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. "[Richard Epstein] never gives only the technical side of the matter, but...always offers intuitive motivations and explains basic decisions which constitute the whole approach, and which build a bridge to the students' experiences, with natural language as well as with standard 'elementary' mathematics. The book is a self-contained textbook, requiring as background only some facility in mathematics...This makes the book particularly suitable as a textbook for self-study." ---Siegfried J. Gottwald, Zentralblatt MATH Richard L. Epstein received his doctorate in mathematics from the University of California, Berkeley. He is the author of eleven books, including two others in the series The Semantic Foundations of Logic (Propositional Logics and Predicate Logic), Five Ways of Saying "Therefore," Critical Thinking , and, with Walter Carnielli, Computability . He is head of the Advanced Reasoning Forum in Socorro, New Mexico. Classical Mathematical Logic The Semantic Foundations of Logic By Richard L. Epstein Princeton University Press Copyright © 2011 Princeton University Press All right reserved. ISBN: 978-0-691-12300-4 Contents Preface...................................................................................................xviiAcknowledgments...........................................................................................xixIntroduction..............................................................................................xxiBibliography..............................................................................................487Index of Notation.........................................................................................495Index.....................................................................................................499 Chapter One Classical Propositional Logic A. Propositions 1 Other views of propositions 2 B. Types 3 ? Exercises for Sections A and B 4 C. The Connectives of Propositional Logic 5 ? Exercises for Section C 6 D. A Formal Language for Propositional Logic 1. Defining the formal language 7 A platonist definition of the formal language 8 2. The unique readability of wffs 8 3. Realizations 11 ? Exercises for Section D 12 E. Classical Propositional Logic 1. The classical abstraction and truth-functions 13 2. Models 17 ? Exercises for Sections E.1 and E.2 17 3. Validity and semantic consequence 18 ? Exercises for Section E.3 20 F. Formalizing Reasoning 20 ? Exercises for Section F 24 Proof by induction 25 To begin our analysis of reasoning we need to be clear about what kind of thing is true or false. Then we will look at ways to reason with combinations of those things. A. Propositions When we argue, when we prove, we do so in a language. And we seem to be able to confine ourselves to declarative sentences in our reasoning. I will assume that what a sentence is and what a declarative sentence is are well enough understood by us to be taken as primitive , that is, undefined in terms of any other fundamental notions or concepts. Disagreements about some particular examples may arise and need to be resolved, but our common understanding of what a declarative sentence is will generally suffice. So we begin with sentences, written (or uttered) concatenations of inscriptions (or sounds). To study these we may ignore certain aspects, such as what color ink they are written in, leaving ourselves only certain features of sentences to consider in reasoning. The most fundamental is whether they are true or false. In general we understand well enough what it means for a simple sentence such as 'Ralph is a dog' to be true or to be false. For such sentences we can regard truth as a primitive notion, one we understand how to use in most applications, while falsity we can understand as the opposite of truth, the not-true. Our goal is to formalize truth and falsity for more