This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.

This volume is number ten in the 11-volume Handbook of the History of Logic. While there are many examples were a science split from philosophy and became autonomous (such as physics with Newton and biology with Darwin), and while there are, perhaps, topics that are of exclusively philosophical interest, inductive logic - as this handbook attests - is a research field where philosophers and scientists fruitfully and constructively interact. This handbook covers the rich history of scientific turning points in Inductive Logic, including probability theory and decision theory. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, cognitive psychology, and artificial intelligence, for whom the historical background of his or her work is a salient consideration.

Chapter on the Port Royal contributions to probability theory and decision theory

Serves as a singular contribution to the intellectual history of the 20th century Contains the latest scholarly discoveries and interpretative insights"

This book examines the true core of philosophy and metaphysics, taking account of quantum and relativity theory as it applies to physical Reality, and develops a line of reasoning that ultimately leads us to Reality as it is currently understood at the most fundamental level - the Standard Model of Elementary Particles. This book develops new formalisms for Logic that are of interest in themselves and also provide a Platonic bridge to Reality. The bridge to Reality will be explored in detail in a subsequent book, Relativistic Quantum Metaphysics: A First Principles Basis for the Standard Model of Elementary Particles. We anticipate that the current "fundamental" level of physical Reality may be based on a still lower level and/or may have additional aspects remaining to be found. However the effects of certain core features such as quantum theory and relativity theory will persist even if a lower level of Reality is found, and these core features suggest the form of a new Metaphysics of physical Reality. We have coined the phrase "Operator Metaphysics" for this new metaphysics of physical Reality. The book starts by describing aspects of Philosophy and Metaphysics relevant to the study of current physical Reality. Part of this development are new Logics, Operator Logic and Quantum Operator Logic, developed in earlier books by this author (and revised and expanded in this book). Using them we are led to develop a connection to the beginnings of The Standard Model of Elementary Particles. While mathematics is essential in the latter stages of the book we have tried to present it with sufficient text discussion to make what it is doing understandable to the non-mathematical reader. Generally we will avoid using the jargon of Philosophy, Logic and Physics as much as possible.

Quantification and modalities have always been topics of great interest for logicians. These two themes emerged from philosophy and
language in ancient times; they were studied by traditional informal
methods until the 20th century. In the last century the tools became
highly mathematical, and both modal logic and quantification found numerous applications in Computer Science. At the same time many other kinds of nonclassical logics were investigated and applied to Computer Science.
Although there exist several good books in propositional modal logics, this book is the first detailed monograph in nonclassical first-order quantification. It includes results obtained during the past thirty years. The field is very large, so we confine ourselves with only two kinds of logics: modal and superintuitionistic. The main emphasis of Volume 1 is model-theoretic, and it concentrates on descriptions of different sound semantics and completeness problem --- even for these seemingly simple questions we have our hands full. The major part of the presented material has never been published before. Some results are very recent, and for other results we either give new proofs or first proofs in full detail.

A hands-on introduction to the tools needed for rigorous and theoretical mathematical reasoning
Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics.
This essential book:
* Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs
* Reinforces the foundations of calculus and algebra
* Explores how to use both a direct and indirect proof to prove a theorem
* Presents the basic properties of real numbers
* Discusses how to use mathematical induction to prove a theorem
* Identifies the different types of theorems
* Explains how to write a clear and understandable proof
* Covers the basic structure of modern mathematics and the key components of modern mathematics

A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs.
Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty.
Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra.

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Godel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central.
In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality.
Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredlyinternational enterprise, whose research programmes reach to every corner of the learned world.
Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value.
Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas.

Quantum mechanics is arguably one of the most successful scientific theories ever and its applications to chemistry, optics, and information theory are innumerable. This book provides the reader with a rigorous treatment of the main mathematical tools from harmonic analysis which play an essential role in the modern formulation of quantum mechanics. This allows us at the same time to suggest some new ideas and methods, with a special focus on topics such as the Wigner phase space formalism and its applications to the theory of the density operator and its entanglement properties. This book can be used with profit by advanced undergraduate students in mathematics and physics, as well as by confirmed researchers.

This book is an attempt to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. Its intended audience consists of logicians, type theorists, category theorists and (theoretical) computer scientists.

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

This proceedings volume documents the contributions presented at the conference held at Fairfield University and at the Graduate Center, CUNY in 2018 celebrating the New York Group Theory Seminar, in memoriam Gilbert Baumslag, and to honor Benjamin Fine and Anthony Gaglione. It includes several expert contributions by leading figures in the group theory community and provides a valuable source of information on recent research developments.

Volume II of "Classical Recursion Theory" describes the universe from a local (bottom-up
or synthetical) point of view, and covers the whole spectrum, from the
recursive to the arithmetical sets.
The first half of the book provides a detailed picture of the computable
sets from the perspective of Theoretical Computer Science. Besides giving a
detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity
classes, ranging from small time and space bounds to the elementary functions,
with a particular attention to polynomial time and space computability. It also
deals with primitive recursive functions and larger classes, which are of
interest to the proof theorist.
The second half of the book starts with the classical theory of recursively
enumerable sets and degrees, which constitutes the core of Recursion or
Computability Theory. Unlike other texts, usually confined to the Turing
degrees, the book covers a variety of other strong reducibilities, studying
both their individual structures and their mutual relationships. The last
chapters extend the theory to limit sets and arithmetical sets. The volume
ends with the first textbook treatment of the enumeration degrees, which
admit a number of applications from algebra to the Lambda Calculus.
The book is a valuable source of information for anyone interested in
Complexity and Computability Theory. The student will appreciate the detailed
but informal account of a wide variety of basic topics, while the specialist
will find a wealth of material sketched in exercises and asides. A massive
bibliography of more than a thousand titles completes the treatment on the
historical side.

In real management situations, uncertainty is inherently present in decision making. As such, it is increasingly imperative to research and develop new theories and methods of fuzzy sets. Theoretical and Practical Advancements for Fuzzy System Integration is a pivotal reference source for the latest scholarly research on the importance of expressing and measuring fuzziness in order to develop effective and practical decision making models and methods. Featuring coverage on an expansive range of perspectives and topics, such as fuzzy logic control, intuitionistic fuzzy set theory, and defuzzification, this book is ideally designed for academics, professionals, and researchers seeking current research on theoretical frameworks and real-world applications in the area of fuzzy sets and systems.

This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth.

The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.

This book is a tribute to Professor Ewa Orlowska, a Polish logician who was celebrating the 60th year of her scientific career in 2017. It offers a collection of contributed papers by different authors and covers the most important areas of her research. Prof. Orlowska made significant contributions to many fields of logic, such as proof theory, algebraic methods in logic and knowledge representation, and her work has been published in 3 monographs and over 100 articles in internationally acclaimed journals and conference proceedings. The book also includes Prof. Orlowska's autobiography, bibliography and a trialogue between her and the editors of the volume, as well as contributors' biographical notes, and is suitable for scholars and students of logic who are interested in understanding more about Prof. Orlowska's work.

Now in a new edition --the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of G del's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics."

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.

There are many proposed aims for scientific inquiry - to explain or predict events, to confirm or falsify hypotheses, or to find hypotheses that cohere with our other beliefs in some logical or probabilistic sense. This book is devoted to a different proposal - that the logical structure of the scientist's method should guarantee eventual arrival at the truth, given the scientist's background assumptions. Interest in this methodological property, called "logical reliability", stems from formal learning theory, which draws its insights not from the theory of probability, but from the theory of computability. Kelly first offers an accessible explanation of formal learning theory, then goes on to develop and explore a systematic framework in which various standard learning-theoretic results can be seen as special cases of simpler and more general considerations. Finally, Kelly clarifies the relationship between the resulting framework and other standard issues in the philosophy of science, such as probability, causation, and relativism. Extensively illustrated with figures by the author, The Logic of Reliable Inquiry assumes only introductory knowledge of basic logic and computability theory. It is a major contribution to the literature and will be essential reading for scientists, statiticians, psychologists, linguists, logicians, and philosophers.

The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay's conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

Berto's highly readable and lucid guide introduces students and the interested reader to Godel's celebrated "Incompleteness Theorem," and discusses some of the most famous - and infamous - claims arising from Godel's arguments.Offers a clear understanding of this difficult subject by presenting each of the key steps of the "Theorem" in separate chaptersDiscusses interpretations of the "Theorem" made by celebrated contemporary thinkersSheds light on the wider extra-mathematical and philosophical implications of Godel's theoriesWritten in an accessible, non-technical style