While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their "Principia mathematica (1910-1913)."

This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schroder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Godel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GodeI.

Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials.

Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since."

Dieses Buch ist aus Skripten der Autoren zu ihrer Vorlesung "Mathe- matische Logik (fUr Informatiker)" entstanden. Diese sechssttindige Lehrveranstaltung, die seit dem Sommersemester 1974 jahrlich an der Technischen Universitat Berlin im Fachbereich Informatik ab- gehalten wird, will Informatik-Studenten etwa yom 4. Semester an mit Logik-Methoden vertraut machen und gleichzeitig einen" Bei- trag zur Mathematik-Ausbildung fUr Informatiker leisten. Dement- sprechend handelt es sich urn einen einfUhrenden Text fUr "krasse" Anfanger in der Logik, der mit elementaren Mathematik -Kenntnissen lesbar ist und an Informatik-Voraussetzungen nur einfachste Kon- zepte von Programmiersprachen benotigt. Anliegen des Buches, das sich gleichermaBen an Mathematik- und Informatik-Studenten wen- det, ist es, einerseits eine mathematisch zufriedenstellende Darstellung der Anfangsgrtinde der Pradikatenlogik der ersten Stufe zu geben, andererseits aber auch Anwendungen dieser Logik innerhalb der Informatik einheitlich in die Logik-Darstellung einzubeziehen. Der Versuch, ein Buch tiber Logik mit Informatik-Anwendungen zu schreiben, ist nicht ohne Probleme, da die Auswahl der Verbin- dungen von Logik und Informatik eine subjektive Entscheidung bleibt, so daB tiber den hier vorliegenden Text hinaus Raum fUr andere Bertihrungspunkte und fUr eine intensivere Gestaltung der hier im Text angefUhrten Anwendungen besteht. Man kann dabei z. B. an engere Verbindungen zur theoretischen Informatik denken oder an eine systematische Abhandlung der angesprochenen Anwen- dungsgebiete. Dieser Text will dazu anregen, Informatik und Lo- gik so aufeinander zu beziehen, daB Logik als Hilfsmittel fUr die Informatik angesehen werden darf, d. h. als eine fruchtbare, Infor- matik-Ergebnisse hervorbringende Methode.

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.

The accelerating development of computer technology and communications can replace many of the functions of human intellectual activity, as well as help them in making decisions in various situations of their lives. To implement intelligent functions for various purposes, numerous models, paradigms, architectures, and hardware and software are being developed. Because the world is constantly evolving, there is a need to constantly study various dynamic processes to determine possible negative situations that can lead to undesirable catastrophic phenomena and changes. Recently, more attention has been paid to the study of natural processes in nature. Scientific works are appearing that describe the behavior and development of living organisms and the processes of their interaction. Cellular automata are increasingly used to describe and model them. New Methods and Paradigms for Modeling Dynamic Processes Based on Cellular Automata is a collection of innovative research that describes the models and paradigms of building cellular automata that allows for the simulation of the dynamics of the interaction of living organisms from a different scientific point of view. For this, asynchronous cellular automata with a dynamically changing number of "living" cells are used. The chapters describe the theoretical concepts of constructing asynchronous cellular automata with active cells. Much attention is paid to the use of the proposed theoretical principles for solving modeling problems and solving specific applied problems of forming pseudorandom sequences and image processing based on modeling of the human visual channel. Featuring research on topics such as colony interaction, image processing and recognition, and influence mode, this book is ideally designed for engineers, programmers, software developers, researchers, academicians, and students.

The publication of Rasiowa and Sikorski's The Mathematics of Metamathematics (1970), Rasiowa's An Algebraic Approach to Non-Classical Logics (1974), and Wojcicki's Theory of Logical Calculi (1988) created a niche in the field of mathematical and philosophical logic. This in-depth study of the concept of a consequence relation, culminating in the concept of a Lindenbaum-Tarski algebra, fills this niche. Citkin and Muravitsky consider the problem of obtaining confirmation that a statement is a consequence of a set of statements as prerequisites, on the one hand, and the problem of demonstrating that such confirmation does not exist in the structure under consideration, on the other hand. For the second part of this problem, the concept of the Lindenbaum-Tarski algebra plays a key role, which becomes even more important when the considered consequence relation is placed in the context of decidability. This role is traced in the book for various formal objective languages. The work also includes helpful exercises to aid the reader's assimilation of the book's material. Intended for advanced undergraduate and graduate students in mathematics and philosophy, this book can be used to teach special courses in logic with an emphasis on algebraic methods, for self-study, and also as a reference work.