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Mathematical Logic for Applications MIKLOS´ FERENCZI { MIKLOS´ SZOTS} MATHEMATICAL LOGIC FOR APPLICATIONS 2011 Abstract Referee Contents Technical editor Sponsorship Copyright Editorship ISBN 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{Sz}ots,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. TAMOP-´ 4.1.2-08/2/A/KMR-2009-0028. Prepared under the editorship of Budapest University of Technology and Eco- nomics, Mathematical Institute. Referee: K´arolyVarasdi Prepared for electronic publication by: Agota´ Busai Title page design: Gergely L´aszl´oCs´ep´any, Norbert T´oth ISBN: 978-963-279-460-0 Copyright: 2011{2016, Mikl´osFerenczi, Mikl´osSz}ots,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{Sz}ots,BME tankonyvtar.ttk.bme.hu Contents 0 INTRODUCTION2 1 ON THE CONCEPT OF LOGIC6 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 1 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 Index 116 tankonyvtar.ttk.bme.hu Ferenczi{Sz}ots,BME 0. INTRODUCTION 3 Chapter 0 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, G¨odel,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{Sz}ots,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
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