Paraconsistent Research Articles

Hilbert-style axiomatization of first-degree entailment and a family of its extensions

This paper presents a Hilbert-style system for the logic of first-degree entailment defined in a Fmla-Fmla framework. The effective use of this formulation as a basis for a whole family of systems extending the logic of first-degree entailment in various directions is shown. By systematizing this family, some new systems are uncovered, and some other well-established logics (such as the first-degree entailment fragment of Priest's Logic of Paradox) obtain new axiomatization. Semantics for the key systems from the family is formulated.

Model Theory in a Paraconsistent Environment

AbstractThe purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.Abstract prepared by Bruno Costa Coscarelli.E-mail: brunocostacoscarelli@gmail.comURL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

Implication, Equivalence, and Negation

A system $HCL_{\overset{\neg}{\leftrightarrow}}$ in the language of {$ \neg, \leftrightarrow $} is obtained by adding a single negation-less axiom schema to $HLL_{\overset{\neg}{\leftrightarrow}}$ (the standard Hilbert-type system for multiplicative linear logic without propositional constants), and changing $ \rightarrow $ to $\leftrightarrow$. $HCL_{\overset{\neg}{\leftrightarrow}}$ is weakly, but not strongly, sound and complete for ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ (the {$ \neg,\leftrightarrow$} – fragment of classical logic). By adding the Ex Falso rule to $HCL_{\overset{\neg}{\leftrightarrow}}$ we get a system with is strongly sound and complete for ${\bf CL}_ {\overset{\neg}{\leftrightarrow}}$ . It is shown that the use of a new rule cannot be replaced by the addition of axiom schemas. A simple semantics for which $HCL_{\overset{\neg}{\leftrightarrow}}$ itself is strongly sound and complete is given. It is also shown that $L_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , the logic induced by $HCL_{\overset{\neg}{\leftrightarrow}}$ , has a single non-trivial proper axiomatic extension, that this extension and ${\bf CL}_{\overset{\neg}{\leftrightarrow}}$ are the only proper extensions in the language of { $\neg$, $\leftrightarrow$ } of $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ , and that $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ and its single axiomatic extension are the only logics in {$ \neg, \leftrightarrow$ } which have a connective with the relevant deduction property, but are not equivalent $\neg$ to an axiomatic extension of ${\bf R}_{\overset{\neg}{\leftrightarrow}}$ (the intensional fragment of the relevant logic ${\bf R}$). Finally, we discuss the question whether $ {\bf L}_{HCL}$$_{\overset{\neg}{\leftrightarrow}}$ can be taken as a paraconsistent logic.

Cenários Prospectivos para a Inovação Social

A Inovação Social tem seu campo fundamentado na criação de soluções frente às necessidades sociais, impulsionada por diversos agentes que se organizam sob os mais diferentes tipos jurídicos, ampliando a ação da inovação social (IS) de um campo local para nacional ou internacional. Estas transformações sociais impõem desafios aos gestores que passam a considerar as IS como uma nova dimensão para a estratégia e o planejamento nas organizações, exigindo a elaboração de cenários futuros. O objetivo deste estudo foi desenvolver cenários prospectivos para a inovação social no curto prazo. A abordagem é qualitativa e quantitativa, utilizando-se da Lógica Paraconsistente Anotada Evidencial (lógica Et). Foi elaborado um questionário, contendo 58 questões, sendo 57 fechadas e 1 aberta. Para validar o questionário a priori foram consultados 3 especialistas, as proposições foram confirmadas por meio de rodadas Delphi e teste de consistência interna medida pelo Alpha de Cronbach. A amostra foi constituída por 26 participantes, distribuídos em dois grupos, grupo A formado por 13 especialistas e grupo B por 13 profissionais gestores ou que tenham participado de projetos sociais. Para medir se existe diferença estatística significativa entre os grupos, foi utilizado o teste de Levene. Os resultados indicam que não existe diferença significativa entre os grupos e validam o modelo para o cenário de curto prazo, que contém os fatores distribuídos em transformação, inovação, geografia da inovação, atores e processos. Lógica Paraconsistente Anota Evidencial: as teorias paraconsistentes permitem teoremas de contradição, em uma mesma proposição pode assumir graus de crença ou descrença. São consideradas não triviais e adequadas ao conhecimento complexo (Costa & Abe, 2000). Alpha de Cronbach: utilizado para quantificar a confiabilidade de um questionário, trata-se de um índice de homogeneidade da amostra (Brown, 2002). Teste de Levene: demonstra a homogeneidade das variações, sendo considerado um instrumento robusto e poderoso (Gastwirth, Gel, & Miao, 2009).

Hölderlin e a poética do paradoxo

This article seeks to clarify and expose the principle of Hölderlin's “poetic logic”, as it appears in the “poetological” texts, mainly in the Observations on Oedipus and Antigone. As we try to show here, Hölderlin's “poetic logic” does not follow the principles of formal logic, but it comes close to dialectical logic, although it cannot be reduced to Hegelian dialectics. The logic here is not aimed at standardizing the rules of the use of thought, but at poetic practice, whose nature, in Hölderlin's sense, consists in the paradoxical apprehension of existence as such. Although it can be more clearly observed in the tragic genre, this paradoxical logic of poetry also permeates Hölderlin's so-called late hymns, such as Patmos and Bread and wine, in which tragic oppositions are coated in the elegiac form.

On the Paraconsistent Logic CG′3

Paraconsistent logical systems are well-known reasoning frameworks aimed to infer new facts or properties under contradictory assumptions. Applications of these systems are well known in wide range of computer science domains. In this article, we study the paraconsistent logic CG ′ 3 , which can be viewed as an extension of the logic G ′ 3 . CG ′ 3 is also 3 -valued, but with two designated values. Main results can be summarized as follows: a Hilbert-type axiomatization, based on Kalmár’s approach; and a new notion of validity, based on also novel Kripke semantics.

Neighbourhood Semantics for FDE-Based Modal Logics

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

La negación paraconsistente a la luz del realismo lógico

In this paper I analyze some of the negation operators from the family of paraconsistent logics from the perspective of logical realism. I start by taking as cases two cases of study: the LP negation operator along with its metaphysical interpretation stated as dialetheism and the FDE negation with the metaphysical interpretation proposed by Beall (2016). I use this operators to draw a comparison with the different level of metaphysical commitments in some other paraconsistent negations. I discuss how LP negation a robust metaphysical commitment such as dialetheism relies on a leading role of logical negation while in Beall’s interpretation the search for a metaphysically balanced picture entails the disappearance of logical negation as a substantive notion. I the argue that it is possible to obtain an intermediate alternative, with a four-valued paraconsistent logic SF, which is compatible with a metaphysically balanced picture as Beall’s but it recovers a substantive notion of logical negation as a fundamental connective.

Non-Boolean classical relevant logics II: Classicality through truth-constants

This paper gives an account of Anderson and Belnap’s selection criteria for an adequate theory of entailment. The criteria are grouped into three categories: criteria pertaining to modality, those pertaining to relevance, and those related to expressive strength. The leitmotif of both this paper and its prequel is the relevant legitimacy of disjunctive syllogism. Relevant logics are commonly held to be paraconsistent logics. It is shown in this paper, however, that both E and R can be extended to explosive logics which satisfy all of Anderson and Belnap’s selection criteria, provided the truth-constant known as the Ackermann constant is available. One of the selection criteria related to expressive strength is having an “enthymematic” conditional for which a deduction theorem holds. I argue that this allows for a new interpretation of Anderson and Belnap’s take on logical consequence, namely as committing them to pluralism about logical consequence.

Fatores críticos de sucesso e sua influência na prática docente

This study analyzed and identified the Critical Success Factors that most influence teaching practice. These factors involved aspects such as: teaching tools, perceived quality, training and psychological well-being. The objective was to contribute data that corroborate the development of teaching practices in the school environment, reflecting on the students' better performance. It was found that the use of Critical Success Factors (FCS) can positively influence teacher performance, and that by improving classroom management, their professional performance is increased. The theoretical references used were Rockart's Critical Success Factors (1979) and the perceived quality of Parasuraman et al. (1990). The instruments used for the analysis were the Kolmogorov-Smirnov (1979), to identify and describe the Critical Success Factors that assisted teaching practice, and the Servqual de Parasuraman et al. (1990), to analyze the quality of teachers' service, perceived by school managers. As the Kolmogorov-Smirnov method has simplified application, PARACONSISTENT LOGIC was also employed, as it allows, according to Bispo and Cazarini (2006) the natural treatment of divergences, inconsistencies and contradictions, as well as to obtain more accurate and close results. This work was restricted to the ten best schools in the city of Niterói, according to the concept in the National High School Exam (ENEM). This delimitation makes the results obtained subject to standards such as behavior and market demands. It was also expected, besides adding value to the current literature, to stimulate studies around the proposed theme, aiming to increasingly help teachers in the effectiveness of their actions in the school environment.

Anna Karenina and the two envelopes problem

SummaryThe Anna Karenina principle is named after the opening sentence in the eponymous novel: Happy families are all alike; every unhappy family is unhappy in its own way. The two envelopes problem (TEP) is a much‐studied paradox in probability theory, mathematical economics, logic and philosophy. Time and again a new analysis is published in which an author claims finally to explain what actually goes wrong in this paradox. Each author (the present author included) emphasises what is new in their approach and concludes that earlier approaches did not get to the root of the matter. We observe that though a logical argument is only correct if every step is correct, an apparently logical argument which goes astray can be thought of as going astray at different places. This leads to a comparison between the literature on TEP and a successful movie franchise: it generates a succession of sequels, and even prequels, each with a different director who approaches the same basic premise in a personal way. We survey resolutions in the literature with a view to synthesis, correct common errors, and give a new theorem on order properties of an exchangeable pair of random variables, at the heart of most TEP variants and interpretations. A theorem on asymptotic independence between the amount in your envelope and the question whether it is smaller or larger shows that the pathological situation of improper priors or infinite expectation values has consequences as we merely approach such a situation.

Axiomatisations of the Genuine Three-Valued Paraconsistent Logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$

Genuine Paraconsistent logics $$\mathbf {L3A}$$ and $$\mathbf {L3B}$$ were defined in 2016 by Beziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernandez-Tello et al, provide implications for both logics and define the logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ . In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ satisfy a restricted version of the Substitution Theorem, and that both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ and among other logics, for instance $${\mathbf {Int}}$$ and some $${\mathbf {LFI}}$$ s.

Proof Systems for 3-valued Logics Based on Gödel’s Implication

Abstract The logic $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ was introduced in Robles and Mendéz (2014, Logic Journal of the IGPL, 22, 515–538) as a paraconsistent logic which is based on Gödel’s 3-valued matrix, except that Kleene–Łukasiewicz’s negation is added to the language and is used as the main negation connective. We show that $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ is exactly the intersection of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. (In $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ the set ${\cal D}$ of designated elements is $\{1\}$, while in $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ ${\cal D}=\{1,0.5\}$.) We then construct a Hilbert-type system which has (MP) for $\to $ as its sole rule of inference, and is strongly sound and complete for $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Then we show how, by adding one axiom (in the case of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$) or one new rule of inference (in the case of $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$), we get strongly sound and complete systems for $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$. Finally, we provide quasi-canonical Gentzen-type systems which are sound and complete for those logics and show that they are all analytic, by proving the cut-elimination theorem for them.

The psychological aspects of paraconsistency

The creation of paraconsistent logics have expanded the boundaries of formal logic by introducing coherent systems that tolerate contradictions without triviality. Thanks to their novel approach and rigorous formalization they have already found many applications in computer science, linguistics and mathematics. As a natural next step, some philosophers have also tried to answer the question if human everyday reasoning could be accurately modelled with paraconsistent logics. The purpose of this article is to argue against the notion that human reasoning is paraconsistent. Numerous findings in the area of cognitive psychology and cognitive neuroscience go against the hypothesis that humans tolerate contradictions in their inferences. Humans experience severe stress and confusion when confronted with contradictions (i.e., the so-called cognitive dissonance). Experiments on the ways in which humans process contradictions point out that the first thing humans do is remove or modify one of the contradictory statements. From an evolutionary perspective, contradiction is useless and even more dangerous than lack of information because it takes up resources to process. Furthermore, it appears that when logicians, anthropologists or psychologists provide examples of contradictions in human culture and behaviour, their examples very rarely take the form of: $$(p \wedge \lnot p)$$ . Instead, they are often conditional statements, probabilistic judgments, metaphors or seemingly incompatible beliefs. At different points in time humans are definitely able to hold contradictory beliefs, but within one reasoning leading to a particular behaviour, contradiction is never tolerated.

A family of genuine and non-algebraisable C-systems

ABSTRACT In 2016, Béziau introduced the notion of genuine paraconsistent logic as logic that does not verify the principle of non-contradiction; as an important example, he presented the genuine paraconsistent logic in terms of three connectives , , and . In this paper, we show that is an axiomatic extension of through the introduction of a non-primitive deductive implication. Furthermore, we prove that is an algebraisable logic with Blok-Pigozzi's method. From the proof that is non-algebraisable logic, we are able to see that is not algebraisable logic and studying the borders of algebrisabilty, we can give an enumerable family of new genuine, paraconsistent and non-algebraisable logics, extensions of . Finally, we introduced n-valued () and infinite-valued logic and show that they are genuine and non-algebraisable paraconsistent ones; in addition, we present semantics for this extensions of by means of Fidel's structures.

Two ways of approaching the paradoxes in logic and mathematics

It is usual to think of the paradoxes appearing inside a particular logical or mathematical theory as the serious obstacles hindering any further development of that theory. Paradoxes are supposed to show that a theory in question is built on an unstable, or even contradictory foundation, and thus point the need for its complete revision. However, a different view on the paradoxes is also possible. They could instead be understood as the arguments which show that some particular assumptions concerning the objects with respect to which they appear, or the ways of reasoning about them, are wrong. If treated in that way, paradoxes or their solutions could lead to some new insights into the nature of objects they concern. They could thus turn out to make a useful focus in developing the understanding of these objects. In this work, the effect that the first approach towards the paradoxes had on development of logic and mathematics, in particular set theory, will be described. Using some examples, we will try to show that the alternative view on the paradoxes or their formal equivalents actually leads to some important results in logic, and at the same time, opens the door to a new logical theory - the so-called theory of concepts.

Myers' paradox

This note is an analysis of the paradox given by Myers (2019). It is shown, assuming that the resources available in paraconsistent logic may be applied, how the conclusion of the paradox may be perfectly acceptable, but that the argument is, nonetheless, invalid. This provides a dialethic solution to the paradox.

Fuzzy Logic in the Quantum Mechanical Worldview; Related to Zadeh, Wittgenstein, Moore, Saussure, Quine, Lewis Carroll, etc.

Recently we proposed the linguistic Copenhagen interpretation of quantum mechanics, which is called quantum language or measurement theory. This theory is valid for both quantum and classical systems. Thus we think that quantum language is one of the most powerful scientific theories, like statistics. In this paper we justify Zadeh’s fuzzy sets theory in quantum language, that is, fuzzy propositions are identified with binary measurements. This implies that the definition of “proposition” is, for the first time, acquired in the field of non-mathematics. Further, we assert that fuzzy logic is more natural than crisp logic in science. And furthermore, we discuss and solve Saussure’s linguistics, Moore’s paradox, Quine’s analytic-synthetic distinction and Lewis Carroll’s logical paradox. Therefore, from the philosophical point of view, our result gives a complete answer to Wittgenstein’s problem: “Why does logic work in our world?” and “What is a scientific proposition?” in his picture theory. That is, we simultaneously justify both Zadeh’s fuzzy sets and Wittgenstein’s picture theory in the quantum mechanical worldview.

The game and the logical paradoxes of justice: from king Solomon to the present day

In the article, the inevitability of logical paradoxes and unsolvable incidents in justice is justified by the incompleteness of formal systems, including such as law and law. The conclusions are illustrated by references to biblical and literary stories, as well as real situations that arise in judicial activity. Elements of game theory and the Gries code of communication are proposed as methods for optimizing judicial decision-making in complex cases, and specific situations of their application are modeled. It is concluded that there is a special type of thinking that a judge should have, and conditions that can block or stimulate its manifestation

Rotary Inverted Pendulum Identification for Control by Paraconsistent Neural Network

Artificial neural networks (ANNs) have been used over the last few decades to perform tasks by learning with comparisons. Fitting input-output models, system identification, control, and pattern recognition are some fields for ANN applications. However, problems involving uncertain situations could be challenging for them. The family of paraconsistent logics (PL) is a powerful tool that can deal with uncertainty and contradictory information, so getting attention from researchers for its implications and applications in artificial intelligence. This investigation describes a novel activation function reasoned on the paraconsistent annotated logic by two-value annotations (PAL2v) rules, a variation of PL, allowing the design of a new paraconsistent neural net (PNN), applied in model identification for control (I4C) of a closed-loop rotary inverted pendulum (RIP) system.