PREFACE
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INTRODUCTION |
PART I-PRINCIPLES OF INFERENCE AND DEFINITION |
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1. THE SENTENTIAL CONNECTIVES |
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1.1 Negation and Conjunction |
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1.2 Disjunction |
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1.3 Implication: Conditional Sentences |
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1.4 Equivalence: Biconditional Sentences |
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1.5 Grouping and Parentheses |
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1.6 Truth Tables and Tautologies |
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1.7 Tautological Implication and Equivalence |
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2. SENTENTIAL THEORY OF INFERENCE |
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2.1 Two Major Criteria of Inference and Sentential Interpretations |
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2.2 The Three Sentential Rules of Derivation |
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2.3 Some Useful Tautological Implications |
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2.4 Consistency of Premises and Indirect Proofs |
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3. SYMBOLIZING EVERYDAY LANGUAGE |
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3.1 Grammar and Logic |
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3.2 Terms |
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3.3 Predicates |
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3.4 Quantifiers |
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3.5 Bound and Free Variables |
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3.6 A Final Example |
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4. GENERAL THEORY OF INFERENCE |
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4.1 Inference Involving Only Universal Quantifiers |
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4.2 Interpretations and Validity |
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4.3 Restricted Inferences with Existential Quantifiers |
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4.4 Interchange of Quantifiers |
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4.5 General Inferences |
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4.6 Summary of Rules of Inference |
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5. FURTHER RULES OF INFERENCE |
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5.1 Logic of Identity |
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5.2 Theorems of Logic |
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5.3 Derived Rules of Inference |
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6. POSTSCRIPT ON USE AND MENTION |
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6.1 Names and Things Named |
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6.2 Problems of Sentential Variables |
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6.3 Juxtaposition of Names |
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7. TRANSITION FROM FORMAL TO INFORMAL PROOFS |
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7.1 General Considerations |
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7.2 Basic Number Axioms |
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7.3 Comparative Examples of Formal Derivations and Informal Proofs |
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7.4 Examples of Fallacious Informal Proofs |
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7.5 Further Examples of Informal Proofs |
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8. THEORY OF DEFINITION |
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8.1 Traditional Ideas |
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8.2 Criteria for Proper Definitions |
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8.3 Rules for Proper Definitions |
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8.4 Definitions Which are Identities |
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8.5 The Problem of Divison by Zero |
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8.6 Conditional Definitions |
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8.7 Five Approaches to Division by Zero |
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8.8 Padoa's Principle and Independence of Primitive Symbols |
PART II-ELEMENTARY INTUITIVE SET THEORY |
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9. SETS |
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9.1 Introduction |
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9.2 Membership |
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9.3 Inclusion |
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9.4 The Empty Set |
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9.5 Operations on Sets |
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9.6 Domains of Individuals |
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9.7 Translating Everyday Language |
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9.8 Venn Diagrams |
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9.9 Elementary Principles About Operations on Sets |
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10. RELATIONS |
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10.1 Ordered Couples |
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10.2 Definition of Relations |
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10.3 Properties of Binary Relations |
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10.4 Equivalence Relations |
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10.5 Ordering Relations |
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10.6 Operations on Relations |
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11. FUNCTIONS |
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11.1 Definition |
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11.2 Operations on Functions |
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11.3 Church's Lambda Notation |
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12. SET-THEORETICAL FOUNDATIONS OF THE AXIOMATIC METHOD |
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12.1 Introduction |
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12.2 Set-Theoretical Predicates and Axiomatizations of Theories |
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12.3 Ismorphism of Models for a Theory |
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12.4 Example: Profitability |
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12.5 Example: Mechanics |
INDEX |
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