Now in a new edition --the classic presentation of the theory of computable functions in the context of the foundations of mathematics. Part I motivates the study of computability with discussions and readings about the crisis in the foundations of mathematics in the early 20th century, while presenting the basic ideas of whole number, function, proof, and real number. Part II starts with readings from Turing and Post leading to the formal theory of recursive functions. Part III presents sufficient formal logic to give a full development of G del's incompleteness theorems. Part IV considers the significance of the technical work with a discussion of Church's Thesis and readings on the foundations of mathematics. This new edition contains the timeline "Computability and Undecidability" as well as the essay "On mathematics."

Propositional Logics presents the history, philosophy, and mathematics of the major systems of propositional logic. Classical logic, modal logics, many-valued logics, intuitionism, paraconsistent logics, and dependent implication are examined in separate chapters. Each begins with a motivation in the originators' own terms, followed by the standard formal semantics, syntax, and completeness theorem. The chapters on the various logics are largely self-contained so that the book can be used as a reference. An appendix summarizes the formal semantics and axiomatizations of the logics. The view that unifies the exposition is that propositional logics comprise a spectrum. As the aspect of propositions under consideration varies, the logic varies. Each logic is shown to fall naturally within a general framework for semantics. A theory of translations between logics is presented that allows for further comparisons, and necessary conditions are given for a translation to preserve meaning. For this third edition the material has been re-organized to make the text easier to study, and a new section on paraconsistent logics with simple semantics has been added which challenges standard views on the nature of consequence relations. The text includes worked examples and hundreds of exercises, from routine to open problems, making the book with its clear and careful exposition ideal for courses or individual study.

This series of books presents the fundamentals of logic in a style accessible to both students and scholars. The text of each essay presents a story, the main line of development of the ideas, while the notes and appendices place the research within a larger scholarly context. The essays overlap, forming a unified analysis of logic as the art of reasoning well, yet each essay is designed so that it may be read independently. The topic of this volume is prescriptive reasoning. Descriptive claims say how the world is, was, or will be; prescriptive claims say how the world should be. We have fairly clear rules for reasoning with descriptive claims. The goal of the first essay, "Reasoning with Prescriptive Claims," is to clarify how to reason with prescriptive ones. The first step in doing so is to justify our viewing prescriptions as true or false. That justification is part of a general approach to reasoning in which many kinds of personal evaluations are taken to be true-false divisions. That view has been implicit if not explicit in analyses of reasoning from formal logic through argument analysis. in "Truth and Reasoning" I set out reasons for adopting that methodology. Theories, too, seem to be descriptive or prescriptive. Some say how the world is, others how the world should be. Yet, as shown in "Prescriptive Theories?," on close examination the distinction evaporates. Unless, that is, one says that certain theories about values use an entirely different notion of truth than is used in science and is codified in our usual methods of reasoning. Absent that, there seems to be no justification for constructing and evaluating what are typically thought of as prescriptive theories differently from descriptive ones. Many discussions of how to evaluate prescriptive claims are given in terms of what is rational or irrational to do. In the final essay, "Rationality," what we mean by the idea of someone being rational is investigated and the limitations of that label in evaluating reasoning or actions is shown.

In "Classical Mathematical Logic," Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.

The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference.

"Classical Mathematical Logic" presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

This series of books is meant to present the fundamentals of reasoning well in a clear manner accessible to both scholars and students. The body of each essay gives the main development of the subject, while the footnotes and appendices place the research within a larger scholarly context. The topic of this volume is the nature and evaluation of reasoning in science and mathematics. Science and mathematics can both be understood as proceeding by a method of abstraction from experience. Mathematics is distinguished from other sciences only in its greater abstraction and its demand for necessity in its inferences. That methodology of abstraction is the main focus here. The study of these subjects is not just of academic interest. First comes clear thinking, then comes clear research and clear writing. The essays: Background Models and Theories Experiments Mathematics as the Art of Abstraction

This series of books presents the fundamentals of logic in a style accessible to both students and scholars. The text of each essay presents a story, the main line of development of the ideas, while the notes and appendices place the research within a larger scholarly context. The essays overlap, forming a unified analysis of logic as the art of reasoning well, yet each essay is designed so that it may be read independently. The question addressed in this volume is how we can justify our beliefs through reasoning. The first essay, "Arguments," investigates what it is that we call true or false and how we reason toward truths through arguments. A general theory of argument analysis is set out on the basis of what we can assume about those with whom we reason. The next essay, "Fallacies," explains how the classification of an argument as a fallacy can be used within that general approach. In contrast, there is no agreement on what the terms "induction" and "deduction" mean, and they are not useful in evaluating arguments, as shown in "Induction and Deduction." In reasoning to truths, in the end we must take some claims as basic, not requiring any justification for accepting them. How we choose those claims and how they affect our reasoning is examined in "Base Claims." The essay "Analogies" considers how comparisons can be used as the basis of arguments, arguing from similar situations to similar conclusions. An important use of analogies is in reasoning about the mental life of other people and things, which is examined in "Subjective Claims," written with Fred Kroon and William S. Robinson. "Generalizing" examines how to argue from part of a collection or mass to the whole or a larger part. The question there is whether we are ever justified in accepting such an argument as good. "Probabilities" sets out the three main ways probability statements have been interpreted: the logical relation view, the frequency view, and the subjective degree of belief view. Each of those is shown to be inadequate to make precise the scale of plausibility of claims and the scale of the likelihood of a possibility. Many discussions of how to reason well and what counts as good reason are given in terms of who or what is rational. In the final essay, "Rationality," it's shown that what we mean by the idea of someone being rational is of very little use in evaluating reasoning or actions. This volume is meant to give a clearer idea of how to reason well, setting out methods of evaluation that are motivated in terms of our abilities and interests. At the ground of our reasoning, though, are metaphysical assumptions, too basic and too much needed in our reasoning for us to justify them through reasoning. But we can try to uncover those assumptions to see how they are important and what depends on them, which is a major them of this volume.

This series of books presents the fundamentals of reasoning well, in a style accessible to both students and scholars. The text of each essay presents a story, the main line of development of the ideas, while the footnotes and appendices place the research within a larger scholarly context. The essays overlap, forming a unified analysis of reasoning, yet each essay is designed so that it may be read independently of the others. The topic of this volume is the evaluation of reasoning about cause and effect, reasoning using conditionals, and reasoning that involves explanations. The essay "Reasoning about Cause and Effect" sets out a way to analyze whether there is cause and effect in terms of whether an inference from a claim describing the purported cause to a claim describing the purported effect satisfies specific conditions. Different notions of cause and effect correspond to placing different conditions on what counts as a good causal inference. An application of that method in "The Directedness of Emotions" leads to a clearer understanding of the issue whether every emotion need be directed at something. In the essay "Conditionals" various ways of analyzing reasoning with claims of the form "if . . . then . . ." are surveyed. Some of those uses are meant to be judged as inferences that are not necessarily valid, and conditions are given for when we can consider such inferences to be good. In "Explanations" verbal answers to a question why a claim is true are evaluated in terms of conditions placed on inferences from the explaining claims to the claim being explained. Recognizing that the direction of inference of such an explanation is the reverse of that for an argument with the very same claims is crucial in their evaluation. Explanations in terms of functions and goals are also investigated.