Understanding Symbolic Logic

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• ISBN10:

0130201421

• Edition: 4th
• Format: Hardcover
• Publisher: Prentice Hall
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Summary

For one-semester/two-quarter/full-year, lower-/upper-level undergraduate courses in Formal Logic and Deductive Logic. This is a comprehensive Each unit is divided into easily comprehended small bites. The book provides extremely detailed explanations of procedures and techniques.

Preface xiii
PART ONE Sentential Logic
 Introduction to Logic
1(17)
 Why Study Logic?
2(2)
 What Logic Is All About
4(1)
 Induction and Deduction
5(2)
 Form and Validity
7(3)
 Truth and Validity
10(2)
 The Nature of Symbolic Logic
12(1)
 The Scope of Symbolic Logic
13(5)
 Definitions
15(1)
 Study Questions
16(1)
 Exercises
16(2)
 The Structure of Sentential Logic
18(12)
 Simple and Compound Sentences
19(4)
 Sentential Operators
23(2)
 The Structure and Symbolism of Sentential Logic
25(5)
 Definitions
27(1)
 Study Questions
28(1)
 Exercises
28(2)
 Computing Truth Values
30(17)
 Truth Tables for the Operators
31(8)
 Computing Truth Values
39(2)
 Truth-functional Operators
41(1)
 Non-truth-functional Operators
42(5)
 Definitions
44(1)
 Study Questions
44(1)
 Exercises
45(2)
 Symbolizing English Sentences
47(19)
 Simple Sentences
48(2)
 Truth-functional and Non-truth-functional Compounds
50(1)
 Symbolizing English Operators
51(8)
 Symbolizing Multiply Complex Sentences
59(7)
 Exercises
63(3)
 Truth Tables for Testing Validity
66(20)
 Constructing Base Columns for Truth Tables
67(4)
 The Truth Table Test for Validity
71(4)
 Shortcut Validity Tests
75(6)
 Mechanical Decision Procedures
81(5)
 Definitions
82(1)
 Study Questions
82(1)
 Exercises
83(3)
 Further Applications of the Truth Table Method
86(16)
 Tautologies, Contradictions, and Contingencies
87(3)
 Logical Implication and Logical Equivalence
90(3)
 Consistency
93(1)
 Statements and Statement Forms; Applying Truth Table Concepts
94(2)
 Four Kinds of Truth Table Problems and the Relations Between Them
96(6)
 Definitions
97(1)
 Study Questions
98(1)
 Exercises
98(4)
 The Proof Method: Eight Basic Inference Rules
102(34)
 Form and Substitution Instance
104(3)
 The Proof Process
107(2)
 Eight Basic Inference Rules
109(8)
 Derivations and Proofs
117(2)
 Constructing Simple Proofs
119(5)
 Constructing More Complex Proofs
124(12)
 Summary of Rules of Inference
127(1)
 Definitions
128(1)
 Exercises
128(8)
 Replacement Rules
136(27)
 The Structure of Replacement Rules
137(1)
 The Ten Replacement Rules
138(9)
 Constructing Simple Proofs with Replacement Rules
147(3)
 Strategies for More Complex Proofs
150(13)
 Summary of Replacement Rules
156(1)
 Exercises
156(7)
 Conditional Proof and Indirect Proof
163(24)
 Conditional Proof
164(4)
 Indirect Proof
168(3)
 Discharging Assumptions; Restrictions on C.P. and I.P.
171(2)
 Using C.P. and I.P.
173(3)
 Proofs of Theorems
176(3)
 Invalidity
179(1)
 Truth and Proof
180(7)
 Summary of Rules of Conditional Proof and Indirect Proof
181(1)
 Definitions
182(1)
 Exercises
183(4)
PART TWO Monadic Predicate Logic
 Singular Sentences
187(10)
 Singular Sentences and Propositional Functions
189(3)
 Symbolizing Singular Sentences
192(5)
 Definitions
195(1)
 Exercises
195(2)
 Quantifiers
197(13)
 Universal and Existential Quantifiers
198(5)
 Free and Bound Variables; Scope of a Quantifier
203(1)
 Negated Quantifiers
203(7)
 Definitions
207(1)
 Exercises
208(2)
 Categorical Propositions
210(24)
 The Four Categorical Propositions
211(3)
 Individuals, Sets, and Properties
214(1)
 Venn Diagrams
215(4)
 Symbolizing Categorical Propositions
219(2)
 Negated Categorical Propositions
221(2)
 Deriving C.Q.N. Rules from Q.N. Rules
223(1)
 Symbolizing English Categorical Sentences
223(11)
 Symmary of Categorical Propositions
229(1)
 Definitions
230(1)
 Exercises
230(4)
 Complex Subjects and Predicates
234(13)
 Complex Subjects and Predicates
235(4)
 Equivalent Symbolizations
239(8)
 Exercises
244(3)
 Quantifier Form and Truth-Functional Compounds of Quantifier Statements
247(8)
 Quantifier Form
248(1)
 Truth-functional Compounds and Quantifier Form
249(2)
 Symbolizing Truth-functional Compounds
251(4)
 Definitions
252(1)
 Exercises
252(3)
 Proofs in Predicate Logic
255(25)
 Preliminary Statement of the Four Quantifier Rules
256(2)
 Instances of Quantified Formulas
258(1)
 The Rules of Universal Instantiation (U.I.) and Existential Generalization (E.G.)
259(1)
 The Rules of Existential Instantiation (E.I.) and Universal Generalization (U.G.); Flagging Restrictions
260(7)
 Constructing Proofs for ``Pure'' Quantifier Arguments
267(5)
 Constructing Proofs for Arguments Containing Truth-functional Compounds
272(2)
 Constructing Proofs of Quantifier Theorems
274(6)
 Statement of the Quantifier Rules, with All Necessary Restrictions
276(1)
 Exercises
277(3)
 Invalidity in Quantifier Logic
280(15)
 The Natural Interpretation Method
281(3)
 Truth Conditions for Quantifier Statements
284(1)
 The Model Universe Method
285(10)
 Definitions
292(1)
 Exercises
292(3)
PART THREE Relational Predicate Logic
 Symbolization in Relational Predicate Logic
295(28)
 Relational Predicates and Singular Sentences
296(3)
 Multiple Quantifiers
299(7)
 Quantifier Negation
306(3)
 Categorical Relational Statements; Complex Subjects and Predicates
309(4)
 Symbolizing English Sentences
313(10)
 Exercises
317(6)
 Proofs and Invalidity for Relational Predicate Logic
323(14)
 Proofs in Relational Predicate Logic
324(7)
 Invalidity in Relational Predicate Logic
331(6)
 Exercises
335(2)
 Identity and Definite Descriptions
337(16)
 Identity Statements and Their Negations
338(1)
 Exceptives and ``Only'' Statements
339(3)
 Superlatives
342(1)
 Numerical Statements
343(4)
 Definite Descriptions
347(6)
 Exercises
348(5)
 Proofs Involving Identity
353(12)
 Rules for Identity
353(4)
 Proofs Containing Identity Statements
357(8)
 Summary of Identity Rules
361(1)
 Exercises
361(4)
PART FOUR Extra Credit Units
 Well-Formed Formulas for Sentential Logic
365(4)
 Exercises
367(2)
 Polish Notation for Sentential Logic
369(4)
 Exercises
371(2)
 Proof Trees for Sentential Logic
373(6)
 Exercises
378(1)
 Using Venn Diagrams to Prove Validity
379(6)
 Exercises
383(2)
 Stroke (nand) and Dagger (nor) Operators
385(4)
 Exercises
388(1)
 Proof Trees for Predicate Logic
389(12)
 Exercises
399(2)
Answers to Starred Exercises 401(42)
Index 443

Excerpts

This book is intended as a comprehensive introduction to symbolic logic. It presupposes no prior acquaintance with either logic or mathematics, and it includes all the standard topics through relational predicate logic with identity. The book was written in the conviction that any student can master symbolic logic, and it is designed to give the student as much help as possible in attaining that mastery. The main part of the book is divided into twenty units, each of which has an introduction and a statement of study objectives so that the student has an overview of what is to come and knows exactly what is required in order to master the unit. The explanatory material for each unit is divided into several subsections, each of which has a specific function and covers one relatively small, clearly defined topic. The clear separation of topics and the division into easily comprehended small "bites" allow the student to master the material step by step without being overwhelmed by an indigestible mass of information. One-variable predicate logic is developed, in detail, independently of relational predicate logic, and identity is presented in two separate units. The semantics of predicate logic is also developed in a separate unit, as is the semantics for sentential logic. In addition to the basic material, there are several "extra credit" units, which provide a glimpse into alternative methods of logic and more advanced topics. I have tried to give as detailed explanations as possible, both for specific techniques, such as drawing up truth tables or constructing proofs, and for the rationale behind these techniques. It seems to me as important for a student to understandwhythings are done in a certain way as to learn the techniques themselves, and in this book I have tried to supply the "why's" as well as the "how's." The book does, however, supply the "how's" in abundance. Aside from the detailed explanations, there are numerous examples worked out in the text: various types of truth tables, a great many detailed, step-by-step symbolizations, and over fifty fully worked out proofs. In addition, there are copious exercises, with answers to fully half of these provided at the back of the book. Problems for which answers are given are indicated by stars. Because of the detailed explanations, the extensive coverage, and the clear division of topics, the book is extremely flexible. It can be used in either freshman courses or upper-division courses and is suitable for quarter, semester, or even two-quarter courses. In one quarter, for instance, one might cover just Units 1 through 14; in a semester course, Units 1 through 15, 17, and 18; and in a two-quarter course one might cover the entire book, including the supplementary units. Because of the step-by-step approach and the numerous examples and exercises, the book can also be used in self-paced classes. Suggestions on how to structure such a course are included in the Instructor's Manual. A new edition has given me the opportunity to make numerous changes that should clarify and streamline the presentation. In addition to updating examples and exercises, I have provided new or expanded explanations for many topics that students might find puzzling and have made scores of relatively minor changes that significantly clarify the material. The most substantial changes are in sections covering logical form and the distinction between form and substitution instance. I have also expanded and clarified various examples and problems in the text and have added new exercises in Unit 19. It is a great pleasure to acknowledge at this point my considerable debt to the many people who helped make this book what it is. My greatest debt, both in general and in particular, is to Nuel D. Belnap, Jr., from whom I absorbed most of what I know about logic and much of my interest in pedagogy. In addition to these general contributions, the rule system