Matematisk logik

'An invaluable companion for anyone who wants a deep understanding of what's under the hood of often inscrutable machines' Melanie Mitchell A rich, narrative explanation of the mathematics that has brought us machine learning and the ongoing explosion of artificial intelligenceMachine-learning systems are making life-altering decisions for us: approving mortgage loans, determining whether a tumour is cancerous, or deciding whether someone gets bail. They now influence discoveries in chemistry, biology and physics - the study of genomes, extra-solar planets, even the intricacies of quantum systems.We are living through a revolution in artificial intelligence that is not slowing down. This major shift is based on simple mathematics, some of which goes back centuries: linear algebra and calculus, the stuff of eighteenth-century mathematics. Indeed by the mid-1850s, a lot of the groundwork was all done. It took the development of computer science and the kindling of 1990s computer chips designed for video games to ignite the explosion of AI that we see all around us today. In this enlightening book, Anil Ananthaswamy explains the fundamental maths behind AI, which suggests that the basics of natural and artificial intelligence might follow the same mathematical rules.As Ananthaswamy resonantly concludes, to make the most of our most wondrous technologies we need to understand their profound limitations - the clues lie in the maths that makes AI possible.

Hej logik giver en pædagogisk indføring i klassisk logik. Med eksempler og øvelser guider forfatteren læseren igennem centrale temaer i doms- og prædikatslogikken: Hvordan gyldighed hænger sammen med logisk form, hvordan man afdækker logiske strukturer ved at formalisere elementerne i et argument, og hvordan man analyserer logisk form ved at bruge sandhedstabeller og semantiske træer. Bogen kan bruges som lærebog til logikkurser i gymnasiet og på universitetet. Samtidig gør en filosofisk perspektivering af grundtemaerne den også interessant for den mere logikkyndige.Titlen henviser til 'Hej logikmaskinen', der er bogens nye måde at forstå logikkens centrale funktionsbegreb på. Mange finder funktionsbegrebet abstrakt og utilgængeligt, men i Hej logik bliver det forståeligt og intuitivt.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

Algebra con elementi di logica matematica e teoria degli insiemi.

This book is an introduction to the logic of partitions on a set as well as the (quantum) logic of partitions (direct-sum decompositions or DSDs) on a vector space. Partitions of a set are categorically dual to subsets of a set. Thus the logic of partitions is, in that sense, the dual to the Boolean logic of subsets (usually presented as the special case of propositional logic).Since partitions can be seen as the inverse image partitions of random variables or numerical attributes, partition logic is the logic of random variables or numerical attributes (abstracted from the actual values). On the lattice of partitions of an arbitrary unstructured set, there is a rich algebraic structure of dual operations of implication and co-implication - resembling a non-distributive version of Heyting and co-Heyting algebras.

Mathematik für die Informatik: Dieses Buch bringt Ihnen die Grundlagen beiDieses Lehrbuch bietet eine Einführung in die grundlegenden Begriffe und Strukturen der Mathematik, welche am Anfang eines Informatikstudiums relevant sind. Weiterhin demonstriert es Anwendungen von mathematischen Konzepten und Methoden in der Informatik. Diese betreffen insbesondere formale Methoden der Programmverifikation und -entwicklung und den Entwurf von generischen Programmen. Ein spezielles Konzept mit einer leicht verständlichen Vermittlung des Stoffs, vielen Beispielen mit Rückgriffen auf die Schulmathematik und detaillierten Beweisen (verbunden mit der Erklärung des logischen Hintergrunds) erleichtert den Einstieg in die Mathematik an einer wissenschaftlichen Hochschule. Dadurch werden die Studierenden auch auf spätere Begriffe und tiefergehende Anwendungen der Mathematik in der Informatik gut vorbereitet. Die 5. Auflage erweitert den Inhalt um die Grundlagen der linearen Algebra. Die Übungsaufgaben zu den einzelnen Kapitel helfen, das Erlernte zu festigen und zu kontrollieren. Zahlreiche Lösungsvorschläge am Ende des Buchs ermöglichen die Überprüfung der eigenen Lösungen.Der InhaltMengentheoretische Grundlagen ¿ Logische Grundlagen ¿ Allgemeine direkte Produkte und Datenstrukturen ¿ Mathematische Beweise ¿ Spezifikation und Programmverifikation ¿ Spezielle Funktionen ¿ Spezielle Relationen und gerichtete Graphen ¿ Elementare Kombinatorik und ungerichtete Graphen ¿ Diskrete Wahrscheinlichkeitstheorie ¿ Generische Programmierung ¿ Grundbegriffe algebraischer Strukturen ¿ Formale Einführung der natürlichen Zahlen - Grundbegriffe der linearen AlgebraMit diesem Buch gelingt der Einstieg ins Informatik-StudiumMit diesem Buch schaffen Sie eine solide Basis für die Mathematikausbildung im Rahmen desInformatikstudiums. Zudem sind Sie durch die vorgestellten Problemstellungen in der Lage, selbstständig mathematische Konzepte und Methoden anzuwenden. Zielgruppen dieses Buchs über Mathematik in der Informatik sind Bachelor-Studierende in den ersten Studiensemestern folgender Fachbereiche:InformatikMathematikIngenieurwissenschaften

This volume of the Logica Yearbook series brings together articles presented at the annual international symposium Logica 2022, Teplá, the Czech Republic. The articles range over mathematical and philosophical logic, history and philosophy of logic, and the analysis of natural language.

This book constitutes the revised selected papers of the 5th International Workshop on Dynamic Logic. New Trends and Applications, DaLí 2023, held in Tbilisi, Georgia, during September 15¿16, 2023. The 8 full papers in this book were carefully reviewed and selected from 10 submissions. They deal with new trends and applications in the area of Dynamic Logic.

Fuzzy logic systems have been a hot topic in the scientific and academic community for more than half a century. The idea of making machines behave and make decisions like humans do is astounding. The development and implementation of fuzzy logic systems can be seen in various real physical applications in daily human life. The methods employed using fuzzy logic have resulted in innovative technologies. This book provides insights into understanding the principles and concepts behind the advances of fuzzy logic systems. It presents ideas concerning fuzzy logic systems and their technological applications. The book is arranged into two sections on theories and foundations of fuzzy logic systems and implementations of fuzzy logic systems in service to the community.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

Lógica Dialógica. Reglas y ejercicios para hacer lógica con diálogos constituye la primera introducción a la Lógica dialógica en castellano dirigida a la práctica de la dialógica y que contiene comentarios precisos sobre ejercicios y soluciones tanto de lógica proposicional como de lógica clásica de primer orden, intuicionista y elementos de lógica modal proposicional. Es la primera parte de la obra Los Diálogos de la Lógica, concebida en dos textos autónomos separados sobre lógica dialógica. Los dos textos deberían ofrecer conjuntamente una visión técnica y filosófica completa del enfoque dialógico de la lógica

Taking an amusing and digestible look at the usually dry world of probability and statistics, this is the ultimate guide to how you can incorporate them into everyday life, from one of the world's most sought-after experts in game theory. This is the only book you need to become a statistics whizz! Numbers are everywhere – food packaging, weather forecasts, social media, adverts, and more. You can’t escape them. But you can learn to understand them – and avoid being fooled! This book breaks down the key fundamentals in statistics in a fun and accessible way so that you can understand the numbers that occupy your life. • Make sense of sports stats – discover who is the greatest scorer of all time • Learn to interpret scientific studies and how they’re reported in the media so you’re never misled again • Discover tips and tricks to make you a more successful gambler • Explore what role stats has to play in flat-earth conspiracy arguments • Read about misunderstood probabilities in the Sally Clarke and OJ Simpson trials With easy-to-follow explanations, tables, graphs, and real-life examples, this book helps you evaluate your options, calculate your chances of success, and make better decisions.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are free open access, and available via the College Publications website. This Journal is open access, puts no limit on the number of pages of any article, puts no limit on the number of papers in an issue and puts no limit on the number of issues per year. We insist only on a very high academic standard, and will publish issues as they come.

CSP-Rules is a generic pattern-based (or rule-based) solver of finite binary Constraint Satisfaction Problems (CSPs). This software is available on GitHub and it includes fully developed applications to various logic puzzles such as Sudoku, Latin Squares (allowing their Pandiagonal variant), Futoshiki, Kakuro, Numbrix, Hidato, Slitherlink and Map Colouring.CSP-Rules produces not only the solution to a CSP (as all the existing CSP solvers) but also a full "resolution path", i.e. a detailed justification for each step; this is a full mathematical proof of the solution. CSP-Rules and the present book are closely related to the author's book "Pattern Based Constraint Satisfaction and Logic Puzzles", where the fundamental theory underlying the approach is presented.This book includes an extended version of the original "Basic User Manual". But it also introduces new powerful generic rules (OR-chains) for taking advantage of almost-contradictory patterns. In particular, it applies these chains to the analysis of a newly discovered pattern (tridagon) and to the classification of a large database of extreme Sudoku puzzles that have it. It also adds the analyses of 630 impossible patterns and explains how to deal with such a large number of patterns. It shows that a handful of them provide almost the same resolution power as the full set. The long final two chapters are written as research notebooks on these very advanced topics.

This book is dedicated to V.A. Yankov's seminal contributions to the theory of propositional logics. His papers, published in the 1960s, are highly cited even today. The Yankov characteristic formulas have become a very useful tool in propositional, modal and algebraic logic.The papers contributed to this book provide the new results on different generalizations and applications of characteristic formulas in propositional, modal and algebraic logics. In particular, an exposition of Yankov's results and their applications in algebraic logic, the theory of admissible rules and refutation systems is included in the book. In addition, the reader can find the studies on splitting and join-splitting in intermediate propositional logics that are based on Yankov-type formulas which are closely related to canonical formulas, and the study of properties of predicate extensions of non-classical propositional logics.The book also contains an exposition of Yankov's revolutionary approach to constructive proof theory. The editors also include Yankov's contributions to history and philosophy of mathematics and foundations of mathematics, as well as an examination of his original interpretation of history of Greek philosophy and mathematics.

Originally published in 1748, this comprehensive treatise on logic was designed to be a primer for young scholars entering the fields of philosophy and mathematics. With its clear and concise explanations of the principles of reasoning and argumentation, 'The Elements of Logic' remains a valuable resource for anyone seeking to enhance their critical thinking skills.This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it.This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

This classic text, written by philosopher and mathematician William Thomson, presents a systematic exposition of the laws of thought and their role in science, logic, and philosophy. The book is still widely used in philosophy and mathematics courses today.This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it.This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

This book gives a comprehensive introduction to Universal Algebraic Logic. The three main themes are (i) universal logic and the question of what logic is, (ii) duality theories between the world of logics and the world of algebra, and (iii) Tarskian algebraic logic proper including algebras of relations of various ranks, cylindric algebras, relation algebras, polyadic algebras and other kinds of algebras of logic. One of the strengths of our approach is that it is directly applicable to a wide range of logics including not only propositional logics but also e.g. classical first order logic and other quantifier logics. Following the Tarskian tradition, besides the connections between logic and algebra, related logical connections with geometry and eventually spacetime geometry leading up to relativity are also part of the perspective of the book. Besides Tarskian algebraizations of logics, category theoretical perspectives are also touched upon. This book, apart from being a monograph containing state of the art results in algebraic logic, can be used as the basis for a number of different courses intended for both novices and more experienced students of logic, mathematics, or philosophy. For instance, the first two chapters can be used in their own right as a crash course in Universal Algebra.

¿Einsatzfertige Lerneinheiten vermitteln fundamentale mathematische Techniken, die weit über die Oberstufe hinaus von Bedeutung sind. Die Lerninhalte eignen sich auch zur gezielten Vorbereitung auf Mathematikwettbewerbe. Schwerpunkte bilden Stochastik und trigonometrische Funktionen. Es wird mit Erwartungswerten gerechnet, in die Spieltheorie eingeführt, und es werden der Sinussatz und Additionstheoreme für Sinus und Cosinus behandelt. Hinzu kommen Exkurse in die Aussagenlogik und die Graphentheorie. Die Schüler*innen führen Beweise in unterschiedlichen Gebieten. Die Aufgaben fördern die mathematische Denkfähigkeit, Phantasie und Kreativität.

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere.The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Gottingen as his main collaborator in foundational studies in the years to come.The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

He [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge- braic geometry as special cases.-Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat- ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con- structions in mathematics is between those-the majority-who believe that to exclude from mathematics all statements that cannot be proved construc- tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc- tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value.