This textbook offers a detailed introduction to the methodology and applications of sequent calculi in propositional logic. Unlike other texts concerned with proof theory, emphasis is placed on illustrating how to use sequent calculi to prove a wide range of metatheoretical results. The presentation is elementary and self-contained, with all technical details both formally stated and also informally explained. Numerous proofs are worked through to demonstrate methods of proving important results, such as the cut-elimination theorem, completeness, decidability, and interpolation. Other proofs are presented with portions left as exercises for readers, allowing them to practice techniques of sequent calculus. After a brief introduction to classical propositional logic, the text explores three variants of sequent calculus and their features and applications. The remaining chapters then show how sequent calculi can be extended, modified, and applied to non-classical logics, including modal, intuitionistic, substructural, and many-valued logics. Sequents and Trees is suitable for graduate and advanced undergraduate students in logic taking courses on proof theory and its application to non-classical logics. It will also be of interest to researchers in computer science and philosophers.

Together, Sets and Proofs and its sister volume Models and Computability will provide readers with a comprehensive guide to the current state of mathematical logic. All the authors are leaders in their fields and are drawn from the invited speakers at "Logic Colloquium "97" (the major international meeting of the Association of Symbolic Logic). It is expected that the breadth and timeliness of these two volumes will prove an invaluable and unique resource for specialists, postgraduate researchers, and the informed and interested nonspecialist.

A compact and easily accessible book, it guides the reader in unravelling the apparent mysteries found in doing mathematical proofs. Simply written, it introduces the art and science of proving mathematical theorems and propositions and equips students with the skill required to tackle the task of proving mathematical assertions. Theoremus - A Student's Guide to Mathematical Proofs is divided into two parts. Part 1 provides a grounding in the notion of mathematical assertions, arguments and fallacies and Part 2, presents lessons learned in action by applying them into the study of logic itself. The book supplies plenty of examples and figures, gives some historical background on personalities that gave rise to the topic and provides reflective problems to try and solve. The author aims to provide the reader with the confidence to take a deep dive into some more advanced work in mathematics or logic.

· Are you more likely to become a professional footballer if your surname is Ball? · How can you be one hundred per cent sure you will win a bet? · Why did so many Pompeiians stay put while Mount Vesuvius was erupting? · How do you prevent a nuclear war? Ever since the dawn of human civilisation, we have been trying to make predictions about what's in store for us. We do this on a personal level, so that we can get on with our lives efficiently (should I hang my laundry out to dry, or will it rain?). But we also have to predict on a much larger scale, often for the good of our broader society (how can we spot economic downturns or prevent terrorist attacks?). For just as long, we have been getting it wrong. From religious oracles to weather forecasters, and from politicians to economists, we are subjected to poor predictions all the time. Our job is to separate the good from the bad. Unfortunately, the foibles of our own biology - the biases that ultimately make us human - can let us down when it comes to making rational inferences about the world around us. And that can have disastrous consequences. How to Expect the Unexpected will teach you how and why predictions go wrong, help you to spot phony forecasts and give you a better chance of getting your own predictions correct.

This edited book focuses on non-classical logics and their applications, highlighting the rapid advances and the new perspectives that are emerging in this area. Non-classical logics are logical formalisms that violate or go beyond classical logic laws, and their specific features make them particularly suited to describing and reason about aspects of social interaction. The richness and diversity of non-classical logics mean that this area is a natural catalyst for ideas and insights from many different fields, from information theory to game theory and business science. This volume is the post-proceedings of the 8th International Conference on Logic and Cognition, held at Sun Yat-Sen University Institute of Logic and Cognition (ILC) in Guangzhou, China in December 2016. The conference series started in 2001, and is organized by the ILC, often in collaboration with various international research groups. This eighth installment was jointly organized by ILC and Alessandra Palmigiano's Applied Logic research group. The conference series aims to foster the development of effective logical tools to study social behavior from a philosophical, cognitive and formal perspective in order to challenge the field of logic in ways that open up new and exciting research directions. Chapter "The Category of Node-and-Choice Forms, with Subcategories for Choice-Sequence Forms and Choice-Set Forms" of this book is available open access under a CC BY 4.0 license at link.springer.com

The most crucial ability for machine learning and data science is mathematical logic for grasping their essence rather than knowledge and experience. This textbook approaches the essence of machine learning and data science by considering math problems and building Python programs. As the preliminary part, Chapter 1 provides a concise introduction to linear algebra, which will help novices read further to the following main chapters. Those succeeding chapters present essential topics in statistical learning: linear regression, classification, resampling, information criteria, regularization, nonlinear regression, decision trees, support vector machines, and unsupervised learning. Each chapter mathematically formulates and solves machine learning problems and builds the programs. The body of a chapter is accompanied by proofs and programs in an appendix, with exercises at the end of the chapter. Because the book is carefully organized to provide the solutions to the exercises in each chapter, readers can solve the total of 100 exercises by simply following the contents of each chapter. This textbook is suitable for an undergraduate or graduate course consisting of about 12 lectures. Written in an easy-to-follow and self-contained style, this book will also be perfect material for independent learning.

Following their introduction in the early 1980s, o-minimal structures have provided an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. This book gives a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. It starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

This text presents methods of modern set theory as tools that can be usefully applied to other areas of mathematics. The author describes numerous applications in abstract geometry and real analysis and, in some cases, in topology and algebra. The book begins with a tour of the basics of set theory, culminating in a proof of Zorn's Lemma and a discussion of some of its applications. The author then develops the notions of transfinite induction and descriptive set theory, with applications to the theory of real functions. The final part of the book presents the tools of "modern" set theory: Martin's Axiom, the Diamond Principle, and elements of forcing. Written primarily as a text for beginning graduate or advanced level undergraduate students, this book should also interest researchers wanting to learn more about set theoretical techniques applicable to their fields.

This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory.

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.

This textbook can serve as a comprehensive manual of discrete mathematics and graph theory for non-Computer Science majors; as a reference and study aid for professionals and researchers who have not taken any discrete math course before. It can also be used as a reference book for a course on Discrete Mathematics in Computer Science or Mathematics curricula. The study of discrete mathematics is one of the first courses on curricula in various disciplines such as Computer Science, Mathematics and Engineering education practices. Graphs are key data structures used to represent networks, chemical structures, games etc. and are increasingly used more in various applications such as bioinformatics and the Internet. Graph theory has gone through an unprecedented growth in the last few decades both in terms of theory and implementations; hence it deserves a thorough treatment which is not adequately found in any other contemporary books on discrete mathematics, whereas about 40% of this textbook is devoted to graph theory. The text follows an algorithmic approach for discrete mathematics and graph problems where applicable, to reinforce learning and to show how to implement the concepts in real-world applications.

This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of a hybrid that "crossbreeds" topology (Stone spaces) and order (Kripke frames), resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia's book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume.

This monograph explores the logical systems of early logicians in the Arabic tradition from a theoretical perspective, providing a complete panorama of early Arabic logic and centering it within an expansive historical context. By thoroughly examining the writings of the first Arabic logicians, al-Farabi, Avicenna and Averroes, the author analyzes their respective theories, discusses their relationship to the syllogistics of Aristotle and his followers, and measures their influence on later logical systems. Beginning with an introduction to the writings of the most prominent Arabic logicians, the author scrutinizes these works to determine their categorical logic, as well as their modal and hypothetical logics. Where most other studies written on this subject focus on the Arabic logicians' epistemology, metaphysics, and theology, this volume takes a unique approach by focusing on the actual technical aspects and features of their logics. The author then moves on to examine the original texts as closely as possible and employs the symbolism of modern propositional, predicate, and modal logics, rendering the arguments of each logician clearly and precisely while clarifying the theories themselves in order to determine the differences between the Arabic logicians' systems and those of Aristotle. By providing a detailed examination of theories that are still not very well-known in Western countries, the author is able to assess the improvements that can be found in the Arabic writings, and to situate Arabic logic within the breadth of the history of logic. This unique study will appeal mainly to historians of logic, logicians, and philosophers who seek a better understanding of the Arabic tradition. It also will be of interest to modern logicians who wish to delve into the historical aspects and progression of their discipline. Furthermore, this book will serve as a valuable resource for graduate students who wish to complement their general knowledge of Arabic culture, logic, and sciences.

This book presents an intuitive picture-oriented approach to the formative processes technique and to its applications. In the first part the authors introduce basic set-theoretic terminology and properties, the decision problem in set theory, and formative processes. The second part of the book is devoted to applications of the technique of formative processes to decision problems. All chapters contain exercises and the book is appropriate for researchers and graduate students in the area of computer science logic.

This book presents a collection of contributions from related logics to applied paraconsistency. Moreover, all of them are dedicated to Jair Minoro Abe,on the occasion of his sixtieth birthday. He is one of the experts in Paraconsistent Engineering, who developed the so-called annotated logics. The book includes important contributions on foundations and applications of paraconsistent logics in connection with engineering, mathematical logic, philosophical logic, computer science, physics, economics, and biology. It will be of interest to students and researchers, who are working on engineering and logic.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection - prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) - show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics - FilMat

This is a self-contained exposition by one of the leading experts in lattice theory, George Gratzer, presenting the major results of the last 70 years on congruence lattices of finite lattices, featuring the author's signature Proof-by-Picture method. Key features: * Insightful discussion of techniques to construct "nice" finite lattices with given congruence lattices and "nice" congruence-preserving extensions * Contains complete proofs, an extensive bibliography and index, and over 140 illustrations * This new edition includes two new parts on Planar Semimodular Lattices and The Order of Principle Congruences, covering the research of the last 10 years The book is appropriate for a one-semester graduate course in lattice theory, and it is a practical reference for researchers studying lattices. Reviews of the first edition: "There exist a lot of interesting results in this area of lattice theory, and some of them are presented in this book. [This] monograph...is an exceptional work in lattice theory, like all the contributions by this author. ... The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. Moreover, the author provides a series of companion lectures which help the reader to approach the Proof-by-Picture sections." (Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica, Vol. LII (1), 2007) "The book is self-contained, with many detailed proofs presented that can be followed step-by-step. [I]n addition to giving the full formal details of the proofs, the author chooses a somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the combinatorial (as opposed to algebraic) nature of many of the presented results. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations of finite lattices, with emphasis on the more 'geometric' aspects." -Mathematical Reviews

This book covers new aspects and frameworks of control, design, and optimization based on the TP model transformation and its various extensions. The author outlines the three main steps of polytopic and LMI based control design: 1) development of the qLPV state-space model, 2) generation of the polytopic model; and 3) application of LMI to derive controller and observer. He goes on to describe why literature has extensively studied LMI design, but has not focused much on the second step, in part because the generation and manipulation of the polytopic form was not tractable in many cases. The author then shows how the TP model transformation facilitates this second step and hence reveals new directions, leading to powerful design procedures and the formulation of new questions. The chapters of this book, and the complex dynamical control tasks which they cover, are organized so as to present and analyze the beneficial aspect of the family of approaches (control, design, and optimization). Additionally, the book aims to convey simple TP modeling; a new convex hull manipulation based possibilities for optimization; a general framework for stability analysis; standardized modeling and system description; relaxed and universal LMI based design framework; and a gateway to time-delayed systems.

This volume offers a wide range of both reconstructions of Nikolai Vasiliev's original logical ideas and their implementations in the modern logic and philosophy. A collection of works put together through the international workshop "Nikolai Vasiliev's Logical Legacy and the Modern Logic," this book also covers foundations of logic in the light of Vasiliev's contradictory ontology. Chapters range from a look at the Heuristic and Conceptual Background of Vasiliev's Imaginary Logic to Generalized Vasiliev-style Propositions. It includes works which cover Imaginary and Non-Aristotelian Logics, Inconsistent Set Theory and the Expansion of Mathematical Thinking, Plurivalent Logic, and the Impact of Vasiliev's Imaginary Logic on Epistemic Logic. The Russian logician, Vasiliev, was widely recognized as one of the forerunners of modern non-classical logic. His "imaginary logic" developed in some of his work at the beginning of 20th century is often considered to be one of the first systems of paraconsistent and multi-valued logic. The novelty of his logical project has opened up prospects for modern logic as well as for non-classical science in general. This volume contains a selection of papers written by modern specialists in the field and deals with various aspects of Vasiliev's logical ideas. The logical legacy of Nikolai Vasiliev can serve as a promising source for developing an impressive range of philosophical interpretations, as it marries promising technical innovations with challenging philosophical insights.

This book offers insight into the nature of meaningful discourse. It presents an argument of great intellectual scope written by an author with more than four decades of experience. Readers will gain a deeper understanding into three theories of the logos: analytic, dialectical, and oceanic. The author first introduces and contrasts these three theories. He then assesses them with respect to their basic parameters: necessity, truth, negation, infinity, as well as their use in mathematics. Analytic Aristotelian logic has traditionally claimed uniqueness, most recently in its Fregean and post-Fregean variants. Dialectical logic was first proposed by Hegel. The account presented here cuts through the dense, often incomprehensible Hegelian text. Oceanic logic was never identified as such, but the author gives numerous examples of its use from the history of philosophy. The final chapter addresses the plurality of the three theories and of how we should deal with it. The author first worked in analytic logic in the 1970s and 1980s, first researched dialectical logic in the 1990s, and discovered oceanic logic in the 2000s. This book represents the culmination of reflections that have lasted an entire scholarly career.

Numerical computation, knowledge discovery and statistical data analysis integrated with powerful 2D and 3D graphics for visualization are the key topics of this book. The Python code examples powered by the Java platform can easily be transformed to other programming languages, such as Java, Groovy, Ruby and BeanShell. This book equips the reader with a computational platform which, unlike other statistical programs, is not limited by a single programming language.The author focuses on practical programming aspects and covers a broad range of topics, from basic introduction to the Python language on the Java platform (Jython), to descriptive statistics, symbolic calculations, neural networks, non-linear regression analysis and many other data-mining topics. He discusses how to find regularities in real-world data, how to classify data, and how to process data for knowledge discoveries. The code snippets are so short that they easily fit into single pages. Numeric Computation and Statistical Data Analysis on the Java Platform is a great choice for those who want to learn how statistical data analysis can be done using popular programming languages, who want to integrate data analysis algorithms in full-scale applications, and deploy such calculations on the web pages or computational servers regardless of their operating system. It is an excellent reference for scientific computations to solve real-world problems using a comprehensive stack of open-source Java libraries included in the DataMelt (DMelt) project and will be appreciated by many data-analysis scientists, engineers and students.

In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets). In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set). It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports. After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi. The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.

This monograph presents an application of concepts and methods from algebraic topology to models of concurrent processes in computer science and their analysis. Taking well-known discrete models for concurrent processes in resource management as a point of departure, the book goes on to refine combinatorial and topological models. In the process, it develops tools and invariants for the new discipline directed algebraic topology, which is driven by fundamental research interests as well as by applications, primarily in the static analysis of concurrent programs. The state space of a concurrent program is described as a higher-dimensional space, the topology of which encodes the essential properties of the system. In order to analyse all possible executions in the state space, more than "just" the topological properties have to be considered: Execution paths need to respect a partial order given by the time flow. As a result, tools and concepts from topology have to be extended to take privileged directions into account. The target audience for this book consists of graduate students, researchers and practitioners in the field, mathematicians and computer scientists alike.