Showing papers on "Paraconsistent logic published in 2021"

Paraconsistent Annotated Logic Algorithms Applied in Management and Control of Communication Network Routes.

Abstract: This paper presents a computational method based on non-classical logic dedicated to routing management and information stream control in communication networks. Paraconsistent logic (PL) was used to create an algorithmic structure whose main property is to accept contradiction. Moreover, a computational structure, the denominated paraconsistent data analyzer (PDAPAL2v), was constructed to perform routing management in communication networks. Direct comparisons of PDAPAL2v with a classical logic system that simulates routing conditions were made in the laboratory. In the conventional system, the paraconsistent algorithms were considered as binary logic gates, and in the tests, the same adjustment limits of PDAPAL2v were applied. Using a database with controlled insertion of noise, we obtained an efficacy of 97% in the detection of deteriorated packets with PDAPAL2v and 72% with the conventional simulation system. Functional tests were carried out, showing that PDAPAL2v is able to assess the conditions and degradation of links and perform the analysis and correlation of various inputs and variables, even if the signals have contradictory values. From practical tests in the laboratory, the proposed method represents a new way of managing and controlling communication network routes with good performance.

A Bridge Between Q-Worlds

Abstract: Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding quantum mechanics by reformulating parts of the theory inside of non-classical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, `Q-worlds'. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.

On the strongest three-valued paraconsistent logic contained in classical logic and its dual

Abstract: LP⊃,F is a three-valued paraconsistent propositional logic that is essentially the same as J3. It has the most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic. However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned. In this paper, properties concerning the logical equivalence relation of a logic are used to distinguish LP⊃,F from the others. As one of the bonuses of focusing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent and commutative laws for conjunction and disjunction. For most properties of LP⊃,F that have been proposed as desirable properties of a reasonable paraconsistent propositional logic, its paracomplete analogue has a comparable property. In this paper, properties concerning the logical equivalence relation of a logic are also used to distinguish the paracomplete analogue of LP⊃,F from the other three-valued paracomplete propositional logics with those comparable properties.

A family of genuine and non-algebraisable C-systems

Abstract: In 2016, Beziau 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 paraconsist...

R-Mingle is Nice, and so is Arnon Avron

Abstract: Arnon Avron has written: “Dunn-McCall logic RM is by far the best understood and the most well-behaved in the family of logics developed by the school of Anderson and Belnap.” I agree. There is the famous saying: “Do not let the perfect become the enemy of the good.” I might say: “good enough.” In this spirit, I will examine the logic R-Mingle, exploring how (in the terminology of Avron) it is only a “semi-relevant logic” but still a paraconsistent logic. I shall discuss the history of RM, and compare RM to Anderson and Belnap’s system R of relevant implication and to classical two-valued logic. There is a “consumer’s guide,” evaluating these logics as “tools,” in the light of my recent work on “Humans as Rational Toolmaking Animals.”

The evidence approach to paraconsistency versus the paraconsistent approach to evidence

Abstract: In this paper, we analyze the epistemic approach to paraconsistency. This approach is advanced as an alternative to dialetheism on what concerns interpreting paraconsistency and contradictions; instead of having to accept that there are true contradictions (as dialetheists argue), it is suggested that we may understand such situations as involving only conflicting evidence, which restricts contradictions to a notion of evidence weaker than truth. In this paper, we first distinguish two conflicting programs entangled in the proposal: (1) interpreting paraconsistency in general through the notion of evidence, and (2) modeling reasoning with evidence by using paraconsistent logic. The first part of the program, we argue, does not succeed, on the grounds that it does not lead to a uniform proposal to the understanding of paraconsistency, and fails to engage with dialetheism in a legitimate dispute about interpretation of paraconsistency. Also, when seen through the lights of the second kind of approach, a ‘logic as modeling’ approach, weaknesses of dealing with evidence through paraconsistency come to light, basically because evidence does not seem to suggest the need of a paraconsistent treatment. As a result, one can neither approach paraconsistency in general through evidence, nor approach evidence with the use of paraconsistent logics.

Relevance and Verification

Abstract: A. J. Ayer’s empiricist criterion of meaning was supposed to have sorted all statements into nonsense on the one hand, and tautologies or genuinely factual statements on the other. Unfortunately for Ayer, it follows from classical logic that his criterion is trivial – it classifies all statements as either tautologies or genuinely factual, but none as nonsense. However, in this paper I argue that Ayer’s criterion of meaning can be defended from classical proofs of its triviality by the adoption of a relevant logic – an idea which is motivated because, according to Ayer, the genuinely factual statements are those which observation is

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

Abstract: 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.

Contradictions and falling bridges: what was Wittgenstein’s reply to Turing?

Abstract: In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wit...

Reflections on Orlov

Abstract: In 1928 Ivan Orlov published a remarkable paper which contains the first formulation of a relevant logic. The paper remained largely unknown to English-speakers until this discovery of relevant log...

Neural Network Explainable AI Based on Paraconsistent Analysis: An Extension

Abstract: This paper explores the use of paraconsistent analysis for assessing neural networks from an explainable AI perspective. This is an early exploration paper aiming to understand whether paraconsistent analysis can be applied for understanding neural networks and whether it is worth further develop the subject in future research. The answers to these two questions are affirmative. Paraconsistent analysis provides insightful prediction visualisation through a mature formal framework that provides proper support for reasoning. The significant potential envisioned is the that paraconsistent analysis will be used for guiding neural network development projects, despite the performance issues. This paper provides two explorations. The first was a baseline experiment based on MNIST for establishing the link between paraconsistency and neural networks. The second experiment aimed to detect violence in audio files to verify whether the paraconsistent framework scales to industry level problems. The conclusion shown by this early assessment is that further research on this subject is worthful, and may eventually result in a significant contribution to the field.

On the Paraconsistent Logic CG′3

Abstract: 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 Kalmar’s approach; and a new notion of validity, based on also novel Kripke semantics.

Islamic Contradictory Theology . . . Is there any such Thing

Abstract: The application of paraconsistent logics to theological contradictions is a fascinating move. Jc Beall’s (J Anal Theol, 7(1): 400–439, 2019) paper entitled ‘Christ—A Contradiction: A Defense of ‘Contradictory Christology’ is a notable example. Beall proposes a solution to the fundamental problem of Christology. His solution aims at making the case, and defending the viability of, what he has termed, ‘Contradictory Christology’. There are at least two essential components of Beall’s ‘Contradictory Christology’. These include the dogmatic statements of Chalcedon and FDE logic. The first is the theological contradiction in question. The second is the type of paraconsistent logic. Both components are integral to a contradictory theology in general. I argue that there can be no such thing as an Islamic contradictory theology. I make the case by establishing two points. These points correspond to both integral components of a contradictory theology in general. The first is that an Islamic theological contradiction does not entail an actual (logical) contradiction. The second is that FDE logic, including alternative sub-classical systems of logic, are not adequate in tolerating an Islamic theological contradiction.

Restricted Rules of Inference and Paraconsistency

Abstract: In this paper, we study two companions to a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems - Intuitionistic Paraconsistent Weak Kleene logic (IPWK) and Paraconsistent Pre-Rough logic (PPRL) - are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of Intuitionistic Propositional logic (IPC) and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of Pre-Rough logic (PRL) but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Plonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.

In Pursuit of the Non-Trivial

Abstract: This paper is about the underlying logical principles of scientific theories. In particular, it concerns ex contradictione quodlibet (ECQ) the principle that anything follows from a contradiction. ECQ is valid according to classical logic, but invalid according to paraconsistent logics. Some advocates of paraconsistency claim that there are ‘real’ inconsistent theories that do not erupt with completely indiscriminate, absurd commitments. They take this as evidence in favor of paraconsistency. Michael (2016) calls this the non-triviality strategy (NTS). He argues that this strategy fails in its purpose. I will show that Michael's criticism significantly over-reaches. The fundamental problem is that he places more of a burden on the advocate of paraconsistency than on the advocate of classical logic. The weaknesses in Michael's argument are symptomatic of this preferential treatment of one viewpoint in the debate over another. He does, however, make important observations that allow us to clarify some of the complexities involved in giving a logical reconstruction of a theory. I will argue that there are abductive arguments deserving of further consideration for the claim that paraconsistent logic offers the best explanation of the practice of inconsistent science. In this sense, the debate is still very much open.

Logical theory revision through data underdetermination: an anti-exceptionalist exercise

Abstract: The anti-exceptionalist debate brought into play the problem of what are the relevant data for logical theories and how such data affects the validities accepted by a logical theory. In the present paper, I depart from Laudan's reticulated model of science to analyze one aspect of this problem, namely of the role of logical data within the process of revision of logical theories. For this, I argue that the ubiquitous nature of logical data is responsible for the proliferation of several distinct methodologies for logical theories. The resulting picture is coherent with the Laudanean view that agreement and disagreement between scientific theories take place at different levels. From this perspective, one is able to articulate other kinds of divergence that considers not only the inferential aspects of a given logical theory, but also the epistemic aims and the methodological choices that drive its development.

A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3 $$_{\text {DH}}$$ DH

Abstract: The logic DHb is the result of extending Sylvan and Plumwood’s minimal De Morgan logic B $$_{\text {M}}$$ with a dual intuitionistic negation of the type Sylvan defined for the extension CC $$\omega $$ of da Costa’s paraconsistent logic C $$\omega $$ . We provide Routley–Meyer ternary relational semantics with a set of designated points for DHb and a wealth of its extensions included in G3 $$_{\text {DH}}$$ , the expansion of G3 $$_{+}$$ with a dual intuitionistic negation of the kind considered by Sylvan (G3 $$_{+}$$ is the positive fragment of Godelian 3-valued logic G3). All logics in the paper are paraconsistent.

Logics of formal inconsistency enriched with replacement: an algebraic and modal account

Abstract: It is customary to expect from a logical system that it can be algebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa's paraconsistent calculi $C_n$, algebraic equivalents for such systems have been sought. It is known, however, that these systems are not self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than $C_1$ , thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with additional operations. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied. In addition, a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. the proposed algebraic semantics.

The psychological aspects of paraconsistency

Abstract: 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 Purpose of a Smart Vegetable Garden Model Based on Paraconsistent Annotated Evidential Logic E τ

Abstract: Due to urban growth, the spaces for planting were being reduced; this growth caused an increase in the consumption of water, energy, and food. Consequently, it will be necessary to rethink a new structure for the use of urban space concerning planting. The information technology has converted the agricultural sector presently profoundly. Many studies are being carried out to facilitate the management and increase the productivity of the crops; the term currently used to name the phenomenon of technological implementation in the field is known as “Precision Agriculture.” This work aims to propose an intelligent garden model that allows the collection of information by the soil sensors that will be sent to the Arduino that controls a hydraulic pump responsible for irrigating the soil monitored by the sensor based on Paraconsistent Logic.

Obtaining infinitely many degrees of inconsistency by adding a strictly paraconsistent negation to classical logic

Abstract: This paper is devoted to a consequence relation combining the negation of Classical Logic ($$\mathbf {CL}$$) and a paraconsistent negation based on Graham Priest’s Logic of Paradox ($$\mathbf {LP}$$). We give a number of natural desiderata for a logic $$\mathbf {L}$$ that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. We devise the logic $$\mathbf {CLP}$$ by means of an axiomatization and three equivalent semantical characterizations (a non-deterministic semantics, an infinite-valued set-theoretic semantics and an infinite-valued semantics with integer numbers as values). By showing that this logic is maximally paraconsistent, we prove that $$\mathbf {CLP}$$ is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations.

Paraconsistent Annotated Evidential Logic Eτ Applied to Autonomous Robots in Logistic Center

Abstract: Due to the momentum of Industry 4.0 in various product and service sectors, academic and business investments in the development of new technologies in the logistics sector also stand out. Based on these trends, bibliographical research was carried out of articles dedicated to robotics with application in the logistics sector. Based on the bibliographic survey results, this work proposes a prototype of an autonomous terrestrial mobile robot that must go through corridors in specific layouts in logistics centres focusing on the aid of control tasks and management information for decision making. The main contribution is testing a para-analyzer algorithm based on non-classical logic such as Paraconsistent Annotated Logic Eτ, as the decision-making tool to avoid obstacles.

Characterization and classification of numerical data patterns using Annotated Paraconsistent Logic and the effect of contradiction

Abstract: This work describes the development of a computational mathematical model that uses Annotated Paraconsistent Logic - APL and a concept derived from it, the effect of contradiction, to identify patterns in numerical data for pattern classification purposes. The APL admits paraconsistent and paracomplete logical principles, which allow the manipulation of inconsistent and contradictory data, and its use allowed the identification and quantization of the attribute related to the contradiction. To validate the model, series of Raman spectroscopies obtained after exposure of proteins, lipids and nucleic acids, collected from cutaneous tissue cell samples previously examined for the detection of cancerous lesions, identified as basal carcinoma, melanoma and normal, were used. Initially, the attributes related to contradiction, derivative and median obtained from spectroscopies were identified and quantified. A machine learning process with approximately 31.6% of each type of samples detects a sequence of spectroscopies capable of characterizing and classifying the type of lesion through the chosen attributes. Approximately 68.4% of the samples are used for classification tests. The proposed model identified a segment of spectroscopies where the classification of test samples had a hit rate of 76.92%. As a differential and innovation of this work, the use of APL principles in a complete process of training, learning and classification of patterns for numerical data sets stands out, with flexibility to choose the attributes used for the characterization of patterns, and a quantity of samples of about one third of the total required for characterization.

Paraconsistent Foundations for Quantum Probability.

Abstract: It is argued that a fuzzy version of 4-truth-valued paraconsistent logic (with truth values corresponding to True, False, Both and Neither) can be approximately isomorphically mapped into the complex-number algebra of quantum probabilities. I.e., p-bits (paraconsistent bits) can be transformed into close approximations of qubits. The approximation error can be made arbitrarily small, at least in a formal sense, and can be related to the degree of irreducible "evidential error" assumed to plague an observer's observations. This logical correspondence manifests itself in program space via an approximate mapping between probabilistic and quantum types in programming languages.

Credal Calculi, Evidence, and Consistency

Abstract: This paper defends the use of possibility and necessity models based on the Logics of Formal Inconsistency, taking advantage of their expressivity in terms of the notions of consistency (\(\circ \)) and inconsistency (\(\bullet \)). The present proposal directly generalizes the approach of Besnard and Lang (Proceedings of 10th Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, San Francisco, pp. 69–76 1994), whose main guidelines we borrow here. Some basic properties of possibility and necessity functions over the Logics of Formal Inconsistency are obtained and it is shown, by revisiting a paradigmatic example, how paraconsistent possibility and necessity reasoning can, in general, attain realistic models for artificial judgement. We will call such models credal calculi, emphasizing some of their appealing consequences.

Use of Paraconsistent Logic Evidential Annotated Eτ in Logistic Systems.

Abstract: Traditional Logistic Systems related to the modules of routes and vehicle tracking were observed, which use classical logic. In this study, whose main objective is a proposal for digital transformation with the implementation of the use of Paraconsistent Logic Evidential Annotated Eτ for optimization and reduction exposure to the risk of the cargo transport process in these types of systems. The methodology used para-analyzer algorithm, with selection of specialists in decision-making issues to improve the safety of the routes and minimize the risks exposed in the cargo’s path from shipment to the destination. An exploratory research was used surveys through applied logical questionnaires. The main results were compared to the Case Study of the Volkswagen Modular Consortium - Trucks and buses, from a logistical point of view, in which the three main factors resulting from this study can prove an alignment regarding the implementation of this innovation. In addition, data on incidents occurred at the Agrobusiness Company regarding the Routes and Tracking Systems were presented, proving the risk of the loads and the need for digital transformation in these types of systems.

Paraconsistent Logic: Consistency, Contradiction and Negation: Walter Carnielli and Marcelo E. Coniglio, New York, Springer International Publishing, 2016, xxiv + 398 pp., US$109.99 (pbk) ISBN-13: 978-3319814537

Abstract: The book Paraconsistent Logic: Consistency, Contradiction and Negation by Walter Carnielli and Marcelo Coniglio (from now on C&C) is the most thorough study of Logics of Formal Inconsistency (LFIs)...

Extended First-Order Step Theories

Abstract: Active Logic is a conceptual system, the principles of which are satisfied with formalisms of reasoning, which allow correlating the results of reasoning with moments in time and ensuring tolerance to contradictions. At present, tolerance to contradictions (paraconsistency) as applied to Active Logic systems has received theoretical justification in the works of the authors of this report and refers to the so-called formalisms of step theories, combining the principles of Active Logic and logical programming. In particular, it was proved that the argumentative semantics of the formalism of extended step theories, which have two types of negation and have more developed expressive capabilities, are paranormal. The formalism introduced in this paper has even greater expressive capabilities. Keeping all the controls for the process of reasoning in time, this formalism allows one to more adequately represent facts about the subject area by using the language of first-order logic instead of the language of propositional logic. For the formalism of extended step theories of the first order, a declarative semantics is proposed, its paraneosistance is shown, and the expressive possibilities of this formalism in relation to the tasks of action planning are demonstrated using the example of the monkey and the banana.