Richard E. Hodel

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This book is based on an interdisciplinary course on logic offered to upper-level undergraduates at Duke University over a period of more than ten years. Why an interdisciplinary course on logic? Although logic has been a discipline of study in philosophy since ancient times, in recent decades it has played an important role in other disciplines as well. For example, logic is at the core of two programming languages, is used in program verification, has enriched philosophy (and computer science) with non-classical logics that can deal constructively with contradictions, and has shaken the foundations of mathematics with insight into non-computable functions and non-provability. Several of these ideas are treated in this book. This book addresses select topics drawn from three different areas of logic: proof theory, computability theory, and philosophical logic. A common thread throughout is the application of logic to computers and computation. Part 1 on Proof Theory introduces a deductive system (resolution logic) that comes from an area of research known as automated deduc- tion. Part 2 on Computability Theory explores the limits of computation using an abstract model of computers called register machines. Part 3 on Philosophical Logic presents a certain non-classical logic (relevance logic) and a semantics for it that is useful for automated reasoning systems that must deal with the possibility of inconsistent information. The three areas from which the special topics are drawn — proof theory, computability theory, and philosophical logic — exhibit the different roles that logic plays in three different disciplines: computer science, mathematics, and philosophy. The three parts of the book were written by a computer scientist, a mathematician, and a philosopher, respectively, and each part was reviewed by the other two authors for accessibility to students in their fields. The three parts of the book are roughly of equal length. The second part, on computability theory, is largely independent of the first, but the third part, on philosophical logic, is best presented after the first two parts.