Extension of Paraconsistent Many-Valued Similarity Method to Group Decision

The Paraconsistent Many-Valued Similarity (PMVS) method for multi-attribute decision making will be incomplete as a decision model if it is not extended to the realm of group decision-making. Therefore, in this article, our primary objective is to show how the paraconsistent many-valued similarity method can be used to solve group decision-making problems involving choice making or ranking of a finite set of decision alternatives. Moreover, since weights are very important parameters in multi-attribute decision-making, we have introduced the Borda rule to calculate the weights of experts and that of every criterion under consideration. To demonstrate how the proposed method works, a numerical example on energy sources of an economy from the points of view of a group of experts is investigated. Further, we compare the results of this new approach with that of fuzzy TOPSIS group decision-making method to illustrate the robustness and effectiveness of the former.

Tableau Systems for Paraconsistency and Minimal Inconsistency

In this paper, the sematics of a paraconsistent logic and its nonmonotonic extension by minimal inconsistency are presented first. And then signed tableaux for paraconsistelit logic and minimal tableaux for logic of minimal inconsistency is proposetl. Finally the reduction of logic of paraconsistency and minimal inconsistency on ordinary semantics which provides new approach to proof procedure and implementation of paraconsistency and minimal inconsistency are provided.

Paraconsistent Annotated Logic in Analysis of Physical Systems:Introducing the Paraquantum Factor of Quantization h_(ψ)

We present in this paper an alternative of modeling physical systems through a non-Classical logic namely the Paraconsistent Logic (PL) whose main feature is the revocation of the principle of non-contradiction. The Paraconsistent Annotated Logic with annotation of two values (PAL2v) is a type of PL and has in its theoretical structure the main feature of dealing with contradictions offering flexibility in drawing conclusions. Several works about applications of PAL2v have shown that such logic is able to provide us with an adequate treatment to uncertainties. Based on the foundations of the PAL2v we presented the ParaQuantum logic (PQL) with the goal of performing analysis of signals from information sources which model physical systems. The formalization of the concepts of the logics PQL, that it is represented in a Lattice, requires the considering of Paraquantum logical states ψ which are propagated through variations of the evidence Degrees μ and λ which come out from measurements performed in Observable Variables in the physical world. When we analyze the lattice of the PQL, we obtain equations which quantify values of physical quantities from where we obtain the effects of propagation of the Paraquantum logical states ψ. In this paper, we introduce the Paraquantum Factor of quantization hψ whose value is associated with a special logical state on the lattice which is identified with the Planck constant h. We conclude through these studies that the Paraquantum Logical Model based on the ParaQuantum logics PQL can link the several fields of the physical sciences by means of quantization of values. It is an innovative approach of formulating natural phenomena.

Analysis of Physical Systems with Paraconsistent Annotated Logic: Introducing the Paraquantum Gamma Factor γ_(ψ)

In this paper we use a non-classical logic called ParaQuantum Logic (PQL) which is based on the foundations of the Paraconsistent Annotated logic with annotation of two values (PAL2v). The formalizations of the PQL concepts, which is represented by a lattice with four vertices, leads us to consider Paraquantum logical states ψ which are propagated by means of variations of the evidence Degrees extracted from measurements performed on the Observable Variables of the physical world. In this work we introduce the Paraquantum Gamma Factor γPψ which is an expansion factor on the PQL lattice that act in the physical world and is correlated with the Paraquantum Factor of quantization hψ whose value is associated with a special logical state on the lattice which is identified with the Planck constant h. Our studies show that the behavior of the Paraquantum Gamma Factor γPψ, at the time of reading the evidence Degrees through measurements of the Observable Variables in the physical world, is identical to that one of the Lorentz Factor γ used in the relativity theory. In the final part of this paper we present results about studies of expansion and contraction of the Paraquantum Logical Model which correlate the factors γPψ, and γ. By applying these correlation factors, the lattice of the PQL suitable for the universe understudy can be contracted or expanded, allowing the quantization model to cover the several study fields of physics.

Analysis of the Spectral Line Emissions of the Hydrogen Atom with Paraquantum Logic

In this work we presented a study of the obtaining of the spectral line emissions of the hydrogen atom using equations that are originated from the foundations of the Paraquantum Logic (PQL). Based on a class of logics called Paraconsistent Logics with annotation of two values (PAL2v), PQL performs a logical treatment on signals obtained by measurements on physical quantities which are considered Observable Variables in the physical world. In the process of application of the PQL the obtained values are transformed in Evidence Degrees and represented on a Lattice of four Vertices where special equations transform these degrees into Paraquantum logical states ψ which propagate. This allows creating Paraquantum logical models of physical systems of the real world. Using the paraquantum equations, we investigated the hydrogen atom spectrum and his main series known. We performed a numerical comparative study that applies the Paraquantum Logical Model to calculate the wavelengths values. The values of wavelengths obtained by the Paraquantum Equations are compared by the results found by the Rydberg formula and are verified that the series of the spectral line emissions of the hydrogen atom can be identified with the representative Lattices of the Paraquantum Logic. Through the application of the Paraquantum equations it was found a numeric value relates the layers of Paraquantum model of the Hydrogen atom. This value represents a constant that relates the Lattices that compose the Paraquantum universe, and it was denominated Paraquantum Structure Constant, whose symbol is αψ. The obtained results of the comparison demonstrate that the Paraquantum Logic comes with good possibilities of being the ideal logic to model our physical reality.

An Introductory Study of the Hydrogen Atom with Paraquantum Logic

Paraquantum Logics (PQL) has its origins in the fundamental concepts of the Paraconsistent Annotated Logics (PAL) whose main feature is to be capable of treating contradictory information. Based on a class of logics called Paraconsistent Logics with annotations of two values (PAL2v), PQL performs a logical treatment on signals obtained by measurements on physical quantities which are considered Observable Variables in the physical world. In the process of application of the PQL the obtained values are transformed in Evidence Degrees and represented on a Lattice of four Ver- tices where special equations transform these degrees into Paraquantum logical states ψ which propagate. The propagation of Paraquantum logical states provides us with results which can be interpreted and modeled through phenomena studied in physics. Using the paraquantum equations, we investigate the effects of balancing of Energies and the quantization and transience properties of the Paraquantum Logical Model in real Physical Systems. As a demonstration of the usage of the paraquantum equations we perform a numerical comparative study that applies the PQL to the Bohr’s model to find the energy levels of the Hydrogen atom. It is verified that the values of energy in each level of the Paraquantum logical model of the Hydrogen atom are close to the values found by the conventional way. The results through the Paraquantum Logic allow considering other important properties of the atom, as the forecast of number of electrons in each layer.

Undulatory Theory with Paraconsistent Logic (Part I): Quantum Logical Model with Two Wave Functions

Paraconsistent logic (PL) is a non-classical logic that accepts contradiction in its foundations. It can be represented in the form of paraconsistent annotated logic with annotation of two values (PAL2v). When used to model quantum phenomena, PAL2v is called paraquantum logic (PQL). In this work, the concept of PQL is applied to create a logical model presenting the fundamental principles of quantum mechanics that support particle-wave theory. This study uses the well-known Young’s double-slit experiment, wherein quantum phenomena appear when a monochromatic light beam passes through the two slits. We focused on a reference point located between the slits, where we observed the effects of two types of wave interferences in a region defined as a two-wave region (2W region). Considering that the effect in this 2W region is very similar to that studied by Huygens, we adopt a paraquantum logical model in which a particle (or quantum) is represented by two wave functions. The two wave functions result in four State Vectors (Ket, Bra,,) in the PQL Lattice that express the symmetry and the entanglement of Quantum Mechanics. The constructed model adapts well to the quantum phenomena, is strongly consistent, and can be considered as an innovative form of analysis in the field of quantum mechanics. Based on this model, we present in two parts (Part I and Part II) the comparative analysis of values found in SchrÖdinger’s equation and probabilistic models of wave-particle theory using Bonferroni inequality.

Relativity Theory and Paraquantum Logic—Part II:Fundamentals of an Unified Calculation

The studies of the PQL are based on propagation of Paraquantum logical states ψ in a representative Lattice of four vertices. Based in interpretations that consider resulting information of measurements in physical systems are found paraquantum equations for computation of the physical quantities in real physical systems. In the first part of this work we presented a study of Relativity theory which involved the time and the space with their characteristics as degrees of evidence applied in Paraquantum Logical Model. Now, in this second Part we present a study of application of the PQL in resolution of phenomena of physical systems that involve concepts of the Relativity Theory and the correlation of these effects with the Newtonian Universe and Quantum Mechanics. Considering physical fundamental quantities varying periodically in amplitude, we introduce the paraquantum equations which consider frequency in the analysis. From of these mathematical relationships obtained in the PQL Lattice some main physical constants related to the studies of De Broglie appeared. With the equations of Energy obtained through the analyses is demonstrated that the Paraquantum Logic is capable to correlate values and to unify the several study areas of the Physical Science.

Undulatory Theory with Paraconsistent Logic (Part II): Schrödinger Equation and Probability Representation

Part I of this study proved that the Paraconsistent Annotated Logic using two values (PAL2v), known as the Paraquantum Logic (PQL), can represent the quantum by a model comprising two wave functions obtained from interference phenomena in the 2W (two-wave) region of Young’s experiment (double slit). With this model represented in one spatial dimension, we studied in the Lattice of the PQL, with their values represented in the set of complex numbers, the state vector of unitary module and its correspondence with the two wave functions. Based on these considerations, we applied the PQL model for obtaining Paraquantum logical states ψ related to energy levels, following the principles of the wave theory through SchrÖdinger’s equation. We also applied the probability theory and Bonferroni’s inequality for demonstrating that quantum wave functions, represented by evidence degrees, are probabilistic functions studied in the PQL Lattice, confirming that the final Paraquantum Logic Model is well suited to studies involving aspects of the wave-particle theory. This approach of quantum theory using Paraconsistent logic allows the interpretation of various phenomena of Quantum Mechanics, so it is quite promising for creating efficient models in the physical analysis and quantum computing processes.

Model Checking over Paraconsistent Temporal Logic

Classical logic cannot be used to effectively reason about concurrent systems with inconsistencies (inconsistencies often occur, especially in the early stage of the development, when large and complex concurrent systems are developed). In this paper, we propose the use of a guasi-classical temporal logic (QCTL) for supporting the verification of temporal properties of such systems even where the consistent model is not available. Our models are paraKripke structures (extended standard Kripke structures), in which both a formula and its negation are satisfied in a same state, and properties to be verified are expressed by QCTL with paraKripke structures semantics. We introduce a novel notion of paraKripke models, which grasps the paraconsistent character of the entailment relation of QCTL. Furthermore, we explore the methodology of model checking over QCTL, and describe the detailed algorithm of implementing QCTL model checker. In the sequel, a simple example is presented, showing how to exploit the proposed model checking technique to verify the temporal properties of inconsistent concurrent systems.

A Comparison of Paraconsistent Description Logics

Description logics (DLs) are a family of logic-based knowledge representation formalisms with a number of computer science applications. DLs are especially well-known to be valuable for obtaining logical foundations of web ontology languages (e.g., W3C’s ontology language OWL). Paraconsistent (or inconsistency-tolerant) description logics (PDLs) have been studied to cope with inconsistencies which may frequently occur in an open world. In this paper, a comparison and survey of PDLs is presented. It is shown that four existing paraconsistent semantics (i.e., four-valued semantics, quasi-classical semantics, single-interpretation semantics and dual-interpretation semantics) for PDLs are essentially the same semantics. To show this, two generalized and extended new semantics are introduced, and an equivalence between them is proved.

An Application of Paraconsistent Annotated Logic for Design Software Testing Strategies

Nowadays, application model systems for decision-making based on non-classical logic such as Paraconsistent Logic are used successfully in the treatment of uncertainties. The method presented in this paper is based on the fundamental concepts of Paraconsistent Annotated Logic with annotation of 2 values (PAL2v). In this study, two algorithms based on PAL2v are presented gradually, to extract the effects of the contradiction in signals of information from a database of uncertain knowledge. The Paraconsistent Extractors Algorithms of Contradiction Effect-Para Extrctr is applied to filters of networks of analyses (PANets) of signal information, where uncertain and contradictory signals may be found. Software test case scenarios are subordinated to an application model of Paraconsistent decision-making, which provides an analysis using Paraconsistent Logic in the treatment of uncertainties for design software testing strategies. This quality-quantity criterion to evaluate the software product quality is based on the characteristics of software testability analysis. The Para consistent reasoning application model system presented in this case study, reveals itself to be more efficient than the traditional methods because it has the potential to offer an appropriate treatment to different originally contradicting source information.

Paraconsistent Differential Calculus(Part I):First-Order Paraconsistent Derivative

A type of Inconsistent Mathematics structured on Paraconsistent Logic (PL) and that has, as the main purpose, the study of common mathematical objects such as sets, numbers and functions, where some contradictions are allowed, is called Paraconsistent Mathematics. The PL is a non-Classical logic and its main property is to present tolerance for contradiction in its fundamentals without the invalidation of the conclusions. In this paper (part 1), we use the PL in its annotated form, denominated Paraconsistent Annotated Logic with annotation of two values—PAL2v for present a first-order Paraconsistent Derivative. The PAL2v has, in its representation, an associated lattice FOUR based on Hasse Diagram. This PAL2v-Lattice allows development of a Para-consistent Differential Calculus based on fundamentals and equations obtained by geometric interpretations. In this first article it is presented some examples applying derivatives of first-order with the concepts of Paraconsistent Mathematics. In the second part of this work we will show the Paraconsistent Derivative of second-order with application examples.

Paraconsistent Differential Calculus(Part II):Second-Order Paraconsistent Derivative

The Paraconsistent Logic (PL) is a non-classical logic and its main property is to present tolerance for contradiction in its fundamentals without the invalidation of the conclusions. In this paper, we use the PL in its annotated form, denominated Paraconsistent Annotated Logic with annotation of two values-PAL2v. This type of paraconsistent logic has an associated lattice that allows the development of a Paraconsistent Differential Calculus based on fundamentals and equations obtained by geometric interpretations. In this paper (Part II), it is presented a continuation of the first article (Part I) where the Paraconsistent Differential Calculus is given emphasis on the second-order Paraconsistent Derivative. We present some examples applying Paraconsistent Derivatives at functions of first and second-order with the concepts of Paraconsistent Mathematics.

An Introduction to Paraconsistent Integral Differential Calculus:With Application Examples

In this paper we show that it is possible to integrate functions with concepts and fundamentals of Paraconsistent Logic (PL). The PL is a non-classical Logic that tolerates the contradiction without trivializing its results. In several works the PL in his annotated form, called Paraconsistent logic annotated with annotation of two values (PAL2v), has presented good results in analysis of information signals. Geometric interpretations based on PAL2v-Lattice associate were obtained forms of Differential Calculus to a Paraconsistent Derivative of first and second-order functions. Now, in this paper we extend the calculations for a form of Paraconsistent Integral Calculus that can be viewed through the analysis in the PAL2v-Lattice. Despite improvements that can develop calculations in complex functions, it is verified that the use of Paraconsistent Mathematics in differential and Integral Calculus opens a promising path in researches developed for solving linear and nonlinear systems. Therefore the Paraconsistent Integral Differential Calculus can be an important tool in systems by modeling and solving problems related to Physical Sciences.

Treatment of Uncertainties with Algorithms of the Paraconsistent Annotated Logic

The method presented in this work is based on the fundamental concepts of Paraconsistent Annotated Logic with annotation of 2 values (PAL2v). The PAL2v is a non-classic Logics which admits contradiction and in this paper we perform a study using mathematical interpretation in its representative lattice. This studies result in algorithms and equations give an effective treatment on signals of information that represent situations found in uncertainty knowledge database. From the obtained equations, algorithms are elaborated to be utilized in computation models of the uncertainty treatment Systems. We presented some results that were obtained of analyses done with one of the algorithms that compose the paraconsistent analyzing system of logical signals with the PAL2v Logic. The paraconsistent reasoning system built according to the PAL2v methodology notions reveals itself to be more efficient than the traditional ones, because it gets to offer an appropriate treatment to contradictory information.

Paraconsistent Algorithm Extractor of Contradiction Effects―ParaExtr_(ctr)

Nowadays networks of analyses based in non-classic logics are used with success in the treatment of uncertainties. The characteristic of accepting the contradiction in his structure is the main cause of the methodologies based in Paraconsistent Logic is ideals for applications in systems of analyses and decision making. In this work we presented an algorithm based in Paraconsistent logic capable to extract in a gradual way the effects of the contradiction in originated signals of information of uncertain knowledge database. The Algorithm Paraconsistent Extractor of Contradiction effects - Paraextrctrctr is formed with base in fundamental concepts of the Paraconsistent Annotated Logic with annotation of two values (PAL2v) it can be applied in filters of networks of analyses of signal information where uncertain and contradictory signals can be present. The process of extraction of the effect of the contradiction is always begun by the largest inconsistency degree among two signals that belong to the group that is in analysis. 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 Paraextrctrctr 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.

Introducing the Paraquantum Equations and Applications

In this paper, we present an equationing method based on non-classical logics applied to resolution of problems which involves phenomena of physical science. A non-classical logic denominated of the Paraquantum Logic (PQL), which is based on the fundamental concepts of the Paraconsistent Annotated logic with annotation of two values (PAL2v), is used. The formalizations of the PQL concepts, which are represented by a lattice with four vertices, lead us to consider Paraquantum logical states ψ which are propagated by means of variations of the evidence Degrees extracted from measurements performed on the Observable Variables of the physical world. The studies on the lattice of PQL give us equations that quantify values of physical largenesses from where we obtain the effects of the propagation of the Paraquantum logical states ψ. The PQL lattice with such features can be extensively studied and we obtain a Paraquantum Logical Model with the capacity of contraction or expansion which can represent any physical universe. In this paper the Paraquantum Logical Model is applied to the Newton Laws where we obtain equations and verify the action of an expansion factor the PQL lattice called Paraquantum Gamma Factor γPψ and its correlation with another important factor called Paraquantum Factor of quantization hψ. We present numerical examples applied to real physical systems through the equations which deal with paraquantum physical largenesses and how these values are transmitted to the physical world. With the results of these studies we can verify that the Paraquantum Logical Model has the property of interconnect several fields of the Physical Science.

Study of the Interactions between Particles Based in Paraquantum Logic

Paraquantum Logic (P QL ) has its origins in the fundamental concepts of the Paraconsistent Annotated Logic (PAL) whose main feature is to be capable of treating contradictory information. In this work we presented a study of application of the P QL in resolution of phenomena of physical systems that involves the interactions between physical bodies or particles. Initially is considered that each particle or physical body that is in the physical world has a representative Lattice in the Paraquantum world. From this consideration is made a study of the phenomena of Paraquantum Entanglement modeling the interaction between particles based in fundamental concepts of the Paraquantum Logic. The mathematical relationships of representative Lattices of the Paraquantum Logic originate models with values that are identified with some physical constants. In this work these paraquantum values are identified with the Universal constant of Gravity, proposed by Newton, and the constant K, that relates the Interaction Force in charged particles in the Coulomb’s Law. The results showed that the Paraquantum Logical Model elaborated starting from the fundamental concepts of the Paraquantum Logic (P QL ) is adequate to support theories based in a Paraquantum Universe built by an infinite amount of Lattices and forming a Paraquantum net of infinite dimensions.

Dark Energy Calculations Using the Paraquantum Gamma Factor(γ_(Pψ)) on the Relativistic Energy Equation

A Paraconsistent Logic (PL) is a non-classical logic which revokes the principle of non-Contradiction and admits the treatment of contradictory information in its theoretical structure. Paraquantum Logic (PQL) is based on a type of PL denominated Paraconsistent Annotated Logic with annotation of two values (PAL2v). The PAL2v have a representative Lattice of four vertices (Lattice FOUR) where are made interpretations with construction of Paraquantum Logical Model and equations capable computation values extract of Observable Variable measurements. The studies of the PQL are based on propagation of Paraquantum logical states ψ in a Paraquantum Universe represented by PQL-Lattice of four vertices. These studies of PQL are based in two Paraquantum factors: the Paraquantum Gamma Factor (γPψ) that has his action in the measurements of Observable Variables in the Physical world and the Paraquantum Factor of quantization hψ, which has his action in the Paraquantum Universe. In this paper we analyze the application of Paraquantum Gamma Factor γPψ and its intrinsic characteristics that add important information into the equation of Einstein’s relativistic Energy (E = MC2). In this article were made several calculations to demonstrate the effects of applying the Paraquantum Gamma Factor (γPψ) in relativistic energy equation. It is found that the factors of using the Paraquantum Logical Model make an adjustment in the equation of Einstein’s relativistic Energy and identify related values with recent results found for the Dark Energy and dark matter. In the Paraquantum/Relativistic Energy equation the γPψ appears as an important factor of transition between the relativistic universe and the Newtonian Universe. The results suggest that its use would be very important in the interpretation of the behavior of other astronomical factors as the cosmological constant and gravitation.