Proofs and Algorithms: An Introduction to Logic and Computability

Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: An Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.

Gilles Dowek is a Professor at École Polytechnique. He is also a Researcher at the Laboratoire d'Informatique de l'École Polytechnique and the Institut National de Recherche en Informatique et en Automatique (INRIA). His research concerns the formalization of mathematics and the mechanization of reasoning. His main contribution is a reformulation of the axiomatic method which provides a central role to the notion of computation.

Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation.

Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself.

Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.

Proofs.-Predictive Logic.-Inductive Definitions.-Languages.-The Languages of Predicate Logic.-Proofs.-Examples of Theories.-Variations on the Principle of the Excluded Middle.-Models.-The Notion of a Model.-The Soundness Theorem.-The Completeness Theorem.-Other Applications of the Notion of Model.-Algorithms.-Computable Functions.-Computable Functions.-Computability over Lists and Trees.-Eliminating Recursion.-Programs.-Computation as a Sequence of Small Steps.-Proofs and Algorithms.-Church's Theorem.-Automated Theorem Proving.-Sequent Calculus.-Proof Search in the Sequent Calculus Without Cuts.-Decidable theories.-Constructivity.-Epilogue.-Index.-Bibliography