Search

Symbolic regression (SR) is one of the most powerful machine learning techniques that produces transparent models, searching the space of mathematical expressions for a model that represents the relationship between the predictors and the dependent variable without the need of taking assumptions about the model structure. Currently, the most prevalent learning algorithms for SR are based on genetic programming (GP), an evolutionary algorithm inspired from the well-known principles of natural selection. This book is an in-depth guide to GP for SR, discussing its advanced techniques, as well as examples of applications in science and engineering.The basic idea of GP is to evolve a population of solution candidates in an iterative, generational manner, by repeated application of selection, crossover, mutation, and replacement, thus allowing the model structure, coefficients, and input variables to be searched simultaneously. Given that explainability and interpretability are key elements for integrating humans into the loop of learning in AI, increasing the capacity for data scientists to understand internal algorithmic processes and their resultant models has beneficial implications for the learning process as a whole.This book represents a practical guide for industry professionals and students across a range of disciplines, particularly data science, engineering, and applied mathematics. Focused on state-of-the-art SR methods and providing ready-to-use recipes, this book is especially appealing to those working with empirical or semi-analytical models in science and engineering.

This book is dedicated to Dick de Jongh's contributions to the theory of intuitionistic and provability logics. Consisting of 13 chapters, written by leading experts, this book discusses de Jongh's original contributions and consequent developments that have helped to shape these fields. The book begins with an autobiographic note by Dick de Jongh, which discusses the main themes of his work and places the other contributions in context. The next four chapters explore the De Jongh-Sambin fixed point theorem and other contributions to provability and interpretability logics. The following four chapters focus on modal, intuitionistic and intuitionistic modal logics. They discuss independence of formulas, unification and de Jongh formulas in intuitionistic and modal logics. Then there follow two chapters on the other two areas to which Dick de Jongh made important contributions: the theory of well-partial orders, and formal learning theory. The second to last chapter on Origami Geometry can be seen as representing the Master of Logic program of the Institute for Logic, Language and Computation (ILLC) in which de Jongh invested a lot of energy. The book ends with a complete bibliography of Dick de Jongh in the last chapter. This volume provides a vital overview – and continuation of - de Jongh's prolfic work in the theory of intuitionistic and provability logics.

This is the first comprehensive textbook on higher-order logic that is written specifically to introduce the subject matter to graduate students in philosophy. The book covers both the formal aspects of higher-order languages—their model theory and proof theory, the theory of λ-abstraction and its generalizations—and their philosophical applications, especially to the topics of modality and propositional granularity. The book has a strong focus on non-extensional higher-order logics, making it more appropriate for foundational metaphysics than other introductions to the subject from computer science, mathematics, and linguistics. A Philosophical Introduction to Higher-order Logics assumes only that readers have a basic knowledge of first-order logic. With an emphasis on exercises, it can be used as a textbook though is also ideal for self-study.Author Andrew Bacon organizes the book's 18 chapters around four main parts:I. Typed LanguageII. Higher-Order LanguagesIII. General Higher-Order LanguagesIV. Higher-Order Model Theory In addition, two appendices cover the Curry-Howard isomorphism and its applications for modeling propositional structure. Each chapter includes exercises that move from easier to more difficult, strategically placed throughout the chapter, and concludes with an annotated suggested reading list providing graduate students with most valuable additional resources. Key Features: Is the first comprehensive introduction to higher-order logic as a grounding for addressing problems in metaphysics Introduces the basic formal tools that are needed to theorize in, and model, higher-order languages Offers an abundance of- Simple exercises throughout the book, serving as comprehension checks on basic concepts and definitions- More difficult exercises designed to facilitate long-term learning Contains annotated sections on further reading, pointing the reader to related literature, learning resources, and historical context

View the abstract.

Standard STEM courses, for all of their value, do not tend to include systematic lectures or treatment about the nature of the scientific method. This book aims to provide a wide reflection on the general principles of physics and explore the foundations of scientific knowledge as a whole. The author delves into the study of what lies at the basis of science in general, and physics in particular. Themes such as the relation between natural phenomena and mathematical language are addressed, highlighting the main hubs of conceptual development in science. The volume also examines the conceptual and practical instruments that have been progressively developed to investigate the nature of physics. Furthermore, the author discusses the importance of “scientific practice” within the scientific community, emphasizing its role in advancing knowledge and how it contributes to physics as a whole. Divided into three parts, each covering different aspects of physics and its foundations, the text, while assuming basic knowledge of physics and mathematics taught in university courses, is accessible to all STEM students, and will be useful for anyone looking to gain valuable insights into the nature of physics and the methods used to acquire knowledge in this field.

Events between which we have no epistemic reason to discriminate have equal epistemic probabilities. Bertrand's chord paradox, however, appears to show this to be false, and thereby poses a general threat to probabilities for continuum sized state spaces. Articulating the nature of such spaces involves some deep mathematics and that is perhaps why the recent literature on Bertrand's Paradox has been almost entirely from mathematicians and physicists, who have often deployed elegant mathematics of considerable sophistication. At the same time, the philosophy of probability has been left out. In particular, left out entirely are the philosophical ground of the principle of indifference, the nature of the principle itself, the stringent constraint this places on the mathematical representation of the principle needed for its application to continuum sized event spaces, and what these entail for rigour in developing the paradox itself. This book puts the philosophy and its entailments back in and in so doing casts a new light on the paradox, giving original analyses of the paradox, its possible solutions, the source of the paradox, the philosophical errors we make in attempting to solve it and what the paradox proves for the philosophy of probability. The book finishes with the author's proposed solution—a solution in the spirit of Bertrand's, indeed—in which an epistemic principle more general than the principle of indifference offers a principled restriction of the domain of the principle of indifference.Bertrand's Paradox and the Principle of Indifference will appeal to scholars and advanced students working in the philosophy of mathematics, epistemology, philosophy of science, probability theory and mathematical physics.

Mathematical Logic: An Introduction is a textbook that uses mathematical tools to investigate mathematics itself. In particular, the concepts of proof and truth are examined. The book presents the fundamental topics in mathematical logic and presents clear and complete proofs throughout the text. Such proofs are used to develop the language of propositional logic and the language of first-order logic, including the notion of a formal deduction. The text also covers Tarski's definition of truth and the computability concept. It also provides coherent proofs of Godel's completeness and incompleteness theorems. Moreover, the text was written with the student in mind and thus, it provides an accessible introduction to mathematical logic. In particular, the text explicitly shows the reader how to prove the basic theorems and presents detailed proofs throughout the book. Most undergraduate books on mathematical logic are written for a reader who is well-versed in logical notation and mathematical proof. This textbook is written to attract a wider audience, including students who are not yet experts in the art of mathematical proof.

“The Use of Abstraction and Logic in Mathematics” is an edited book consisting of 16 contemporaneous open-access articles that are essentially devoted to mathematical logic research, from classical to non-classical logical systems, from algebraic logic to fuzzy logic. The book addresses the following mathematical logic topics: first-order and higher-order logic; as well as infinitary, description, modal; fixed-point, algebraic and fuzzy logic. This book also includes examples of practical applications of logical systems in link prediction and image processing tasks, as well as in the training of neural networks and artificial intelligence. The intended audience of this book is undergraduate and graduate students, as well as junior researchers. Familiarity with first-order and higher-order logics, as well as set theory, and algebra is essential to grasp the concepts and methods described in this book.

Kurt Gödel (1906-1978) gained world-wide fame by his incompleteness theorem of 1931. Later, he set as his aim to solve what are known as Hilbert's first and second problems, namely Cantor's continuum hypothesis about the cardinality of real numbers, and secondly the consistency of the theory of real numbers and functions. By 1940, he was halfway through the first problem, in what was his last published result in logic and foundations. His intense attempts thereafter at solving these two problems have remained behind the veil of a forgotten German shorthand he used in all of his writing. Results on Foundations is a set of four shorthand notebooks written in 1940-42 that collect results Gödel considered finished. Its main topic is set theory in which Gödel anticipated several decades of development. Secondly, Gödel completed his 1933 program of establishing the connections between intuitionistic and modal logic, by methods and results that today are at the same time new and 80 years old.The present edition of Gödel's four notebooks encompasses the 368 numbered pages and 126 numbered theorems of the Results on Foundations, together with a list of 74 problems on set theory Gödel prepared in 1946, and a list of an unknown date titled'The grand program of my research in ca. hundred questions.''

Laws of Form is a seminal work in foundations of logic, mathematics and philosophy published by G Spencer-Brown in 1969. The book provides a new point of view on form and the role of distinction, markedness and the absence of distinction (the unmarked state) in the construction of any universe. A conference was held August 8-10, 2019 at the Old Library, Liverpool University, 19 Abercromby Square, L697ZN, UK to celebrate the 50th anniversary of the publication of Laws of Form and to remember George Spencer-Brown, its author. The book is a collection of papers introducing and extending Laws of Form written primarily by people who attended the conference in 2019.

This book provides a collection of chapters on the development of scientific philosophy and symbolic logic in the early twentieth century. The turn of the last century was a key transitional period for the development of symbolic logic and scientific philosophy. The Peano school, the editorial board of the Revue de Métaphysique et de Morale, and the members of the Vienna Circle are generally mentioned as champions of this transformation of the role of logic in mathematics and in the sciences. The scholarship contained provides a rich historical and philosophical understanding of these groups and research areas. Specifically, the contributions focus on a detailed investigation of the relation between structuralism and modern mathematics. In addition, this book provides a closer understanding of the relation between symbolic logic and previous traditions such as syllogistics. This volume also informs the reader on the relation between logic, the history and didactics in the Peano School. This edition appeals to students and researchers working in the history of philosophy and of logic, philosophy of science, as well as to researchers on the Vienna Circle and the Peano School.

El libro Introducción a la Lógica Matemática parte del reconocimiento de la importancia de la lógica, en el sentido de contribuir con el lector en mejorar sus procesos de argumentación en contextos tanto científicos como coloquiales. Entre los propósitos de este libro está el de servir de texto de consulta para asignaturas como Lógica Matemática o áreas a fines y de ayudar a las personas que se interesan por esta temática a evitar la excesiva axiomatización, de tal manera que les permita hacer uso racional de la duda. El contenido desarrollado en el libro se presenta en siete capítulos que tratan asuntos como la descripción de los sistemas axiomáticos en una ruta histórica de la evolución de la lógica, la relación de las proposiciones con el idioma y bases para una argumentación adecuada, los procesos de deducción proposicional y cuantificacional, estructura lógica de teoremas y métodos de demostración. Cada capítulo del libro inicia con una actividad exploratoria que le permitirá al lector valorar qué tanto conoce sobre el tema. Al final de los capítulos se proponen numerosas situaciones o ejercicios cuyo propósito es lograr un dominio conceptual y contribuir con el desarrollo de destrezas en los temas correspondientes.

Calculi of temporal logic are widely used in modern computer science. The temporal organization of information flows in the different architectures of laptops, the Internet, or supercomputers would not be possible without appropriate temporal calculi. In the age of digitalization and High-Tech applications, people are often not aware that temporal logic is deeply rooted in the philosophy of modalities. A deep understanding of these roots opens avenues to the modern calculi of temporal logic which have emerged by extension of modal logic with temporal operators. Computationally, temporal operators can be introduced in different formalisms with increasing complexity such as Basic Modal Logic (BML), Linear-Time Temporal Logic (LTL), Computation Tree Logic (CTL), and Full Computation Tree Logic (CTL•). Proof-theoretically, these formalisms of temporal logic can be interpreted by the sequent calculus of Gentzen, the tableau-based calculus, automata-based calculus, game-based calculus, and dialogue-based calculus with different advantages for different purposes, especially in computer science.The book culminates in an outlook on trendsetting applications of temporal logics in future technologies such as artificial intelligence and quantum technology. However, it will not be sufficient, as in traditional temporal logic, to start from the everyday understanding of time. Since the 20th century, physics has fundamentally changed the modern understanding of time, which now also determines technology. In temporal logic, we are only just beginning to grasp these differences in proof theory which needs interdisciplinary cooperation of proof theory, computer science, physics, technology, and philosophy.

This is an expository account of work on strong forms of the Axiom of Determinacy (AD) by a group of set theorists in Southern California, in particular by W. Hugh Woodin. The first half of the book reviews necessary background material, including the Moschovakis Coding Lemma, the existence of strong partition cardinals, and the analysis of pointclasses in models of determinacy. The second half of the book introduces Woodin's axiom system $\mathrm{AD}^{+}$ and presents his initial analysis of these axioms. These results include the consistency of $\mathrm{AD}^{+}$ from the consistency of AD, and its local character and initial motivation. Proofs are given of fundamental results by Woodin, Martin, and Becker on the relationships among AD, $\mathrm{AD}^{+}$, the Axiom of Real Determinacy, and the Suslin property. Many of these results are proved in print here for the first time. The book briefly discusses later work and fundamental questions which remain open. The study of models of $\mathrm{AD}^{+}$ is an active area of contemporary research in set theory. The presentation is aimed at readers with a background in basic set theory, including forcing and ultrapowers. Some familiarity with classical results on regularity properties for sets of reals under AD is also expected.

Samson Abramsky's wide-ranging contributions to logical and structural aspects of Computer Science have had a major influence on the field. This book is a rich collection of papers, inspired by and extending Abramsky's work. It contains both survey material and new results, organised around six major themes: domains and duality, game semantics, contextuality and quantum computation, comonads and descriptive complexity, categorical and logical semantics, and probabilistic computation. These relate to different stages and aspects of Abramsky's work, reflecting its exceptionally broad scope and his ability to illuminate and unify diverse topics.Chapters in the volume include a review of his entire body of work, spanning from philosophical aspects to logic, programming language theory, quantum theory, economics and psychology, and relating it to a theory of unification of sciences using dual adjunctions. The section on game semantics shows how Abramsky's work hasled to a powerful new paradigm for the semantics of computation. The work on contextuality and categorical quantum mechanics has been highly influential, and provides the foundation for increasingly widely used methods in quantum computing. The work on comonads and descriptive complexity is building bridges between currently disjoint research areas in computer science, relating Structure to Power.The volume also includes a scientific autobiography, and an overview of the contributions. The outstanding set of contributors to this volume, including both senior and early career academics, serve as testament to Samson Abramsky's enduring influence. It will provide an invaluable and unique resource for both students and established researchers.

The book explores Peirce's non standard thoughts on a synthetic continuum, topological logics, existential graphs, and relational semiotics, offering full mathematical developments on these areas. More precisely, the following new advances are offered: (1) two extensions of Peirce's existential graphs, to intuitionistic logics (a new symbol for implication), and other non-classical logics (new actions on nonplanar surfaces); (2) a complete formalization of Peirce's continuum, capturing all Peirce's original demands (genericity, supermultitudeness, reflexivity, modality), thanks to an inverse ordinally iterated sheaf of real lines; (3) an array of subformalizations and proofs of Peirce's pragmaticist maxim, through methods in category theory, HoTT techniques, and modal logics. The book will be relevant to Peirce scholars, mathematicians, and philosophers alike, thanks to thorough assessments of Peirce's mathematical heritage, compact surveys of the literature, and new perspectives offered through formal and modern mathematizations of the topics studied.

This book covers topics from numerical and symbolic computing, including the main numerical and symbolic methods, schemes and applications. Section 1 focuses on numerical computing methods, describing matrix differential equations for solving systems of linear algebraic equations, a numerical problem encryption for high-performance computing applications, a numerical verification method of solutions for elliptic variational inequalities, a trigonometric numerical integrator for solving first order ordinary differential equation, and augmented Lagrangian methods for numerical solutions to higher order differential equations. Section 2 focuses on symbolic computing methods, describing a learning network assisted by means of symbolic computation, multiple factorial analysis of symbolic data, combining symbolic tools with interval analysis, and application to solve robust control problems, symbolic and graphical computations of a class of slightly perturbed equations. Section 3 focuses on numerical computing applications, describing non-negativity preserving numerical algorithms for problems in mathematical finance, a continuum approach using numerical simulations of microscale gas flows, numerical methods in electro-cardiology, numerical simulation of the blood flow through a brain vascular aneurysm with an artificial stent using the SPH method, and numerical modeling of soil water flow and nitrogen dynamics in a tomato field irrigated with municipal wastewater. Section 4 focuses on symbolic computing applications, describing symbolic time series analysis and its application in social sciences, application of symbolic computation in nonlinear differential-difference equations, symbolic modelling of dynamic human motions, and a framework for bridging the gap between symbolic and non-symbolic AI.

The overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not.Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency.Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software.While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory.The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.

This open access book is the first ever collection of Karl Popper's writings on deductive logic.Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ('falsificationism') and his social and political philosophy ('open society') have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics.This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work.

Introduction à la métalogique du calcul propositionnel, cet ouvrage contient non seulement l'initiation aux concepts de base du domaine (syntaxe, sémantique, consistance, complétude, etc.), mais également des développements utiles (logiques intuitionniste et modale) et aussi des approches originales (probabilités). Les auteurs accordent une attention rigoureuse à la présentation et à l'explication des concepts des calculs propositionnels. Formellement très simples, les calculs propositionnels constituent un terrain idéal pour s'initier aux manipulations symboliques. En ce qui concerne, plus généralement, l'usage du formalisme, les auteurs privilégient une attitude pédagogique sans pour autant faire de concessions. La logique est, et restera, formelle, mais il est possible de l'apprivoiser ; et le meilleur outil pour ce faire est la pratique consistant à résoudre des problèmes. On trouvera donc dans cet ouvrage de nombreux exercices dont la plupart sont accompagnés de solutions.

During the Iraq War, American soldiers were sent to both fight an enemy and to recover a “failed state” in pixelated camouflage uniforms, accompanied by robots, and armed with satellite maps and biometric hand-held scanners. The Iraq War, however, was no digital game: massive-scale physical death and destruction counter the vision of a clean replayable war. The military policy of the United States, and not the actual experience of war, has been rooted in the logic of digital, and nascent algorithmic technology. This logic attempted to reduce culture, society, as well as the physical body and environment into visual data that lacks cultural and historical context. This book details the emergence of a nascent algorithmic war culture in the context of the Iraq War (2003-2010) in relation to the data-driven early 20th century British Mandate for Iraq. Through a series of five inquiries into the ways in which the Iraq War attempted to and often failed tosee population and territory as digital and further proto-algorithmic entities, it offers an insight into the digitization and further unmanned automaton of war. It does so through a comparative historical framework reaching back to the quantification techniques harnessed during the British Mandate for Iraq (1918-1932) in order to explicate the parallels and complicated the diversions between the numerical logics that have driven both military state-building enterprises.

One of the most difficult problems challenging the human mind is knowledge of the world in its human, natural, and supra-natural dimensions: what is the nature of this multidimensional reality? How do we know and verify the truth of our knowledge claims of this reality? A contemporary Polish philosopher, Malgorzata Czarnocka, has advanced one of the most comprehensive and insightful studies of the cognitive act and the conditions under which it takes place. The proposition explicated in this book is that Czarnocka's analysis of perception functions as a model of explanation in our attempt to know the nature of the being that underlies the universe. This analysis becomes the basis of the author's discussion of symbolic truth as a model of explanation and its other applications.

The book contains a collection of chapters written by experts from the fields of philosophy, law, logic, computer science and artificial intelligence who pay tribute to Professor Risto Hilpinen's impressive work on the logic of induction, on deontic logic and epistemology, and on philosophy of science.In addition to an introduction by the editors, a section on Professor Hilpinen's positions, professional services and honors, as well as a complete bibliography of his writings, the editors, McNamara, Jones and Brown, have compiled a multidisciplinary global cross-section of academic contemporaries that provides insights and perspectives on Hilpinen's influence and legacy.The essays reflect central aspects of Risto Hilpinen's research interests, and offer further contributions to some of the philosophical fields for which he is best known: applied modal logic, including deontic logic (from the ancient Greek δέον déon, pertaining to the concepts of duty and obligation), the semantics of normative language, the logic of action, and the theory of practical reasoning; the analysis of the concept of artifact; and the theory of semiotics in the tradition of Charles Peirce. The presence in the collection of several papers relating to deontic logic underlines Hilpinen's importance in that area, in which his publications have long been recognized as standard works. The book is an essential collection of ideas for all those who feel at home in a variety of formal disciplines, from propositional logic to the logic of artificial intelligence.

Those who want to understand logic, if they manage to pass at least an initial, though far from simple, modern course of study, eventually conclude that practically logic consists in formulating premises and a taken-from-nowhere assertion in an incomprehensible language and then proving or disproving cause-consequence links between them. Conversely, many topical tasks of logical analysis, such as forming and testing hypotheses, inferring consequences with predefined properties, and searching for, and analysis of, logical errors and inconsistencies in reasoning, among others, are outside the scope of this discourse. They are scattered haphazardly in works on theory of argumentation, non-classical logics, and artificial intelligence.This book demonstrates the capabilities of two relatively simple mathematical systems developed by the authors, namely E-structures and n-tuple algebra, which allow the modelling of various types of reasoning and solve the above and some other tasks of logical analysis.

This book gathers together a colorful set of problems on classical Mathematical Logic, selected from over 30 years of teaching. The initial chapters start with problems from supporting fields, like set theory (ultrafilter constructions), full-information game theory (strategies), automata, and recursion theory (decidability, Kleene's theorems). The work then advances toward propositional logic (compactness and completeness, resolution method), followed by first-order logic, including quantifier elimination and the Ehrenfeucht– Fraïssé game; ultraproducts; and examples for axiomatizability and non-axiomatizability. The Arithmetic part covers Robinson's theory, Peano's axiom system, and Gödel's incompleteness theorems. Finally, the book touches universal graphs, tournaments, and the zero-one law in Mathematical Logic. Instructors teaching Mathematical Logic, as well as students who want to understand its concepts and methods, can greatly benefit from this work. The style andtopics have been specially chosen so that readers interested in the mathematical content and methodology could follow the problems and prove the main theorems themselves, including Gödel's famous completeness and incompleteness theorems. Examples of applications on axiomatizability and decidability of numerous mathematical theories enrich this volume.

This book provides a comprehensive introduction to the essential elements of standard (classical) symbolic logic. Key topics covered include: · The characteristic nature and scope of logic as a discipline · The construction of a series of distinctly named formal languages suitable for formal translation · Semantic models · The construction of decision procedures · The execution of proof-theoretic arrangements like natural deduction and proof-sequent systems The book covers both the semantics and proof theory of the standard sentential (propositional) logic and predicate (first-order) logic. Other topics covered include: parsing trees,extraction of alternative notations (for instance, Polish notation), Fitch-style proof-theory, sequent and ‘tree'proof systems, comparisons and contrasts with intuitionistic logic, and presentations of predicate logic models. An ancillary chapter on elements of set theory is conveniently placed at the end and includes insights into the Zermelo-Fraenkel systematization of set theory. The philosophy of logic is also explored. Exercises in the text provide instruction on mathematical induction for the construction of formula, tests for the well-formedness of Polish notation, and functional completeness. Symbolic Logic is essential reading for all philosophy students taking intermediate level formal logic courses and will also appeal to diligent first year students of logic. The text is replete with exercises on both the formal machinery and the philosophical aspects of logic.

This monograph shows that, through a recourse to the concepts and methods of abstract algebraic logic, the algebraic theory of regular varieties and the concept of analyticity in formal logic can profitably interact. By extending the technique of Plonka sums from algebras to logical matrices, the authors investigate the different classes of models for logics of variable inclusion and they shed new light into their formal properties.The book opens with the historical origins of logics of variable inclusion and on their philosophical motivations. It includes the basics of the algebraic theory of regular varieties and the construction of Plonka sums over semilattice direct systems of algebra. The core of the book is devoted to an abstract definition of logics of left and right variable inclusion, respectively, and the authors study their semantics using the construction of Plonka sums of matrix models. The authors also cover Paraconsistent Weak Kleene logic and survey its abstract algebraic logical properties. This book is of interest to scholars of formal logic.

This volume is dedicated to Hiroakira Ono life's work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science.It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.

Ludwig Wittgenstein's brief Tractatus Logico-Philosophicus (1922) is one of the most important philosophical works of the twentieth century, yet it offers little orientation for the reader. The first-time reader is left wondering what it could be about, and the scholar is left with little guidance for interpretation.In Tractatus in Context, James C. Klagge presents the vital background necessary for appreciating Wittgenstein's gnomic masterpiece. Tractatus in Context contains the early reactions to the Tractatus, including the initial reviews written in 1922-1924. And while we can't talk with Wittgenstein, we can do the next best thing—hear what he had to say about the Tractatus. Klagge thus presents what Wittgenstein thought about germane issues leading up to his writing the book, in discussions and correspondence with others about his ideas, and what he had to say about the Tractatus after it was written—in letters, lectures and conversations. It offers, you might say, Wittgenstein's own commentary on the book.Key Features: Illuminates what is at stake in the Tractatus, by providing the views of others that engaged Wittgenstein as he was writing it. Includes Wittgenstein's earlier thoughts on ideas in the book as recorded in his notebooks, letters, and conversations as well as his later, retrospective comments on those ideas. Draws on new or little-known sources, such as Wittgenstein's coded notebooks, Hermine's notes, Frege's letters, Hänsel's diary, Ramsey's notes, and Skinner's dictations. Draws connections between the background context and specific passages in the Tractatus, using a proposition-by-proposition commentary.

In these essays Geoffrey Hellman presents a strong case for a healthy pluralism in mathematics and its logics, supporting peaceful coexistence despite what appear to be contradictions between different systems, and positing different frameworks serving different legitimate purposes. The essays refine and extend Hellman's modal-structuralist account of mathematics, developing a height-potentialist view of higher set theory which recognizes indefinite extendability of models and stages at which sets occur. In the first of three new essays written for this volume, Hellman shows how extendability can be deployed to derive the axiom of Infinity and that of Replacement, improving on earlier accounts; he also shows how extendability leads to attractive, novel resolutions of the set-theoretic paradoxes. Other essays explore advantages and limitations of restrictive systems - nominalist, predicativist, and constructivist. Also included are two essays, with Solomon Feferman, on predicative foundations of arithmetic.

Architecture of Mathematics describes the logical structure of Mathematics from its foundations to its real-world applications. It describes the many interweaving relationships between different areas of mathematics and its practical applications, and as such provides unique reading for professional mathematicians and nonmathematicians alike. This book can be a very important resource both for the teaching of mathematics and as a means to outline the research links between different subjects within and beyond the subject.Features All notions and properties are introduced logically and sequentially, to help the reader gradually build understanding. Focusses on illustrative examples that explain the meaning of mathematical objects and their properties. Suitable as a supplementary resource for teaching undergraduate mathematics, and as an aid to interdisciplinary research. Forming the reader's understanding of Mathematics as a unified science, the book helps to increase his general mathematical culture.

This open access book constitutes the thoroughly refereed post-conference proceedings of the 6th International Workshop on Graph Structures for Knowledge Representation and Reasoning, GKR 2020, held virtually in September 2020, associated with ECAI 2020, the 24th European Conference on Artificial Intelligence.The 7 revised full papers presented together with 2 invited contributions were reviewed and selected from 9 submissions. The contributions address various issues for knowledge representation and reasoning and the common graph-theoretic background, which allows to bridge the gap between the different communities.

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.

This renewed edition of the Tractatus Logico-Philosophicus, exactly a century after Wittgenstein's release, presents the text in a hierarchical manner, “which is the way in which the book was composed and in which Wittgenstein arranged (selected and supplemented) the best of the philosophical remarks that he had been writing since 1913” (Peter Hacker). That tree-like reading is recommended by Wittgenstein himself in the sole footnote of his book, in which he suggests that the inner logical structure of the text is set by the decimal numbers of its propositions. “They alone – the Author will add – give the book perspicuity and clearness, and without this numbering it would be an incomprehensible jumble”. Indeed, the compact and intricate sequence of the traditional presentation is only a rigorous logical bet, but only a logical machine or a robot can unravel the tangle: for an ordinary human understanding that does not exploit its numbering, the book remains “an incomprehensible jumble”. In the present disposition, instead, all horizontal and vertical references become directly manifest and any reader can enjoy the fine architecture and the elegant reasoning of Wittgenstein's work. Every page is an actual reading unit, perfectly coherent and complete. The Tractatus becomes comprehensible also to unskilled readers, of course at more or less deep levels, while a scholar or a more practised reader can detect suggestions and meanings that had remained, until now, completely hidden. A historical note shows in which manner the new structural perspective sheds new light also in the compositional manuscript we have, which “writing units” are very similar, actually, to the pages of the present edition. Besides, this allows to rebuild the list of “Supplements” (here in the Appendix) that Wittgenstein gathered after he roughly finished his manuscript, but that he used very little in the final book. Printing the Tractatus following Wittgenstein's decimal prescriptions required meticulous philological care and some discretional conventions: for instance, at the top of each page the commented-upon proposition is printed again, to make the sight complete and self-sufficient. On the other hand, some forcing of the text by the translators in their sequential reading could be eliminated, restoring a more literal translation. Also the famous and intriguing picture of the eye and its visual field (5.6331) has been restored as Wittgenstein drafted it, making the entire page perfectly understandable and coherent. This documented and editorial work on one of the most referenced books of the last century was conceived to obtain, and in fact gained, a perspicuous and crystal clear text, philologically faithful and relaxingly readable at the same time.

A remarkable account of Kurt Gödel, weaving together creative genius, mental illness, political corruption, and idealism in the face of the turmoil of war and upheaval. At age 24, a brilliant Austrian-born mathematician published a mathematical result that shook the world. Nearly a hundred years after Kurt Gödel's famous 1931 paper'On Formally Undecidable Propositions'appeared, his proof that every mathematical system must contain propositions that are true - yet never provable within that system - continues to pose profound questions for mathematics, philosophy, computer science, and artificial intelligence. His close friend Albert Einstein, with whom he would walk home every day from Princeton's famous Institute for Advanced Study, called him'the greatest logician since Aristotle.'He was also a man who felt profoundly out of place in his time, rejecting the entire current of 20th century philosophical thought in his belief that mathematical truths existed independent of the human mind, and beset by personal demons of anxiety and paranoid delusions that would ultimately lead to his tragic end from self-starvation. Drawing on previously unpublished letters, diaries, and medical records, Journey to the Edge of Reason offers the most complete portrait yet of the life of one of the 20th century's greatest thinkers. Stephen Budiansky's account brings to life the remarkable world of philosophical and mathematical creativity of pre-war Vienna, and documents how it was barbarically extinguished by the Nazis. He charts Gödel's own hair's-breadth escape from Nazi Germany to the scholarly idyll of Princeton; and the complex, gently humorous, sensitive, and tormented inner life of this iconic but previously enigmatic giant of modern science. Weaving together Gödel's public and private lives, this is a tale of creative genius, mental illness, political corruption, and idealism in the face of the turmoil of war and upheaval.

This volume examines problem solving and applied systems aimed at improving performance and management of organizations. The book's eight chapters are integrated into two parts: methodologies and techniques that discuss complex dynamic analysis of the organizations, participative processes for building trend scenarios, consultancy as a systemic intervention process, processes to promote innovative goals in organizations, and analytical processes and solid mathematical representation systems. The authors also include a model to urban parks location, an analytic model to urban services location, and a system to forecast demand with fussy sets.Describes methodologies to analyze processes in complex dynamic organizations, including as participative, interventional, innovative, and analytical approaches;Clarifies a strategies for providing structure to complex organizations and applying analytical methods to decision making;Illustrates problem holistic solving strategies;Explains how to approach several problems from a holistic point of view and how analyze the subjacent processes to make decisions.

Logical paradoxes – like the Liar, Russell's, and the Sorites – are notorious. But in Paradoxes and Inconsistent Mathematics, it is argued that they are only the noisiest of many. Contradictions arise in the everyday, from the smallest points to the widest boundaries. In this book, Zach Weber uses “dialetheic paraconsistency” – a formal framework where some contradictions can be true without absurdity – as the basis for developing this idea rigorously, from mathematical foundations up. In doing so, Weber directly addresses a longstanding open question: how much standard mathematics can paraconsistency capture? The guiding focus is on a more basic question, of why there are paradoxes. Details underscore a simple philosophical claim: that paradoxes are found in the ordinary, and that is what makes them so extraordinary.

This book is about Gottlob Frege. The guiding thought is that Frege left philosophy a legacy which has been largely ignored, not least of all by his admirers. In order of logical priority, Frege's first concern was to locate the law-like behaviour of truths and falsehoods merely by virtue of their being such (in his terms, the structure of Wahrsein). The just-mentioned legacy lies in his first step towards that goal. It consists in winnowing the'logical'from the'psychological', the business of being true as such from that of holding, or holding forth as true-and to keep these separate. A first lesson: what belongs to what is thus abstracted cannot be read directly back into what it was abstracted from. This is what is most widely ignored. The book is divided in three parts. The first presents Frege's general picture of the business of being true-of what belongs to the abstraction. The second is primarily concerned with steps Frege takes (in print) between 1891 and 1895, to pave the way for what became, after logic itself, his central project, that whose attempted carrying out is contained in Grundgesetze I. The third part concerns views of logic, truth, the inexorableness of logic, which Frege eventually came to hold, and what it might be to study'The Mind'as opposed to minds.

This renewed edition of the Tractatus Logico-Philosophicus, exactly a century after Wittgenstein's release, presents the text in a hierarchical manner, “which is the way in which the book was composed and in which Wittgenstein arranged (selected and supplemented) the best of the philosophical remarks that he had been writing since 1913” (Peter Hacker). That tree-like reading is recommended by Wittgenstein himself in the sole footnote of his book, in which he suggests that the inner logical structure of the text is set by the decimal numbers of its propositions. “They alone – the Author will add – give the book perspicuity and clearness, and without this numbering it would be an incomprehensible jumble”. Indeed, the compact and intricate sequence of the traditional presentation is only a rigorous logical bet, but only a logical machine or a robot can unravel the tangle: for an ordinary human understanding that does not exploit its numbering, the book remains “an incomprehensible jumble”. In the present disposition, instead, all horizontal and vertical references become directly manifest and any reader can enjoy the fine architecture and the elegant reasoning of Wittgenstein's work. Every page is an actual reading unit, perfectly coherent and complete. The Tractatus becomes comprehensible also to unskilled readers, of course at more or less deep levels, while a scholar or a more practised reader can detect suggestions and meanings that had remained, until now, completely hidden. A historical note shows in which manner the new structural perspective sheds new light also in the compositional manuscript we have, which “writing units” are very similar, actually, to the pages of the present edition. Besides, this allows to rebuild the list of “Supplements” (here in the Appendix) that Wittgenstein gathered after he roughly finished his manuscript, but that he used very little in the final book. Printing the Tractatus following Wittgenstein's decimal prescriptions required meticulous philological care and some discretional conventions: for instance, at the top of each page the commented-upon proposition is printed again, to make the sight complete and self-sufficient. On the other hand, some forcing of the text by the translators in their sequential reading could be eliminated, restoring a more literal translation. Also the famous and intriguing picture of the eye and its visual field (5.6331) has been restored as Wittgenstein drafted it, making the entire page perfectly understandable and coherent. This documented and editorial work on one of the most referenced books of the last century was conceived to obtain, and in fact gained, a perspicuous and crystal clear text, philologically faithful and relaxingly readable at the same time.

Eleven papers collected in the volume Philosophical Approaches to the Foundations of Logic and Mathematics address various aspects of the “roots”, basic concepts and the nature of logic and mathematics. Taken together, these papers reveal how many serious philosophical problems lie at the foundations of logic and mathematics. The topics discussed in this volume include: transcending anti-foundationalism and two concurrent trends of'anthropological'and'practical'understanding of the foundations of mathematics, new approaches to mathematical realism, the “roots” of logic in a genetic perspective, the primacy of truth or satisfaction, and the “effectiveness” of mathematics in terms of categorical semantics.

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike.

This book is dedicated to the life and work of the mathematician Joachim Lambek (1922–2014). The editors gather together noted experts to discuss the state of the art of various of Lambek's works in logic, category theory, and linguistics and to celebrate his contributions to those areas over the course of his multifaceted career. After early work in combinatorics and elementary number theory, Lambek became a distinguished algebraist (notably in ring theory). In the 1960s, he began to work in category theory, categorical algebra, logic, proof theory, and foundations of computability. In a parallel development, beginning in the late 1950s and for the rest of his career, Lambek also worked extensively in mathematical linguistics and computational approaches to natural languages. He and his collaborators perfected production and type grammars for numerous natural languages. Lambek grammars form an early noncommutative precursor to Girard's linear logic. In a surprising development (2000), he introduced a novel and deeper algebraic framework (which he called pregroup grammars) for analyzing natural language, along with algebraic, higher category, and proof-theoretic semantics. This book is of interest to mathematicians, logicians, linguists, and computer scientists.

This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen's systematical ideas in today's debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen's work on lattice-groups and divisibility theory, and modern set theory and Lorenzen's critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen's consistency proof and Hilbert's larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz.

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to (abstract) algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions.Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics.The bookmay be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics.

Learn more about the history, foundations, and applications of fuzzy logic in this comprehensive resource by an academic leader Introduction to Fuzzy Logic delivers a high-level but accessible introduction to the rapidly growing and evolving field of fuzzy logic and its applications. Distinguished engineer, academic, and author James K. Peckol covers a wide variety of practical topics, including the differences between crisp and fuzzy logic, the people and professions who find fuzzy logic useful, and the advantages of using fuzzy logic. While the book assumes a solid foundation in embedded systems, including basic logic design, and C/C++ programming, it is written in a practical and easy-to-read style that engages the reader and assists in learning and retention. The author includes introductions of threshold and perceptron logic to further enhance the applicability of the material contained within. After introducing readers to the topic with a brief description of the history and development of the field, Introduction to Fuzzy Logic goes on to discuss a wide variety of foundational and advanced topics, like: A review of Boolean algebra, including logic minimization with algebraic means and Karnaugh maps A discussion of crisp sets, including classic set membership, set theory and operations, and basic classical crisp set properties A discussion of fuzzy sets, including the foundations of fuzzy sets logic, set membership functions, and fuzzy set properties An analysis of fuzzy inference and approximate reasoning, along with the concepts of containment and entailment and relations between fuzzy subsetsPerfect for mid-level and upper-level undergraduate and graduate students in electrical, mechanical, and computer engineering courses, Introduction to Fuzzy Logic covers topics included in many artificial intelligence, computational intelligence, and soft computing courses. Math students and professionals in a wide variety of fields will also significantly benefit from the material covered in this book.