Introduction to Logic

Introduction to Logic

Introduction to Logic

Introduction to Logic

Excerpt

This book has been written primarily to serve as a textbook for a first course in modern logic. No background in mathematics or philosophy is supposed. My main objective has been to familiarize the reader with an exact and complete theory of logical inference and to show how it may be used in mathematics and the empirical sciences. Since several books already available have aims closely related to the one just stated, it may be well to mention the major distinguishing features of the present book.

Part I (the first eight chapters) deals with formal principles of inference and definition. Beginning with the theory of sentential inference in Chapter 2 there is continual emphasis on application of the method of interpretation to prove arguments invalid, premises consistent, or axioms of a theory independent. There is a detaimed attempt (Chapter 7) to relate the formal theory of inference to the standard informal proofs common throughout mathematics. The theory of definition is presented (Chapter 8) in more detail than in any other textbook known to the author; a discussion of the method of Padoa for proving the independence of primitive concepts is included.

Part II (the last four chapters) is devoted to elementary intuitive set theory, with separate chapters on sets, relations, and functions. The treatment of ordering relations in Chapter 10 is rather extensive. Part II is nearly self-contained and can be read independently of Part I. The last chapter (Chapter 12) is concerned with the set-theoretical foundations of the axiomatic method. The idea that the best way to axiomatize a branch of mathematics is to define appropriate set-theoretical predicates is familiar to modern mathematicians and certainly does not originate with the author, but the exposition of this idea, which provides a sharp logical foundation for the axiomatic method, has been omitted from the excellent elementary textbooks on modern mathematics which have appeared in recent years.

Beginning with Chapter 4, numerous examples of axiomatically formulated theories are introduced in the discussion and exercises. These ex-

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