Justification Logic: Reasoning with Reasons
Cambridge University Press, 2019
Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.
Published June 2019
Set Theory and the Continuum Problem
Revised and updated from the 1996 edition, this volume is a lucid, elegant, and complete survey of set theory drawn from the authors' substantial teaching experience. The book is intended to provide the reader with a complete foundation in modern set theory, explaining how the subject is axiomatized and the role of the axiom of choice, and elucidating ordinal and cardinal numbers, constructible sets, and forcing. The first of three parts focuses on axiomatic set theory, examining problems related to size comparisons between infinite sets, basics of class theory, and natural numbers, as well as Smullyan's double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The second part addresses Mostowski-Shepherdson mappings, reflection principles, constructible sets and constructibility, and the continuum hypothesis. The text concludes with an extensive exploration of forcing and independence results.
Published April 2010
Incompleteness in the Land of Sets
This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. Gödel numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to Gödel's Second Incompleteness Theorem. Exercises are scattered throughout.
Published February 2007
College Publications, 2007