 𝒮 (que chamamos de 𝒮-algebras), bem como um calculo estilo Hilbert finito equivalente a apresentacao de Nelson. Fornecemos um algoritmo para construir 𝒮-algebras a partir de 𝒮-algebras ou reticulados implicativos e demonstramos alguns resultados sobre a classe de algebras...

Negation: towards an integrative definition from philosophy and cognitive semiotics

Resumen:
La negación es un fenómeno común que se halla en todos los registros semióticos como las lenguas naturales y las lógicas (lenguas formales). En las lenguas naturales la negación aparece de muchas maneras (adverbio, conjunción copulativa, morfema prefijo o aun implícito semánticamente en pares antónimos). En las distintas lógicas (clásicas, paraconsistentes, intuicionistas...) funciona como operador unario, cuyas propiedades son muy variantes y particulares. Desde esta generalidad, podemos preguntar: ¿es la negación un fenómeno puramente semiótico? ¿Cuáles son las condiciones y precondiciones para que un signo se considere negación? ¿Hay diferentes negaciones o son todas variantes de un solo fenómeno? ¿Hay una ontología propia de la negación? Con el acontecer de las lógicas no clásicas, podemos hacer preguntas un poco más específicas: ¿la negación se relaciona directa e indisociablemente con la falsedad? ¿Hay clases de proposiciones, o juicios, propiamente negativas? ¿Hay tipos de negación, o bien sólo hay uno? Preguntas más filosóficas son estas otras: ¿cuándo algo es negativo o se considera una negación? ¿La negación es propia del psiquismo o existe en la realidad como hecho negativo? Dado que estas preguntas moran entre la filosofía, la lingüística y la lógica, una respuesta prudente precisa de un trasfondo interdisciplinar cuyo alcance tenga minuciosidad crítica y rigurosa: la teoría semiótico-cognitiva de Raymond Duval tiene este alcance. Así, pues, se hará un análisis de la negación en sus distintas dimensiones semióticas y en distintos registros; luego se propondrá una definición y se comprobará su efectividad descriptiva y analítica.

Come possono le cose mutare e al tempo stesso rimanere sé stesse? Una vexata quaestio metafisica che, ancora oggi, continua a far discutere. Achille Varzi propone il seguente esempio. Il tavolo della cucina ieri era perfettamente pulito, oggi invece è sporco. Ci sono due intuizioni: a. Il tavolo di ieri è il tavolo di oggi; cioè si tratta dello stesso tavolo; b. Il tavolo di ieri è diverso dal tavolo di oggi: le molecole che lo compongono sono diverse. Come ben si vede, a) e b) sono in conflitto. C'è una contraddizione: a) esclude b), e b) esclude a). E tuttavia, noi ci sentiamo portati a ritenere sia a) che b) entrambe vere. Ci sembra vero che si tratti dello stesso tavolo e, al tempo stesso, ci sembra vero che il tavolo sia mutato. Ci sono tre possibilità: 1) si accetta la contraddizione, dicendo che è il tavolo è lo stesso ma è anche diverso; 2) si accetta a) e si respinge b); 3) si accetta b) e si respinge a). Ognuna di esse ha un costo. La prima, la violazione del principio di non contraddizione (PNC), cioè non si dà il caso che sia vero A e insieme non-A. La seconda il mancato riscontro dell'esperienza: il tavolo, effettivamente, è diverso. La terza, ancora una violazione dell'esperienza: in fondo, si tratta dello stesso tavolo.

Recently there has been much interest in the possible quantum-like behavior of the human brain in such functions as cognition, the mental lexicon, memory, etc., producing a vast literature. These studies are both empirical and theoretical, the tenets of the theory in question being mainly, and apparently inevitably, those of quantum physics itself, for lack of other arenas in which quantum-like properties are presumed to obtain. However, attempts to explain this behavior on the basis of actual quantum physics going on at the atomic or molecular level within some element of brain or neuronal anatomy (other than the ordinary quantum physics that underlies everything), do not seem to survive much scrutiny. Moreover, it has been found empirically that the usual physics-like Hilbert space model seems not to apply in detail to human cognition in the large. In this paper we lay the groundwork for a theory that might explain the provenance of quantum-like behavior in complex systems whose internal structure is essentially hidden or inaccessible. The approach is via the logic obeyed by these systems which is similar to, but not identical with, the logic obeyed by actual quantum systems. The results reveal certain effects in such systems which, though quantum-like, are not identical to the kinds of quantum effects found in physics. These effects increase with the size of the system.

A major motivating concern of the relevant logics has been to devise a decent theory of the logical behavior of the operation of implication, expressed in the object language by '->' and corresponding to the relation of implication which is naturally expressed in the metalanguage. That being so, it should be expected that P-+Q be true if and only if p implies q> and more generally that the conditional be true in model m iff p implies q according to m. A conditional is true in m iff it holds at the base world (the real world) in m, and its antecedent implies its consequent according to m iff the consequent holds in m at every world at which the antecedent holds. The motivationally natural thought just sketched emerges in the Routley-Meyer semantics as the requirement that the base world O should satisfy, for all a and b, a < Z? iff ROab. A frame in which this requirement is met is said to be reduced. Note that a reduced frame satisfies in particular ROOO, whence the truth in any model in such a frame is closed under detachment for the arrow as one might expect. It is well known that the system R is characterized by reduced frames (see for example). What is less well known is the scandal that the arguments establishing reduced modelings for R and its close relatives do not extend to the weaker systems, lacking the theorem scheme WI supported as insightful by such as Brady [2], Priest , Routley [8,, and the author. The object of the present paper is to prove a reduced modeling theorem