Um estudo sobre as origens da Lógica Matemáitca

The present study has as objective to explaining about the origins of the mathematical logic. This has its beginning attributed to the autodidactic English mathematician George Boole (1815-1864), especially because his books The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854) are recognized as the inaugural works of the referred branch. However, surprisingly, in the same time another mathematician called Augutus of Morgan (1806-1871) it also published a book, entitled Formal Logic (1847), in defense of the mathematic logic. Even so, times later on this same century, another work named Elements of Logic (1875) it appeared evidencing the Aristotelian logic with Richard Whately (1787-1863), considered the better Aristotelian logical of that time. This way, our research, permeated by the history of the mathematics, it intends to study the logic produced by these submerged personages in the golden age of the mathematics (19th century) to we compare the valid systems in referred period and we clarify the origins of the mathematical logic. For that we looked for to delineate the panorama historical wrapper of this study. We described, shortly, biographical considerations about these three representatives of the logic of the 19th century formed an alliance with the exhibition of their point of view as for the logic to the light of the works mentioned above. In this sense, we aspirated to present considerations about what effective Aristotelian´s logic existed in the period of Boole and De Morgan comparing it with the new emerging logic (the mathematical logic). Besides of this, before the textual analysis of the works mentioned above, we still looked for to confront the systems of Boole and De Morgan for we arrive to the reason because the Boole´s system was considered better and more efficient. Separate of this preponderance we longed to study the flaws verified in the logical system of Boole front to their contemporaries' production, verifying, for example, if they repeated or not. We concluded that the origins of the mathematical logic is in the works of logic of George Boole, because, in them, has the presentation of a new logic, matematizada for the laws of the thought similar to the one of the arithmetic, while De Morgan, in your work, expand the Aristotelian logic, but it was still arrested to her

A integração do tutorial interativo TryLogic via IMS Learning Tools Interoperability: construindo uma infraestrutura para o ensino de Lógica através de estratégias de demonstração e refutação

Logic courses represent a pedagogical challenge and the recorded number of cases of failures and of discontinuity in them is often high. Amont other difficulties, students face a cognitive overload to understand logical concepts in a relevant way. On that track, computational tools for learning are resources that help both in alleviating the cognitive overload scenarios and in allowing for the practical experimenting with theoretical concepts. The present study proposes an interactive tutorial, namely the TryLogic, aimed at teaching to solve logical conjectures either by proofs or refutations. The tool was developed from the architecture of the tool TryOcaml, through support of the communication of the web interface ProofWeb in accessing the proof assistant Coq. The goals of TryLogic are: (1) presenting a set of lessons for applying heuristic strategies in solving problems set in Propositional Logic; (2) stepwise organizing the exposition of concepts related to Natural Deduction and to Propositional Semantics in sequential steps; (3) providing interactive tasks to the students. The present study also aims at: presenting our implementation of a formal system for refutation; describing the integration of our infrastructure with the Virtual Learning Environment Moodle through the IMS Learning Tools Interoperability specification; presenting the Conjecture Generator that works for the tasks involving proving and refuting; and, finally to evaluate the learning experience of Logic students through the application of the conjecture solving task associated to the use of the TryLogic

Uma visita ao universo matemático de Lewis Carrol e o (re)encontro com sua lógica do nonsense

Notable mathematics teacher, Lewis Carroll, pseudonym of Charles Lutwidge Dodgson (1832-1898), made the mixture of mathematics with literature a ludic environment for learning that discipline. Author of Alice s Adventures In Wonderland and its sequel Alice Through The Looking Glass, he eventually created a real and complex universe which uses what we call the logic of the nonsense as an element to motivate the development of mathematical thinking of the reader, taking it as well, learn by establishing a link between the concrete (mathematics) and the imaginary (their universe). In order to investigate and discuss the educational potential of their works and state some elements that can contribute to a decentralized math education from the traditional method of following the models and decorate formulas, we visited his works based on the studies of archeology of knowledge (FOUCAULT, 2007), the rational thought and symbolic thinking (VERGANI, 2003) and about the importance of stories and narratives to the development of human cognition (FARIAS, 2006). Through a descriptive, analytical study, we used the literary construction and presented part of our study in form of a mathematical novel, to give the mathematical school a particular charm, without depriving it of its basics properties as discipline and content. Our study showed how the works of Carroll have a strong didactic element that can deploy in various activities of study and teaching for mathematics classes

Uma reavaliação do pensamento lógico de George Boole à luz da história da matemática

The aim of the present study is to reevaluate the logical thought of the English mathematician George Boole (1815 - 1864). Thus, our research centers on the mathematical analysis of logic in the context of the history of mathematics. In order to do so, we present various biographical considerations about Boole in the light of events that happened in the 19th century and their consequences for mathematical production. We briefly describe Boole's innovations in the areas of differential equations and invariant theory and undertake an analysis of Boole's logic, especially as formulated in the book The Mathematical Analysis of Logic, comparing it not only with the traditional Aristotelian logic, but also with modern symbolic logic. We conclude that Boole, as he intended, expanded logic both in terms of its content and also in terms of its methods and formal elaboration. We further conclude that his purpose was the mathematical modeling of deductive reasoning, which led him to present an innovative formalism for logic and, because the different ways it can be interpreted, a new conception of mathematics

Um estudo sobre a concepção de paradoxo segundo o pensamento de Augustus de Morgan

This work aims to analyze the concept of "paradox" posed in the work of The Budget Paradox (1872) of mathematical and logical English Augustus De Morgan (1806-1871). Here it is important to note that a large part of this book consists of re-prints of a series of writings by the author in journal Athenaeum, where its performance as auditor of literature. The tests refer to some scientific work produced between the years 1489 and 1866 and the rules of selection for the composition of the work is, basically, the methodological aspects used in the completion or disclosed by such scholars. The concept of paradox is presented in two distinct moments. At first, we found a study of definitions for the term in a philosophical approach, characterizing it as something that requires further investigation; which was complemented with the classic examples of a scientific context. In the second, we present a concept advocated by De Morgan and, under this perspective, that he conceptualized the "paradox" is directly related to the non-usual methods employed in the formulation of new scientific theories. In this study some of these scientific concepts are detailed, where, through the redemption history, engaging in issues of our study Mathematics, Physics, of Logic, among others. Possession of the preliminary analysis and comparison with the design of De Morgan, it became possible to diagnose some limitations in the conceptualization suggested by the author. Further, evidenced, in front of the cases, the nonlinearity of the process of production of knowledge and hence the progress of science

Estudo de estratégias de otimização para poços de petróleo com elevação por bombeio de cavidades progressivas

The progressing cavity pump artificial lift system, PCP, is a main lift system used in oil production industry. As this artificial lift application grows the knowledge of it s dynamics behavior, the application of automatic control and the developing of equipment selection design specialist systems are more useful. This work presents tools for dynamic analysis, control technics and a specialist system for selecting lift equipments for this artificial lift technology. The PCP artificial lift system consists of a progressing cavity pump installed downhole in the production tubing edge. The pump consists of two parts, a stator and a rotor, and is set in motion by the rotation of the rotor transmitted through a rod string installed in the tubing. The surface equipment generates and transmits the rotation to the rod string. First, is presented the developing of a complete mathematical dynamic model of PCP system. This model is simplified for use in several conditions, including steady state for sizing PCP equipments, like pump, rod string and drive head. This model is used to implement a computer simulator able to help in system analysis and to operates as a well with a controller and allows testing and developing of control algorithms. The next developing applies control technics to PCP system to optimize pumping velocity to achieve productivity and durability of downhole components. The mathematical model is linearized to apply conventional control technics including observability and controllability of the system and develop design rules for PI controller. Stability conditions are stated for operation point of the system. A fuzzy rule-based control system are developed from a PI controller using a inference machine based on Mandami operators. The fuzzy logic is applied to develop a specialist system that selects PCP equipments too. The developed technics to simulate and the linearized model was used in an actual well where a control system is installed. This control system consists of a pump intake pressure sensor, an industrial controller and a variable speed drive. The PI control was applied and fuzzy controller was applied to optimize simulated and actual well operation and the results was compared. The simulated and actual open loop response was compared to validate simulation. A case study was accomplished to validate equipment selection specialist system

Uma Fundamentação Intervalar Aplicada à Morfologia Matemática

This work present a interval approach to deal with images with that contain uncertainties, as well, as treating these uncertainties through morphologic operations. Had been presented two intervals models. For the first, is introduced an algebraic space with three values, that was constructed based in the tri-valorada logic of Lukasiewiecz. With this algebraic structure, the theory of the interval binary images, that extends the classic binary model with the inclusion of the uncertainty information, was introduced. The same one can be applied to represent certain binary images with uncertainty in pixels, that it was originated, for example, during the process of the acquisition of the image. The lattice structure of these images, allow the definition of the morphologic operators, where the uncertainties are treated locally. The second model, extend the classic model to the images in gray levels, where the functions that represent these images are mapping in a finite set of interval values. The algebraic structure belong the complete lattices class, what also it allow the definition of the elementary operators of the mathematical morphology, dilation and erosion for this images. Thus, it is established a interval theory applied to the mathematical morphology to deal with problems of uncertainties in images

Desenvolvimento de um sistema de automação para uma planta de inertização de resíduos por plasma

The traditional processes for treatment of hazardous waste are questionable for it generates other wastes that adversely affect people s health. As an attempt to minimize these problems, it was developed a system for treatment of hazardous waste by thermal plasma, a more appropriate technology since it produces high temperatures, preventing the formation of toxic pollutants to human beings. The present work brings out a solution of automation for this plant. The system has local and remote monitoring resources to ensure the operators security as well as the process itself. A special attention was given to the control of the main reactor temperature of the plant as it is the place where the main processing occurs and because it presents a complex mathematical model. To this, it was employed cascaded controls based on Fuzzy logic. A process computer, with a particular man-machine interface (MMI), provides information and controls of the plant to the operator, including by Internet. A compact PLC module is in charge of the central element of management automation and plant control which receives information from sensors, and sends it to the MMI

Uma proposta de regulador nebuloso para geradores eólicos de indução

This work presents a proposal for a voltage and frequency control system for a wind power induction generator. It has been developed na experimental structure composes basically by a three phase induction machine, a three phase capacitor and a reactive static Power compensator controlled by histeresys. lt has been developed control algorithms using conventional methods (Pl control) and linguistic methods (using concepts of logic and fuzzy control), to compare their performances in the variable speed generator system. The control loop was projected using the ADJDA PCL 818 model board into a Pentium 200 MHz compu ter. The induction generator mathematical model was studied throught Park transformation. It has been realized simulations in the Pspice@ software, to verify the system characteristics in transient and steady-state situations. The real time control program was developed in C language, possibilish verify the algorithm performance in the 2,2kW didatic experimental system

Arquitetura de controladores fuzzy em redes foundation fieldbus

Foundation Fieldbus Industrial networks are the high standard technology which allows users to create complex control logic and totally decentralized. Although being so advanced, they still have some limitations imposed by their own technology. Attempting to solve one of these limitations, this paper describes how to design a Fuzzy controller in a Foundation Fieldbus network using their basic elements of programming, the functional blocks, so that the network remains fully independent of other devices other than the same instruments that constitute it. Moreover, in this work was developed a tool that aids this process of building the Fuzzy controller, setting the internal parameters of functional blocks and informing how many and which blocks should be used for a given structure. The biggest challenge in creating this controller is exactly the choice of blocks and how to arrange them in order to effectuate the same functions of a Fuzzy controller implemented in other kind of environment. The methodology adopted was to divide each one of the phases of a traditional Fuzzy controller and then create simple structures with the functional blocks to implement them. At the end of the work, the developed controller is compared with a Fuzzy controller implemented in a mathematical program that it has a proper tool for the development and implementation of Fuzzy controllers, obtaining comparatives graphics of performance between both

Deciding difference logic in a Nelson-Oppen combination framework

O método de combinação de Nelson-Oppen permite que vários procedimentos de decisão, cada um projetado para uma teoria específica, possam ser combinados para inferir sobre teorias mais abrangentes, através do princípio de propagação de igualdades. Provadores de teorema baseados neste modelo são beneficiados por sua característica modular e podem evoluir mais facilmente, incrementalmente. Difference logic é uma subteoria da aritmética linear. Ela é formada por constraints do tipo x − y ≤ c, onde x e y são variáveis e c é uma constante. Difference logic é muito comum em vários problemas, como circuitos digitais, agendamento, sistemas temporais, etc. e se apresenta predominante em vários outros casos. Difference logic ainda se caracteriza por ser modelada usando teoria dos grafos. Isto permite que vários algoritmos eficientes e conhecidos da teoria de grafos possam ser utilizados. Um procedimento de decisão para difference logic é capaz de induzir sobre milhares de constraints. Um procedimento de decisão para a teoria de difference logic tem como objetivo principal informar se um conjunto de constraints de difference logic é satisfatível (as variáveis podem assumir valores que tornam o conjunto consistente) ou não. Além disso, para funcionar em um modelo de combinação baseado em Nelson-Oppen, o procedimento de decisão precisa ter outras funcionalidades, como geração de igualdade de variáveis, prova de inconsistência, premissas, etc. Este trabalho apresenta um procedimento de decisão para a teoria de difference logic dentro de uma arquitetura baseada no método de combinação de Nelson-Oppen. O trabalho foi realizado integrando-se ao provador haRVey, de onde foi possível observar o seu funcionamento. Detalhes de implementação e testes experimentais são relatados

Formal verification of PLC programs using the B Method

PLCs (acronym for Programmable Logic Controllers) perform control operations, receiving information from the environment, processing it and modifying this same environment according to the results produced. They are commonly used in industry in several applications, from mass transport to petroleum industry. As the complexity of these applications increase, and as various are safety critical, a necessity for ensuring that they are reliable arouses. Testing and simulation are the de-facto methods used in the industry to do so, but they can leave flaws undiscovered. Formal methods can provide more confidence in an application s safety, once they permit their mathematical verification. We make use of the B Method, which has been successfully applied in the formal verification of industrial systems, is supported by several tools and can handle decomposition, refinement, and verification of correctness according to the specification. The method we developed and present in this work automatically generates B models from PLC programs and verify them in terms of safety constraints, manually derived from the system requirements. The scope of our method is the PLC programming languages presented in the IEC 61131-3 standard, although we are also able to verify programs not fully compliant with the standard. Our approach aims to ease the integration of formal methods in the industry through the abbreviation of the effort to perform formal verification in PLCs