The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.

For thousands of years, mathematicians have used the timeless art of logic to see the world more clearly. In The Art of Logic, Royal Society Science Book Prize nominee Eugenia Cheng shows how anyone can think like a mathematician - and see, argue and think better.

Learn how to simplify complex decisions without over-simplifying them. Discover the power of analogies and the dangers of false equivalences. Find out how people construct misleading arguments, and how we can argue back.

Eugenia Cheng teaches us how to find clarity without losing nuance, taking a careful scalpel to the complexities of politics, privilege, sexism and dozens of other real-world situations. Her Art of Logic is a practical and inspiring guide to decoding the modern world.

What link might connect two far worlds like quantum theory and music? There is something universal in the mathematical formalism of quantum theory that goes beyond the limits of its traditional physical applications. We are now beginning to understand how some mysterious quantum concepts, like superposition and entanglement, can be used as a semantic resource.

This book is the third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.

Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities as well as their interactions with different ideational spectra. In all classical algebraic structures, the law of compositions on a given set are well-defined, but this is a restrictive case because there are situations in science where a law of composition defined on a set may be only partially defined and partially undefined, which we call NeutroDefined, or totally undefined, which we call AntiDefined. Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebra introduces NeutroAlgebra, an emerging field of research. This book provides a comprehensive collection of original work related to NeutroAlgebra and covers topics such as image retrieval, mathematical morphology, and NeutroAlgebraic structure. It is an essential resource for philosophers, mathematicians, researchers, educators and students of higher education, and academicians.

Fuzzy logic, which is based on the concept of fuzzy set, has enabled scientists to create models under conditions of imprecision, vagueness, or both at once. As a result, it has now found many important applications in almost all sectors of human activity, becoming a complementary feature and supporter of probability theory, which is suitable for modelling situations of uncertainty derived from randomness. Fuzzy mathematics has also significantly developed at the theoretical level, providing important insights into branches of traditional mathematics like algebra, analysis, geometry, topology, and more. With such widespread applications, fuzzy sets and logic are an important area of focus in mathematics. Advances and Applications of Fuzzy Sets and Logic studies recent theoretical advances of fuzzy sets and numbers, fuzzy systems, fuzzy logic and their generalizations, extensions, and more. This book also explores the applications of fuzzy sets and logic applied to science, technology, and everyday life to further provide research on the subject. This book is ideal for mathematicians, physicists, computer specialists, engineers, practitioners, researchers, academicians, and students who are looking to learn more about fuzzy sets, fuzzy logic, and their applications.

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.

Ultrafilters and ultraproducts provide a useful generalization of the ordinary limit processes which have applications to many areas of mathematics. Typically, this topic is presented to students in specialized courses such as logic, functional analysis, or geometric group theory. In this book, the basic facts about ultrafilters and ultraproducts are presented to readers with no prior knowledge of the subject and then these techniques are applied to a wide variety of topics. The first part of the book deals solely with ultrafilters and presents applications to voting theory, combinatorics, and topology, while also dealing also with foundational issues. The second part presents the classical ultraproduct construction and provides applications to algebra, number theory, and nonstandard analysis. The third part discusses a metric generalization of the ultraproduct construction and gives example applications to geometric group theory and functional analysis. The final section returns to more advanced topics of a more foundational nature. The book should be of interest to undergraduates, graduate students, and researchers from all areas of mathematics interested in learning how ultrafilters and ultraproducts can be applied to their specialty.

In the world of mathematics, the study of fuzzy relations and its theories are well-documented and a staple in the area of calculative methods. What many researchers and scientists overlook is how fuzzy theory can be applied to industries outside of arithmetic. The framework of fuzzy logic is much broader than professionals realize. There is a lack of research on the full potential this theoretical model can reach. Emerging Applications of Fuzzy Algebraic Structures provides emerging research exploring the theoretical and practical aspects of fuzzy set theory and its real-life applications within the fields of engineering and science. Featuring coverage on a broad range of topics such as complex systems, topological spaces, and linear transformations, this book is ideally designed for academicians, professionals, and students seeking current research on innovations in fuzzy logic in algebra and other matrices.

This book is the second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices. Volume I, first published in the 1980s, built the foundations of the theory and is considered to be a classic in this field. The long-awaited volumes II and III are now available. Taken together, the three volumes provide a comprehensive picture of the state of art in general algebra today, and serve as a valuable resource for anyone working in the general theory of algebraic systems or in related fields. The two new volumes are arranged around six themes first introduced in Volume I. Volume II covers the Classification of Varieties, Equational Logic, and Rudiments of Model Theory, and Volume III covers Finite Algebras and their Clones, Abstract Clone Theory, and the Commutator. These topics are presented in six chapters with independent expositions, but are linked by themes and motifs that run through all three volumes.

Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

Are you smarter than a Singaporean ten-year-old? Can you beat Sherlock Holmes? If you think the answer is yes - I challenge you to solve my problems. Here are 125 of the world's best brainteasers from the last two millennia, taking us from ancient China to medieval Europe, Victorian England to modern-day Japan, with stories of espionage, mathematical breakthroughs and puzzling rivalries along the way. Pit your wits against logic puzzles and kinship riddles, pangrams and river-crossing conundrums. Some solutions rely on a touch of cunning, others call for creativity, others need mercilessly logical thought. Some can only be solved be 2 per cent of the population. All are guaranteed to sharpen your mind. Let's get puzzling!

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.

"Inspiring and informative...deserves to be widely read." -Wall Street Journal "This fun book offers a philosophical take on number systems and revels in the beauty of math." -Science News Because we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages. Paul Lockhart presents arithmetic not as rote manipulation of numbers-a practical if mundane branch of knowledge best suited for filling out tax forms-but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher. "A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education... Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting." -Jonathon Keats, New Scientist "What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind's most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story... A wonderful book." -Keith Devlin, author of Finding Fibonacci

The book is about strong axioms of infi nity in set theory (also known as large cardinal axioms), and the ongoing search for natural models of these axioms. Assuming the Ultrapower Axiom, a combinatorial principle conjectured to hold in all such natural models, we solve various classical problems in set theory (for example, the Generalized Continuum Hypothesis) and uncover a theory of large cardinals that is much clearer than the one that can be developed using only the standard axioms.