Symbolic Logic: Syntax, Semantics, and Proof

✍ Scribed by David Agler

• Publisher: Rowman & Littlefield Publishers
• Year: 2012
• Tongue: English
• Leaves: 397
• Edition: 1
• Category: Library

✦ Synopsis

Brimming with visual examples of concepts, derivation rules, and proof strategies, this introductory text is ideal for students with no previous experience in logic. Symbolic Logic: Syntax, Semantics, and Proof introduces students to the fundamental concepts, techniques, and topics involved in deductive reasoning. Agler guides students through the basics of symbolic logic by explaining the essentials of two classical systems, propositional and predicate logic. Students will learn translation both from formal language into English and from English into formal language; how to use truth trees and truth tables to test propositions for logical properties; and how to construct and strategically use derivation rules in proofs. This text makes this often confounding topic much more accessible with step-by-step example proofs, chapter glossaries of key terms, hundreds of homework problems and solutions for practice, and suggested further readings.

✦ Table of Contents

Cover Page
Title Page
Copyright Page
Dedication Page
Brief Table of Contents
Extended Table of Contents
Acknowledgements
Introduction
I.1 What Is Symbolic Logic?
I.2 Why Study Logic?
I.3 How Do I Study Logic?
I.4 How Is the Book Structured?
1 Propositions, Arguments, and Logical Properties
1.1 Propositions
1.2 Arguments
1.2.1 What Is an Argument?
1.2.2 Identifying Arguments
1.2.3 Argument Indicators
1.2.4 Types of Arguments
Exercise Set #1
Solutions to Starred Exercises from Exercise Set #1
1.3 Deductively Valid Arguments
1.3.1 Deductive Validity Defined
1.3.2 Testing for Deductive Validity
1.4 Summary
End-of-Chapter Exercises
Solutions to Starred Exercises from End-of-Chapter Exercises
Definitions
2 Language, Syntax, and Semantics
2.1 Truth Functions
2.2 The Symbols of PL and Truth-Functional Operators
2.2.1 Conjunction
2.2.2 Negation
2.2.3 Scope Indicators and the Main Operator
2.3 Syntax of PL
2.3.1 Metalanguage, Object Language, Use, and Mention
2.3.2 Metavariables, Literal Negation, and the Main Operator
2.3.3 The Language and Syntax of PL
2.3.4 Literal Negation and the Main Operator
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
2.4 Disjunction, Conditional, Biconditional
2.4.1 Disjunction
2.4.2 Material Conditional
2.4.3 Material Biconditional
2.5 Advanced Translation
End-of-Chapter Exercises
Solutions to Starred Exercises in End-of-Chapter Exercises
Definitions
Guide to Translation from English to PL
3 Truth Tables
3.1 Valuations (Truth-Value Assignments)
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
3.2 Truth Tables for Propositions
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
3.3 Truth Table Analysis of Propositions
3.3.1 Tautology
3.3.2 Contradiction
3.3.3 Contingency
Exercise Set #3
Solutions to Starred Exercises in Exercise Set #3
3.4 Truth Table Analysis of Sets of Propositions
3.4.1 Equivalence
3.4.2 Consistency
3.5 The Material Conditional Explained (Optional)
3.6 Truth Table Analysis of Arguments
3.6.1 Validity
3.6.2 Validity and Inconsistency
3.7 Short Truth Table Test for Invalidity
End-of-Chapter Exercises
Solutions to Starred Exercises in End-of-Chapter Exercises
Definitions
4 Truth Trees
4.1 Truth-Tree Setup and Basics in Decomposition
4.1.1 Truth-Tree Setup
4.1.2 Truth-Tree Decomposition
Exercise Set #1
4.2 Truth-Tree Decomposition Rules
4.2.1 Conjunction Decomposition Rule (∧D)
4.2.2 Disjunction Decomposition Rule (∨D)
4.2.3 Decompose Propositions under Every Descending Open Branch
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
4.3 The Remaining Decomposition Rules
4.3.1 Conditional Decomposition (→D)
4.3.2 Biconditional Decomposition (↔D)
4.3.3 Negated Conjunction Decomposition (¬∧D)
4.3.4 Negated Disjunction Decomposition (¬∨D)
4.3.5 Negated Conditional Decomposition (¬→D)
4.3.6 Negated Biconditional Decomposition (¬↔D)
4.3.7 Double Negation Decomposition (¬¬D)
4.3.8 Truth-Tree Rules
Exercise Set #3
Solutions to Starred Exercises in Exercise Set #3
4.4 Basic Strategies
4.4.1 Strategic Rule 1
4.4.2 Strategic Rule 2
4.4.3 Strategic Rule 3
4.4.4 Strategic Rule 4
Exercise Set #4
Solutions to Starred Exercises in Exercise Set #4
4.5 Truth-Tree Walk Through
4.6 Logical Properties of Truth Trees
4.6.1 Semantic Analysis of Truth Trees
Exercise Set #5
Solutions to Starred Exercises in Exercise Set #5
4.6.2 Consistency
4.6.3 Tautology, Contradiction, and Contingency
4.6.4 Logical Equivalence
4.6.5 Validity
End-of-Chapter Exercises
Solutions to Starred Exercises in End-of-Chapter Exercises
Definitions
Truth-Tree Vocabulary
Key for Truth-Tree Analysis
Decomposable Propositions
Truth-Tree Decomposition Rules
5 Propositional Logic Derivations
5.1 Proof Construction
5.2 Premises and the Goal Proposition
5.3 Intelim Derivation Rules
5.3.1 Conjunction Introduction (∧I)
5.3.2 Conjunction Elimination (∧E)
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
5.3.3 Assumptions (A) and Subproofs
5.3.4 Conditional Introduction (→I)
5.3.5 Conditional Elimination (→E)
5.3.6 Reiteration (R)
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
5.3.7 Negation Introduction (¬I) and Elimination (¬E)
Exercise Set #3
Solutions to Starred Exercises in Exercise Set #3
5.3.8 Disjunction Introduction (∨I)
5.3.9 Disjunction Elimination (∨E)
5.3.10 Biconditional Elimination and Introduction (↔E and ↔I)
Exercise Set #4
Solutions to Starred Exercises in Exercise Set #4
5.4 Strategies for Proofs
5.4.1 Strategies Involving Premises
Exercise Set #5
Solutions to Starred Exercises in Exercise Set #5
5.4.2 Strategies Involving Assumptions
Exercise Set #6
Solutions to Starred Exercises in Exercise Set #6
5.5 Additional Derivation Rules (PD+)
5.5.1 Disjunctive Syllogism (DS)
5.5.2 Modus Tollens (MT)
5.5.3 Hypothetical Syllogism (HS)
5.5.4 Double Negation (DN)
5.5.5 De Morgan’s Laws (DeM)
5.5.6 Implication (IMP)
5.6 Additional Derivation Strategies
5.6.1 Sample Problem
End-of-Chapter Exercises
Easy Proofs
Solutions to Starred Easy Proofs
Medium Proofs
Solutions to Starred Medium Proofs
Hard Proofs
Solutions to Starred Hard Proofs
Zero-Premise Deductions
Solutions to Starred Zero-Premise Deductions
Conceptual and Application Exercises
List of Strategic Rules
Step-by-Step Use of Strategic Rules for Assumptions
List of Derivation Rules for PD+
Definition
6 Predicate Language, Syntax, and Semantics
6.1 The Expressive Power of Predicate Logic
6.2 The Language of RL
6.2.1 Individual Constants (Names) and n-Place Predicates
6.2.2 Domain of Discourse, Individual Variables, and Quantifiers
6.2.3 Parentheses and Scope of Quantifiers
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
6.3 The Syntax of RL
6.3.1 Free and Bound Variables
6.3.2 Main Operator in Predicate Wffs
6.3.3 The Formal Syntax of RL: Formation Rules
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
6.4 Predicate Semantics
6.4.1 A Little Set Theory
6.4.2 Predicate Semantics
Exercise Set #3
Solutions to Starred Exercises in Exercise Set #3
6.5 Translation from English to Predicate Logic
6.5.1 Translation Keys
6.5.2 The Universal Quantifier (∀)
6.5.3 The Existential Quantifier (∃)
6.5.4 Translation Walk Through
6.5.5 Sample Translations
Exercise Set #4
Solutions to Starred Exercises in Exercise Set #4
6.6 Mixed and Overlapping Quantifiers
Exercise Set #5
Solutions to Starred Exercises in Exercise Set #5
Definitions
7 Predicate Logic Trees
7.1 Four New Decomposition Rules
7.2 Strategies for Decomposing Trees
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
7.3 Logical Properties
7.3.1 Semantic Analysis of Predicate Truth Trees
7.3.2 Consistency and Inconsistency
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
7.3.3 Tautology, Contradiction, and Contingency
Exercise Set #3
Solutions to Starred Exercises in Exercise Set #3
7.3.4 Logical Equivalence
Exercise Set #4
Solutions to Starred Exercises in Exercise Set #4
7.3.5 Validity
Exercise Set #5
Solutions to Starred Exercises in Exercise Set #5
7.4 Undecidability and the Limits of the Predicate Tree Method
End-of-Chapter Exercises
Solutions to Starred Exercises in End-of-Chapter Exercises
Definitions and Tree Rules
8 Predicate Logic Derivations
8.1 Four Quantifier Rules
8.1.1 Universal Elimination (∀E)
8.1.2 Existential Introduction (∃I)
Exercise Set #1
Solutions to Starred Exercises in Exercise Set #1
8.1.3 Universal Introduction (∀I)
8.1.4 Existential Elimination (∃E)
Exercise Set #2
Solutions to Starred Exercises in Exercise Set #2
8.2 Quantifier Negation (QN)
8.3 Sample Proofs
End-of-Chapter Exercises
Solutions to End-of-Chapter Exercises
Summary of Derivation and Strategic Rules
Appendix
Further Reading
Index
About the Author