They adopt throughout a threefold approach. Semantically, they use possible world models; the formal proof machinery is tableaus; and full philosophical discussions are provided of the way that technical developments bear on well-known philosophical problems. The book covers quantification itself, including the difference between actualist and possibilist quantifiers; equality, leading to a treatment of Frege's morning star/evening star puzzle; the notion of existence and the logical problems surrounding it; non-rigid constants and function symbols; predicate abstraction, which abstracts a predicate from a formula, in effect providing a scoping function for constants and function symbols, leading to a clarification of ambiguous readings at the heart of several philosophical problems; the distinction between nonexistence and nondesignation; and definite descriptions, borrowing from both Fregean and Russellian paradigms.

    This text attempts to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category.

    Kant distinguishes things in themselves from phenomena, thus making a metaphysical distinction between intrinsic and relational properties of substances. Langton argues that his claim that we have no knowledge of things in themselves is not idealism, but epistemic humility; we have no knowledge of the intrinsic properties of substances. This interpretation vindicates Kant's scientific realism and shows his primary/secondary quality distinction to be superior even to modern day competitors. And it answers the famous charge that Kant's tale of things in themselves is one that makes itself untellable.

    Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic. Additional results are presented in an appendix.

    Baader and Nipkow cover all the basic material--abstract reduction systems, termination, confluence, completion, and combination problems--but also some important and closely connected subjects: universal algebra, unification theory, Grobner bases, and Buchberger's algorithm. They present the main algorithms both informally and as programs in the functional language Standard ML (An appendix contains a quick and easy introduction to ML). Key chapters cover crucial algorithms such as unification and congruence closure in more depth and develop efficient Pascal programs. The book contains many examples and over 170 exercises. This is also an ideal reference book for professional researchers: results spread over many conference and journal articles are collected here in a unified notation, detailed proofs of almost all theorems are provided, and each chapter closes with a guide to the literature.

    Ones and Zeros follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. Readers will learn about the interesting history of the development of symbolic logic in particular, and the often misunderstood process of mathematical invention and scientific discovery, in general. Ones and Zeros also features practical exercises with answers, real-world examples of digital circuit design, and a reading list. Ones and Zeros will be of particular interest to software engineers who want to gain a comprehensive understanding of computer hardware. Outstanding features include: a history of mathematical logic, an explanation of the logic of digital circuits, and hands-on exercises and examples.

    

    This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; on Frege, Dedekind, Cantor, and Russell; and on miscellaneous topics in logic and proof theory, including three papers on various aspects of the GOdel theorems. Boolos is universally recognized as the leader in the renewed interest in studies of Frege's work on logic and the philosophy of mathematics. John Burgess has provided introductions to each of the three parts of the volume, and also an afterword on Boolos's technical work in provability logic, which is beyond the scope of this volume.

    This book is a relatively self-contained introduction to the subject, which includes the necessary background material, as well as numerous examples and exercises.