This book is recommended for those readers who have completed some intro. ductory course in Logic It can be used from the level MSc It is recommended. also to specialists who wish to apply Logic software engineers computer sci. entists physicists mathematicians philosophers linguists etc Our aim is to. give a survey of Logic from the abstract level to the applications with an em. phasis on the latter one An extensive list of references is attached As regards. problems or proofs for the lack of space we refer the reader to the literature. in general We do not go into the details of those areas of Logic which are. bordering with some other discipline e g formal languages algorithm theory. database theory logic design artificial intelligence etc We hope that the book. helps the reader to get a comprehensive impression on Logic and guide him or. her towards selecting some specialization, Key words and phrases Mathematical logic Symbolic logic Formal lan. guages Model theory Proof theory Non classical logics Algebraic logic Logic. programming Complexity theory Knowledge based systems Authomated the. orem proving Logic in computer science Program verification and specification. tankonyvtar ttk bme hu Ferenczi Szo ts BME, Acknowledgement of support. Prepared within the framework of the project Scientific training matemathics. and physics in technical and information science higher education Grant No. TA MOP 4 1 2 08 2 A KMR 2009 0028, Prepared under the editorship of Budapest University of Technology and Eco. nomics Mathematical Institute, Ka roly Varasdi, Prepared for electronic publication by. A gota Busai, Title page design, Gergely La szlo Cse pa ny Norbert To th. ISBN 978 963 279 460 0, Copyright 2011 2016 Miklo s Ferenczi Miklo s Szo ts BME. Terms of use of This work can be reproduced circulated published and. performed for non commercial purposes without restriction by indicating the. author s name but it cannot be modified, Ferenczi Szo ts BME tankonyvtar ttk bme hu. 0 INTRODUCTION 2, 1 ON THE CONCEPT OF LOGIC 6, 1 1 Syntax 6. 1 2 Basic concepts of semantics 8, 1 3 Basic concepts of proof theory 11. 1 4 On the connection of semantics and proof theory 13. 2 CLASSICAL LOGICS 16, 2 1 First order logic 16, 2 1 1 Syntax 16. 2 1 2 Semantics 18, 2 1 3 On proof systems and on the connection of semantics and. proof theory 21, 2 2 Logics related to first order logic 22. 2 2 1 Propositional Logic 22, 2 2 2 Second order Logic 24. 2 2 3 Many sorted logic 26, 2 3 On proof theory of first order logic 27. 2 3 1 Natural deduction 27, 2 3 2 Normal forms 30, 2 3 3 Reducing the satisfiability of first order sentences to propo. sitional ones 31, 2 3 4 Resolution calculus 33, 2 3 5 Automatic theorem proving 36. 2 4 Topics from first order model theory 37, 2 4 1 Characterizing structures non standard models 38. 2 4 2 Reduction of satisfiability of formula sets 41. 2 4 3 On non standard analysis 42, 3 NON CLASSICAL LOGICS 46. 3 1 Modal and multi modal logics 46, 3 2 Temporal logic 49. 3 3 Intuitionistic logic 51, 3 4 Arrow logics 54, 3 4 1 Relation logic RA 54. 3 4 2 Logic of relation algebras 54, 3 5 Many valued logic 55. 2 MATHEMATICAL LOGIC FOR APPLICATIONS, 3 6 Probability logics 57. 3 6 1 Probability logic and probability measures 57. 3 6 2 Connections with the probability theory 60, 4 LOGIC AND ALGEBRA 62. 4 1 Logic and Boolean algebras 63, 4 2 Algebraization of first order logic 66. 5 LOGIC in COMPUTER SCIENCE 68, 5 1 Logic and Complexity theory 68. 5 2 Program verification and specification 72, 5 2 1 General introduction 72. 5 2 2 Formal theories 74, 5 2 3 Logic based software technologies 78. 5 3 Logic programming 81, 5 3 1 Programming with definite clauses 82. 5 3 2 On definability 84, 5 3 3 A general paradigm of logic programming 87. 5 3 4 Problems and trends 88, 6 KNOWLEDGE BASED SYSTEMS 93. 6 1 Non monotonic reasoning 94, 6 1 1 The problem 94. 6 1 2 Autoepistemic logic 95, 6 1 3 Non monotonic consequence relations 97. 6 2 Plausible inference 99, 6 3 Description Logic 102. Bibliography 106, tankonyvtar ttk bme hu Ferenczi Szo ts BME. 0 INTRODUCTION 3, INTRODUCTION, 1 Logic as an applied science The study of logic as a part of philosophy has. been in existence since the earliest days of scientific thinking Logic or math. ematical logic from now logic was developed in the 19th century by Gottlob. Frege Logic has been a device to research foundations of mathematics based. on results of Hilbert Go del Church Tarski and main areas of Logic became. full fledged branches of Mathematics model theory proof theory etc The. elaboration of mathematical logic was an important part of the process called. revolution of mathematics at the beginning of the 20th century Logic had. an important effect on mathematics in the 20th century for example on alge. braic logic non standard analysis complexity theory set theory. The general view of logic has changed significantly over the last 40 years. or so The advent of computers has led to very important real word appli. cations To formalize a problem to draw conclusions formally to use formal. methods have been important tasks Logic started playing an important role in. software engineering programming artificial intelligence knowledge represen. tation database theory linguistics etc Logic has become an interdisciplinary. language of computer science, As with such applications this has in turn led to extensive new areas of. logic e g logic programming special non classical logics as temporal logic or. dynamic logic Algorithms have been of great importance in logic Logic has. come to occupy a central position in the repertory of technical knowledge and. various types of logic started playing a key roles in the modelling of reasoning. and in other special fields from law to medicine All these developments assign. a place to Applied Logic within the system of science as firm as that of applied. mathematics, As an example for comparing the applications and developing theoretical. foundations of logic let us see the case of artificial intelligence AI for short. AI is an attempt to model human thought processes computationally Many. non classical logics such as temporal dynamic arrow logics are investigated. nowadays intensively because of their possible applications in AI But many. among these logics had been researched by mathematicians philosophers and. linguists before the appearance of AI only from a theoretical viewpoint and the. results were applied in AI later besides new logics were also developed to meet. the needs of AI In many respects the tasks of the mathematician and the AI. worker are quite similar They are both concerned with the formalization of. Ferenczi Szo ts BME tankonyvtar ttk bme hu, 4 MATHEMATICAL LOGIC FOR APPLICATIONS. certain aspects of reasoning needed in everyday practice Philosopher mathe. matician and engineers all use the same logical techniques i e formal languages. structures proof systems classical and non classical logics the difference be. tween their approaches residing in where exactly they put the emphasis when. applying the essentially same methods, 2 Classical and non classical logics Chapter 2 is devoted to classi. cal first order logic and to logics closely related to it called classical logics. Classical first order logic serves as a base for every logic therefore it is consid. ered as the most important logic Its expressive power is quite strong contrary. to propositional logic for example and it has many nice properties e g com. pleteness compactness etc in contrast to second order logic for example. It is said to be the logic of mathematics and its language is said to be the. language of mathematics The reader is advised to understand the basic con. cepts of logic by studying classical first order logic to prepare the study of other. areas of logic, However classical logics describe only static aspects of the modelled seg. ment of the world To develop a more comprehensive logical model multiple. modalities are to be taken into consideration what is necessary and what. is occasional what is known and what is believed what is obligatory and. what is permitted past present future sources of information and their. reliability uncertainty and incompleteness of information among others. A wide variety of logics have been developed and put to use to model the. aspects mentioned above in artificial intelligence computer science linguistics. etc Such logics are called non classical logics We sketch some important. ones among them in Chapter 3 without presenting the whole spectrum of these. logics which would be far beyond the scope of this book. 3 On the concept of logic Since many kinds of special logics are used in. applications a general frame has been defined for logic see Chapter 1 which. is general but efficient enough to include many special logics and to preserve. most of their nice properties, It is worth understanding logic at this general level for a couple of reasons. First we need to distinguish the special and general features of the respective. concrete logics anyway Second it often happens that researchers have to form. their own logical model for a situation in real life In this case they can specialize. a general logic in a way suitable for the situation in question The general theory. of logic or Universal Algebraic Logic is a new and quickly developing area inside. logic see Andre ka H Ne meti I Sain I Universal Algebraic Logic Springer. Chapters 1 and 3 are based among others on the papers Andre ka H. Ne meti I General algebraic logic A perspective on What is logic in What is. logical system Oxford Ed D Gabbay 1994 11 and Andre ka H Ne meti. I Sain I Kurucz A Applying Algebraic Logic A General Methodology. Lecture Notes of the Summer School Algebraic Logic and the Methodology of. Applying it Budapest 1994 67 pages 13, We note that there is a clear difference between a concrete logic with fixed. non logical symbols and a class of concrete logics only the logical symbols. are fixed The latter is a kind of generalization of the concrete ones of course. Usually by logic we understand a class of logics but the reader should be. careful the term logic because the term is used also for a concrete logic We. must not confuse the different degrees of generalizations. tankonyvtar ttk bme hu Ferenczi Szo ts BME, 0 INTRODUCTION 5. 4 Areas of mathematics connected with logic An important aspect of. this study is the connection between Logic and the other areas of mathematics. There are areas of mathematics which are traditionally close to Logic Such. areas are algebra set theory algorithm theory, For example modern logic was defined originally in algebraic form by Boole. De Morgan and Peirce An efficient method in Algebra in Logic for problem. solving is the following translate the problem to Logic to Algebra and solve it. in logical in algebraic form The scientific framework of this kind of activity is. the discipline called Algebraic Logic founded in the middle of the 20th century. by Tarski Henkin Sikorski etc This area is treated in Chapter 4. There are areas in mathematics which originally seemed fairly remote from. Logic but later important and surprising logical connections were discovered. between them For example such an area is Analysis In the sixties Abraham. Robinson worked out the exact interpretation of infinitesimals through a sur. prising application of the Compactness Theorem of First Order Logic Many. experts believe this theory to be a more natural model for differential and in. tegral calculus than the traditional model the more traditional method. besides analysis Robinson s idea was applied to other areas of Mathematics. too and this is called non standard mathematics This connection is discussed. in Section 2 4 3, We also sketch some connections between Logic and Probability theory. 5 The two levels of logics Every logic has two important levels the. level of semantics and that of proof theory or proof systems or syntax For most. logics these two levels two approaches are equivalent in a sense It is important. to notice that both levels use the same formal language as a prerequisite So. every logic has three basic components formal language semantics and proof. theory see 11 13 We make some notices on these components respectively. The concept of language is of great importance in any area of logics When. we model a situation in real life the first thing we choose is a suitable language. more or less describing the situation in question We note that today the theory. of formal languages is an extensive complex discipline and only a part of this. theory is used in Logic directly, Logical semantics is the part of logic which is essentially based on the the. ory of infinite sets In general in the definitions of semantics there are no. algorithms Nevertheless it is extraordinarily important in many areas of ap. plications Semantics is a direct approach to the physical reality. Proof theory is the part of logic which is built on certain formal manipula. tions of given symbols It is a generalization of a classical axiomatic method. The central concept of proof theory is the concept of a proof system or calcu. lus Setting out from proof systems algorithms can be developed for searching. 5 3 Logic programming elaboration of mathematical logic was an 4 MATHEMATICAL LOGIC FOR APPLICATIONS certain aspects of reasoning

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