DOCSLIB.ORG

Logic and Logic Programming J.A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Logic and Logic Programming J.A Logic Programming Logic and Logic Programming J.A. Robinson ogic has been around for a all mathematical concepts and for ery of the logical and set-theoretic very long time [23]. It was the formulation of exact deductive paradoxes (such as Bertrand Rus- already an old subject 23 reasoning about them. It seems to sell's set of all sets which are not centuries ago, in Aristotle's be so. The principal feature of the members of themselves, which day (384-322 BC). While predicate calculus is that it offers a therefore by definition both is, and Aristotle was not its origina- precise characterization of the con- also is not, a member of itself); and Ltor, despite a widespread impres- cept of proof. Its proofs, as well as its the huge reductionist work Prin- sion to the contrary, he was cer- sentences and its other formal ex- cipia Mathematica by Bertrand Rus- tainly its first important figure. He pressions, are mathematically de- sell and Alfred North Whitehead. placed logic on sound systematic fined objects which are intended All of these developments had ei- foundations, and it was a major not only to express ideas meaning- ther shown what could be done, or course of study in his own univer- fully--that is, to be used as one uses had revealed what needed to be sity in Athens. His lecture notes on a language--but also to be the sub- done, with the help of this new logic can still be read today. No ject matter of mathematical analy- logic. But it was necessary first for doubt he taught logic to the future sis. They are also capable of being mathematicians to master its tech- Alexander the Great when he manipulated as the data objects of niques and to explore its scope and served for a time as the young construction and recognition algo- its limits. prince's personal tutor. In Alexan- rithms. Significant early steps toward dria a generation later (about 300 At the end of the nineteenth cen- this end were taken by Leopold B.C.), Euclid played a similar role tury, mathematics had reached a Lowenheim (1915), [29] and in systematizing and teaching the stage in which it was more than Thoralf Skolem [45], who studied geometry and number theory of ready to exploit Frege's powerful the symbolic "satisfiability" of for- that era. Both Aristotle's logic and new instrument. Mathematicians mal expressions. They showed that Euclid's geometry have endured were opening up new areas of re- sets of abstract logical conditions and prospered. In some high search that demanded much could be proved consistent by being schools and colleges, both are still deeper logical understanding and given specific interpretations con- taught in a form similar to their far more careful handling of structed from the very symbolic original one. The old logic, how- proofs, than had previously been expressions in which they are for- ever, like the old geometry, has by required. Some of these were David mulated. Their work opened the now evolved into a much more gen- Hiibert's abstract axiomatic recast- way for Kurt G6del (1930, [17]) and eral and powerful form. ing of geometry and Giuseppe Jacques Herbrand (1930, [19]) to Modern ('symbolic' or 'mathe- Peano's of arithmetic, as well as prove, in their doctoral disserta- matical') logic dates back to 1879, Georg Cantor's intuitive explora- tions, the first versions of what is when Frege published the first ver- tions of general set theory, espe- now called the completeness of the sion of what today is known as the cially his elaboration of the dazzling predicate calculus. G6del and predicate calculus [14]. This system theory of transfinite ordinal and Herbrand both demonstrated that provides a rich and comprehensive cardinal numbers. Others were the proof machinery of the predi- notation, which Frege intended to Ernst Zermelo's axiomatic analysis cate calculus can provide a formal be adequate for the expression of of set theory following the discov- proof for every logically true prop- COMMUNICATIONS OF THE ACM/March 1992/Vol.35, No.3 41 Logic Programming osition, and indeed they each gave property of the predicate calculus. time British code-breaking project a constructive method for finding There had until then been an in- included his participation in the the proof, given the proposition. tense search for a positive solution actual design, construction and G6del's more famous achievement, to what Hilbert called the decision operation of several electronic ma- his discovery in 1931 of the amaz- problem--the problem to devise an chines of this kind, and thus he ing 'incompleteness theorems' algorithm for the predicate calculus must surely be reckoned as one of about formalizations of arithmetic, which would correctly determine, the major pioneers in their early has tended to overshadow this im- for any formal sentence B and any development. portant earlier work of his, which is set A of formal sentences, whether a result about pure logic, whereas or not B is a logical consequence of Logic on the Computer his incompleteness results are about A. Church and Turing found that Apart from this enormously impor- certain applied logics (formal axio- despite the existence of the proof tant cryptographic intelligence matic theories of elementary num- procedure, which correctly recog- work and its crucial role in ballistic ber theory, and similar systems) nizes (by constructing a proof of B computations and nuclear physics and do not directly concern us from A) all cases where B is in fact a simulations, the war-time develop- here. logical consequence of A, there is ment of electronic digital comput- The completeness of the predi- not and cannot be an algorithm ing technology had relatively little cate calculus links the syntactic which can similarly correctly recog- impact on the outcome of the war property of formal provability with nize all cases in which B is not a logi- itself. After the war, however, its the conceptually quite different cal consequence of A. Their discov- rapid commercial and scientific semantic property of logical truth. ery bears directly on all attempts to exploitation quickly launched the It assures us that each property be- write theorem-proving software. It current computer era. By 1950, longs to exactly the same sentences. means that it is pointless to try to much-improved versions of some Formal syntax and formal seman- program a computer to answer 'yes' of the war-time general-purpose tics are both needed, but for a time or 'no' correctly to every question of electronic digital computers be- the spotlight was on formal syntax, the form 'is this a logically true sen- came available to industry, univer- and formal semantics had to wait tence?' The most that can be done is sities and research centers. By the until Alfred Tarski (1934, [46]) in- to identify useful subclasses of sen- mid-1950s it had become apparent troduced the first rigorous semanti- tences for which a decision proce- to many logicians that, at last, suffi- cal theory for the predicate calcu- dure can be found. Many such sub- cient computing power was now at lus, by precisely defining satisfi- classes are known. They are called hand to support computational ability, truth (in a given 'solvable subcases of the decision experiments with the predicate cal- interpretation), logical consequence, problem', but as far as I know none culus proof procedure. It was just a and other related notions. Once it of them have turned out to be of matter of programming it and try- was filled out by the concepts of much practical interest. ing it on some real examples. Sev- Tarski's semantics, the theory of the When World War II began in eral papers describing projects for predicate calculus was no longer 1939 all the basic theoretical foun- doing this were given at a Summer unbalanced. Shortly afterward Ger- dations of today's computational School in Logic held at Cornell hard Gentzen (1936, [15]) further logic were in place. What was still University in 1957. One of these sharpened the syntactical results on lacking was any practical way of ac- [37, pp. 74-76] was by Abraham provability by showing that if a sen- tually carrying out the vast symbolic Robinson, the logician who later tence can be proved at all, then it computations called for by the surprised the mathematical estab- can be proved in a 'direct' way, proof procedure. Only the very lishment by applying logical 'non- without the need to introduce any simplest of examples could be done standard' model theory to legiti- extraneous 'clever' concepts; those by hand. Already there were those, mize infinitesimals in the occurring already in the sentence however--Turing himself for one-- foundations of the integral and dif- itself are always sufficient. who were making plans which ferential calculus. Other published All of these positive discoveries would eventually fill this gap. Tur- accounts of results in the first wave of the 1920s and 1930s laid the ing's method in negatively solving of such experiments were [12, 16, foundations on which today's pred- the decision problem had been to 35, 49]. There had also been, in icate calculus theorem-proving pro- design a highly theoretical, abstract 1956, a strange experiment by [33] grams, and thus logic program- version of the modern stored- which attracted a lot of attention at ming have been built. program, general-purpose univer- the time. It has since been cited as a Not all the great logical discover- sal digital computer (the 'universal milestone of the early stages of arti- ies of this period were positive.
Load more

Recommended publications
  • Infinite Computation, Co-Induction and Computational Logic
    Infinite Computation, Co-induction and Computational Logic Gopal Gupta Neda Saeedloei, Brian DeVries, Richard Min, Kyle Marple, Feliks Klu´zniak Department of Computer Science, University of Texas at Dallas, Richardson, TX 75080. Abstract. We give an overview of the coinductive logic programming paradigm. We discuss its applications to modeling ω-automata, model checking, verification, non-monotonic reasoning, developing SAT solvers, etc. We also discuss future research directions. 1 Introduction Coinduction is a technique for reasoning about unfounded sets [12], behavioral properties of programs [2], and proving liveness properties in model checking [16]. Coinduction also provides the foundation for lazy evaluation [9] and type inference [21] in functional programming as well as for interactive computing [33]. Coinduction is the dual of induction. Induction corresponds to well-founded structures that start from a basis which serves as the foundation: e.g., natural numbers are inductively defined via the base element zero and the successor function. Inductive definitions have 3 components: initiality, iteration and mini- mality. For example, the inductive definition of lists of numbers is as follows: (i) [] (empty list) is a list (initiality); (ii) [H|T] is a a list if T is a list and H is some number (iteration); and, (iii) the set of lists is the smallest set satisfying (i) and (ii) (minimality). Minimality implies that infinite-length lists of numbers are not members of the inductively defined set of lists of numbers. Inductive definitions correspond to least fixed point (LFP) interpretations of recursive definitions. Coinduction eliminates the initiality condition and replaces the minimality condition with maximality. Thus, the coinductive definition of a list of numbers is: (i) [H|T] is as a list if T is a list and H is some number (iteration); and, (ii) the set of lists is the largest set of lists satisfying (i) (maximality).
    [Show full text]
  • Resources for Further Study
    Resources for Further Study A number of valuable resources are available for further study of philosophical logic. In addition to the books and articles cited in the references at the end of each chapter included in this volume, there are four general categories of resources that can be consulted for information about the history and current research developments in philosophical logic. Additional materials can be found by soliciting advice from logicians, philosophers, and mathematicians at local colleges and universities. Logic Handbooks Many university and independent presses publish books on or related to mathematical and philo- sophical logic. There are also several special series of original monographs in logic that are worth investigating. The literature is too vast to justify a selection of the individual books that have con- tributed to the development of logic. We can nevertheless identify special categories of texts of special interest, beginning with handbooks and book series dedicated to logic and philosophical logic. Here are some recent relevant publications: Boyer, Robert S. (1988) A Computational Logic Handbook. Boston, MA: Academic Press. Sherwood, John C. (1960) Discourse of Reason: A Brief Handbook of Semantics and Logic. New York: Harper & Row. Handbook of Fuzzy Computation, ed. Enrique H. Ruspini, Piero P. Bonissone and Witold Pedrycz. Philadelphia, PA: Institute of Physics Publications, 1998. Handbook of Logic and Language, ed. Johan van Benthem and Alice ter Meulen. Cambridge, MA: MIT Press, 1997. Handbook of Logic in Artificial Intelligence and Logic Programming, ed. Dov Gabbay, C. J. Hogger, and J. A. Robinson. Oxford: Clarendon Press, 1993–8. Emmet, E. R. (1984) Handbook of Logic.
    [Show full text]
  • Types in Logic, Mathematics, and Programming from the Handbook of Proof Theory
    CHAPTER X Types in Logic, Mathematics and Programming Robert L. Constable Computer Science Department, Cornell University Ithaca, New York 1~853, USA Contents 1. Introduction ..................................... 684 2. Typed logic ..................................... 692 3. Type theory ..................................... 726 4. Typed programming languages ........................... 754 5. Conclusion ...................................... 766 6. Appendix ...................................... 768 References ........................................ 773 HANDBOOK OF PROOF THEORY Edited by S. R. Buss 1998 Elsevier Science B.V. All rights reserved 684 R. Constable 1. Introduction Proof theory and computer science are jointly engaged in a remarkable enter- prise. Together they provide the practical means to formalize vast amounts of mathematical knowledge. They have created the subject of automated reasoning and a digital computer based proof technology; these enable a diverse community of mathematicians, computer scientists, and educators to build a new artifact a globally distributed digital library of formalized mathematics. I think that this artifact signals the emergence of a new branch of mathematics, perhaps to be called Formal Mathematics. The theorems of this mathematics are completely formal and are processed digitally. They can be displayed as beautifully and legibly as journal quality mathematical text. At the heart of this library are completely formal proofs created with computer assistance. Their correctness is based on the axioms and rules of various foundational theories; this formal accounting of correctness supports the highest known standards of rigor and truth. The need to formally relate results in different foundational theories opens a new topic in proof theory and foundations of mathematics. Formal proofs of interesting theorems in current foundational theories are very large rigid objects. Creating them requires the speed and memory capacities of modern computer hardware and the expressiveness of modern software.
    [Show full text]
  • Contribution of Warsaw Logicians to Computational Logic
    axioms Article Contribution of Warsaw Logicians to Computational Logic Damian Niwi ´nski Institute of Informatics, University of Warsaw, 02-097 Warsaw, Poland; [email protected]; Tel.: +48-22-554-4460 Academic Editor: Urszula Wybraniec-Skardowska Received: 22 April 2016; Accepted: 31 May 2016; Published: 3 June 2016 Abstract: The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal m-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the m-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations.
    [Show full text]
  • Computational Logic: Memories of the Past and Challenges for the Future
    Computational Logic: Memories of the Past and Challenges for the Future John Alan Robinson Highland Institute, 96 Highland Avenue, Greenfield, MA 01301, USA. Abstract. The development of computational logic since the introduction of Frege's modern logic in 1879 is presented in some detail. The rapid growth of the field and its proliferation into a wide variety of subfields is noted and is attributed to a proliferation of subject matter rather than to a proliferation of logic itself. Logic is stable and universal, and is identified with classical first order logic. Other logics are here considered to be first order theories, syntactically sugared in notationally convenient forms. From this point of view higher order logic is essentially first order set theory. The paper ends by presenting several challenging problems which the computational logic community now faces and whose solution will shape the future of the field. 1 Introduction Although logic and computing are each very old subjects, going back very much more than a mere hundred years, it is only during the past century that they have merged and become essentially one subject. As the logican Quine wrote in 1960, "The basic notions of proof theory converge with those of machine computing. ... The utterly pure theory of mathematical proof and the utterly technological theory of machine computation are thus at bottom one, and the basic insights of each are henceforth insights of the other" ([1], p. 41). The aptly-named subject of computational logic - no pun intended - subsumes a wide variety of interests and research activities. All have something to do with both logic and computation.
    [Show full text]
  • Finding Unity in Computational Logic
    Finding Unity in Computational Logic Dale Miller INRIA Saclay and LIX/Ecole´ polytechnique Route de Saclay, 91128 PALAISEAU Cedex, France [email protected] While logic was once developed to serve philosophers and mathematicians, it is increasingly serving the varied needs of computer scientists. In fact, recent decades have witnessed the creation of the new discipline of Computational Logic. While Computation Logic can claim involvement in many, diverse areas of computing, little has been done to systematize the foundations of this new discipline. Here, we envision a unity for Computational Logic organized around recent developments in the theory of sequent calculus proofs. We outline how new tools and methodologies can be developed around a boarder approach to computational logic. Computational logic, unity of logic, proof theory 1. SOFTWARE AND HARDWARE CORRECTNESS Our ability to provide at least some formal guarantees IS CRITICALLY IMPORTANT about software systems will be directly related to our ability to deploy new functionality and services. The de- Computer systems are everywhere in our societies manding requirements in safety critical systems matched and their integration with all parts of our lives is with our inability to formally establish their correct behav- constantly increasing. There are a host of computer ior means that developments and deployments of, say, systems—such as those in cars, airplanes, missiles, avionic systems or high-speed train switches, moves at hospital equipment—where correctness of software is a glacial pace and clings to old technological solutions paramount. In the area of consumer electronics, big that have no history of failing. If we cannot distinguish changes in the attitude towards correctness has taken applets from viruses, we cannot expect people to really place.
    [Show full text]
  • Computational Logic in the First Semester of Computer Science: an Experience Report
    Computational Logic in the First Semester of Computer Science: An Experience Report David M. Cerna1;2 a, Martina Seidl1 b, Wolfgang Schreiner2 c, Wolfgang Windsteiger2 d and Armin Biere1 e 1Institute of Formal Methods and Verification, Johannes Kepler University, Austria 2Research Institute for Symbolic Computation, Johannes Kepler University, Austria Keywords: Logic, Education, Automated Reasoning. Abstract: Nowadays, logic plays an ever-increasing role in modern computer science, in theory as well as in practice. Logic forms the foundation of the symbolic branch of artificial intelligence and from an industrial perspective, logic-based verification technologies are crucial for major hardware and software companies to ensure the correctness of complex computing systems. The concepts of computational logic that are needed for such pur- poses are often avoided in early stages of computer science curricula. Instead, classical logic education mainly focuses on mathematical aspects of logic depriving students to see the practical relevance of this subject. In this paper we present our experiences with a novel design of a first-semester bachelor logic course attended by about 200 students. Our aim is to interlink both foundations and applications of logic within computer science. We report on our experiences and the feedback we got from the students through an extensive survey we performed at the end of the semester. 1 INTRODUCTION 2009; Calcagno et al., 2015; Cook, 2018) driving investments in advancing logical reasoning tools. Recently, J. A. Makowsky and A. Zaman- While logic remains fundamental and pervasive, sky (Makowsky and Zamansky, 2017) reported its importance is not directly obvious in the way it is on an undesirable, but ubiquitous fact: courses on classically taught.
    [Show full text]
  • The Scientific Contribution of Marek Sergot
    The Technical Contribution of Marek Sergot Steve Barker1, Andrew J. I. Jones1, Tony Kakas2, Bob Kowalski3, Rob Miller4, Stephen Muggleton3, and Giovanni Sartor5 1Department of Computer Science, King's College London, UK 2Department of Computer Science, University of Cyprus 3Department of Computing, Imperial College London, UK 4Department of Information Studies, University College London, UK 5CIRSFID, University of Bologna, Italy Marek Sergot's technical contributions range over different subjects. He has developed a series of novel ideas and formal methods bridging different research domains, such as artificial intelligence, computational logic, philosophical logic, legal theory, artificial intelligence and law, multi-agent systems and bioinformat- ics. By combining his background in logic and computing, with his interest for the law and his capacity to understand the subtleties of social interactions and normative reasoning, Marek has been able to open new directions of research, and has been a reference, an inspiration, and a model for many researchers in logic, computing and law. 1 Early work in Logic Programming Marek's early research focused on logic programming, deductive databases and legal reasoning. This led to his development of the query-the-user extension of logic programming [29], in which the user provides information during the execution of a logic program, if and when the program requires it. With query- the-user, interaction between the computer and the user is symmetric | each can ask questions and provide answers to the other. Marek collaborated with Peter Hammond to augment Prolog with query-the- user and explanation facilities, developing the expert system shell APES [13], which was marketed by their small company, Logic Based Systems Ltd.
    [Show full text]
  • Computability Logic გამოთვლადობის ლოგიკა Логика Вычислимости 可计算性逻辑 (Col)
    12/14/2016 Giorgi Japaridze A Survey of Computability Logic გამოთვლადობის ლოგიკა Логика вычислимости 可计算性逻辑 (col) Computability is one of the most interesting and fundamental concepts in mathematics and computer science, and it is natural to ask what logic it induces. This is where Computability Logic (CoL) comes in. It is a formal theory of computability in the same sense as classical logic is a formal theory of truth. In a broader and more proper sense, CoL is not just a particular theory but an ambitious and challenging program for redeveloping logic following the scheme “from truth to computability”. Under the approach of CoL, logical operators stand for operations on computational problems, formulas represent such problems, and their “truth” is seen as algorithmic solvability. In turn, computational problems ­­­ understood in their most general, interactive sense ­­­ are defined as games played by a machine against its environment, with “algorithmic solvability” meaning existence of a machine which wins the game against any possible behavior of the environment. With this semantics, CoL provides a systematic answer to the question “what can be computed?”, just like classical logic is a systematic tool for telling what is true. Furthermore, as it happens, in positive cases “what can be computed” always allows itself to be replaced by “how can be computed”, which makes CoL of potential interest in not only theoretical computer science, but many applied areas as well, including constructive applied theories, interactive knowledge base systems, resource oriented systems for planning and action, or declarative programming languages. Currently CoL is still at an early stage of development, with open problems prevailing over answered questions.
    [Show full text]
  • Logicism and the Development of Computer Science*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UCL Discovery Logicism and the Development of Computer Science* By Donald Gillies, King’s College London A Paper for Bob Kowalski’s 60th Birthday. Published in Antonis C. Kakas and Fariba Sadri (eds,) Computational Logic: Logic Programming and Beyond, Part II, Springer, 2002, pp. 588-604. Abstract This paper argues for the thesis that ideas originating in the philosophy of mathematics have proved very helpful for the development of computer science. In particular, logicism, the view that mathematics can be reduced to logic, was developed by Frege and Russell, long before computers were invented, and yet many of the ideas of logicism have been central to computer science. The paper attempts to explain how this serendipity came about. It also applies Wittgenstein’s later theory of meaning to human-computer interaction, and draws the conclusion that computers do understand the meaning of the symbols they process. The formal language of logic is suitable for humans trying to communicate with computers. Contents 1. Introduction 2. Philosophy of Mathematics in the Foundational Period 3. Logicism and Computer Science 4. How Computer Science has affected Logic 5. A Criticism of Logicism by Wittgenstein and its Significance 1. Introduction Philosophy is often thought of as an activity, which may have considerable theoretical interest, but which is of little practical importance. Such a view of philosophy is, in my opinion, profoundly mistaken. On the contrary, I would claim that philosophical ideas and some kind of philosophical orientation are necessary for many quite practical activities.
    [Show full text]
  • Acl2s: “The ACL2 Sedan”
    Electronic Notes in Theoretical Computer Science 174 (2007) 3–18 www.elsevier.com/locate/entcs ACL2s: “The ACL2 Sedan” Peter C. Dillinger Panagiotis Manolios Daron Vroon College of Computing Georgia Institute of Technology Atlanta, Georgia 30332-0280, USA {peterd,manolios,vroon}@cc.gatech.edu J Strother Moore Department of Computer Sciences University of Texas at Austin Austin, TX 78712-1188, USA [email protected] Abstract ACL2 is the latest inception of the Boyer-Moore theorem prover, the 2005 recipient of the ACM Software System Award. In the hands of experts it feels like a finely tuned race car, and it has been used to prove some of the most complex theorems ever proved about commercially designed systems. Unfortunately, ACL2 has a steep learning curve. Thus, novices tend have a very different experience: they crash and burn. As part of a project to make ACL2 and formal reasoning safe for the masses, we have developed ACL2s, the ACL2 sedan. ACL2s includes many features for streamlining the learning process that are not found in ACL2. In general, the goal is to develop a tool that is “self-teaching,” i.e., it should be possible for an undergraduate to sit down and play with it and learn how to program in ACL2 and how to reason about the programs she writes. Keywords: ACL2, Eclipse, theorem proving, script management 1 Introduction “ACL2” stands for “A Computational Logic for Applicative Common Lisp.” It is the name of a programming language, a first-order mathematical logic based on recursive functions, and a mechanical theorem prover for that logic [9,5,4].
    [Show full text]
  • Computational Logic and Human Thinking: How to Be Artificially Intelligent
    Computational Logic and Human Thinking: How to be Artificially Intelligent Robert Kowalski Department of Computing Imperial College London http://www.doc.ic.ac.uk/~rak/ 21 November 2010 http://www.doc.ic.ac.uk/~rak/papers/newbook.pdf) To be published by Cambridge University Press 0 Preface The mere possibility of Artificial Intelligence (AI) – of machines that can think and act as intelligently as humans – can generate strong emotions. While some enthusiasts are excited by the thought that one day machines may become more intelligent than people, many of its critics view such a prospect with horror. Partly because these controversies attract so much attention, one of the most important accomplishments of AI has gone largely unnoticed: the fact that many of its advances can also be used directly by people, to improve their own human intelligence. Chief among these advances is Computational Logic. Computational Logic builds upon traditional logic, which was originally developed to help people think more effectively. It employs the techniques of symbolic logic, which has been used to build the foundations of mathematics and computing. However, compared with traditional logic, Computational Logic is much more powerful; and compared with symbolic logic, it is much simpler and more practical. Although the applications of Computational Logic in AI require the use of mathematical notation, its human applications do not. As a consequence, I have written the main part of this book informally, to reach as wide an audience as possible. Because human thinking is also the subject of study in many other fields, I have drawn upon related studies in Cognitive Psychology, Linguistics, Philosophy, Law, Management Science and English Composition.
    [Show full text]