Understanding Symbolic Logic

by Klenk, Virginia

• ISBN13: 9780130201423
• ISBN10: 0130201421
• Edition: 4th
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Prentice Hall

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.

Table of Contents

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