Introduction to Symbolic Logic and Its Applications

Rudolf Carnap

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486143491
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

This book is one of the clearest, most comprehensive and rigorous introductions to modern symbolic logic available in any language. Professor Carnap, a world authority on symbolic logic, develops the subject from elementary concepts and simple exercises through the construction and analysis of a number of relatively complex logical languages. He then considers, in great detail, the application of symbolic logic to the clarification and axiomatization of various theories in mathematics, physics, and biology.
Such topics as the nature and use of constants and variables, predicates, sentential connectives, truth-tables, universal and existential sentences, definitions, identity, isomorphism, syntactical and semantical systems and the relations between them, the system of types, varieties of relations, linear order, special operators, structures and cardinal numbers, descriptions, finite and infinite concepts, continuity, thing languages, coordinate languages, axiom systems for set theory, arithmetic, geometry, space-time topology, biological concepts, and many other subjects, are covered in detail. The logic of relations is given a particularly extensive treatment. Hundreds of problems, examples, and exercises are included to give students practice in the techniques of symbolic logic and their usage.

PART ONE
SYSTEM OF SYMBOLIC LOGIC

Chapter A

The simple language A

1. THE PROBLEM OF SYMBOLIC LOGIC

1a. The purpose of symbolic language. Symbolic logic (also called mathematical logic or logistic) is the modern form of logic developed in the last hundred years. This book presents a system of symbolic logic, together with illustrations of its use. Such a system is not a theory (i.e. a system of assertions about objects), but a language (i.e. a system of signs and of rules for their use). We will so construct this symbolic language that into it can be translated the sentences of any given theory about any objects whatever, provided only that some signs of the language have received determinate interpretations such that the signs serve to designate the basic concepts of the theory in question. So long as we remain in the domain of pure logic (i.e. so long as we are concerned with building this language, and not with its application and interpretation respecting a given theory), the signs of our language remain uninterpreted. Strictly speaking, what we construct is not a language but a schema or skeleton of a language: out of this schema we can produce at need a proper language (conceived as an instrument of communication) by interpretation of certain signs.

Part Two of this book sees a variety of such interpretations, and the symbolic formulation (axiomatically, for the most part) of theories from various domains of science. All this is applied logic. Part One of the book attends to pure logic: here we describe the structure of the symbolic language by specifying its rules. In the present Chapter A, the first of the three chapters comprising Part One, we describe a simple symbolic language A containing the following sorts of signs (to be explained later): sentential constants and variables, individual constants and variables, predicate constants and variables of various levels and types, functor constants and variables, sentential connectives, and quantifiers. The third chapter, Chapter C, presents a more comprehensive language C. In Chapter B a symbolic language B is represented both as a syntactical system and as a semantical system.

If certain scientific elements—concepts, theories, assertions, derivations, and the like—are to be analyzed logically, often the best procedure is to translate them into the symbolic language. In this language, in contrast to ordinary word-language, we have signs that are unambiguous and formulations that are exact: in this language, therefore, the purity and correctness of a derivation can be tested with greater ease and accuracy. A derivation is counted as pure when it utilizes no other presuppositions than those specifically enumerated. A derivation in a word-language often involves presuppositions which were not made explicitly, but which entered unnoticed. Numerous examples of this are afforded by the history of geometry, especially in connection with attempts to derive Euclid’s axiom of parallels from his other axioms.

A further advant...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Preface to the English Edition
• Preface to the German Edition
• Contents
• Part One: System of symbolic logic
• Part Two: Application of symbolic logic
• Index
• Symbols of the symbolic language and of the metalanguage