Elementary Symbolic Logic: Concepts, Techniques, and Concepts

Preface
Motivation for Elementary Symbolic Logic
Remarks on Style and Approach
Advice for Instructors
Advice for Students
Acknowledgments

Part I: Basics of Logic
Chapter 1: Introduction
1.1 Arguments and Truth
1.2 Logic and Form
1.3 Logic and Symbolic Logic
1.4 The Logical Core of Language

Chapter 2: Basic Concepts of Logic
2.1 Arguments
2.2 Deductive and Inductive Criteria
2.2.1 Two Ways to Evaluate an Argument
2.2.2 Deductive Criteria: Validity and Invalidity
2.2.3 Validity and Form
2.2.4 Degenerate Validity
2.2.5 Inductive Criteria: Strong and Weak

Part II: Sentential Logic
Chapter 3: The Operators of Sentential Logic
3.1 Remarks on Symbolic Logic and Logical Languages
3.2 Simple Sentences, Compound Sentences, and Sentential Operators
3.3 Truth-Functional and Non-Truth-Functional Operators
3.4 The Caret, Tilde, and Wedge: Syntax, Semantics, and Translation
3.4.1 The Caret
3.4.2 The Tilde
3.4.3 The Wedge
3.5 The Official Grammar of SL
3.5.1 Formation Rules for SL
3.5.2 Syntax Proofs and Syntax Trees for SL
3.5.3 Scope and Main Operator
3.6 The Arrow and Double Arrow: Syntax, Semantics, and Translation
3.6.1 The Arrow
3.6.2 The Double Arrow
3.7 Alternative Operator Sets

Chapter 4: Truth Tables and Semantic Properties in Sentential Logic
4.1 Calculating Truth Values
4.2 Constructing Truth Tables
4.3 Semantic Properties of Individual Sentences in SL
4.3.1 Logical Truths
4.3.2 Contradictions
4.3.3 Contingent Sentences
4.4 Some Simple Metalogic
4.5 Semantic Properties of Multiple Sentences in SL
4.5.1 Equivalence
4.5.2 Contradictory Pairs
4.5.3 Consistency
4.6 Validity in SL
4.7 More Simple Metalogic
4.8 Validity and the Semantics for the Arrow

Chapter 5: Natural Deduction in Sentential Logic
5.1 Proofs and Proof Construction
5.2 Rules for the Caret
5.2.1 ∧I Rule
5.2.2 ∧E Rule
5.2.3 Motivation for the Rules
5.3 Rules for the Arrow
5.3.1 →E Rule
5.3.2 →I Rule
5.3.3 R Rule
5.4 Rules for the Wedge
5.4.1 ∨I Rule
5.4.2 ∨E Rule
5.5 Rules for the Double Arrow
5.5.1 ↔E Rule
5.5.2 ↔I Rule
5.6 Rules for the Tilde
5.6.1 ~E and ~I Rules
5.7 Theorems and Zero-Premise Proofs in SL
5.8 Soundness and Completeness of SL
5.9 Additional Rules: MT, DS, HS, DN, DEM
5.10 Alternative Proof Systems

Part III: Predicate Logic
Chapter 6: Translation with Names, Predicates, and Quantifiers
6.1 Motivation for Predicate Logic
6.2 Basics of the PL Vocabulary
6.3 Symbolization with Names and Predicates
6.4 Basic Symbolization with Quantifiers
6.4.1 Basic Symbolization with “∀” and “∃”
6.4.2 Bound and Free Variables, Open and Closed Formulas
6.4.3 Quantifiers and Negations
6.4.4 Quantifiers and Multi-Place Predicates
6.5 Categorical Sentences
6.5.1 All Fs Are Gs
6.5.2 No Fs Are Gs
6.5.3 Some Fs Are/Are Not Gs
6.5.4 Categorical Sentences with Multi-Place Predicates
6.6 Overlapping Quantifiers
6.7 Identity and Numerical Sentences

Chapter 7: Syntax and Semantics for Predicate Logic
7.1 The Official Grammar of PL
7.1.1 Formation Rules for PL
7.1.2 Syntax Proofs and Syntax Trees for PL
7.2 PL Models
7.3 Semantics for PL
7.4 Constructing PL Models
7.5 Semantic Properties of Individual Sentences in PL
7.5.1 Logical Truths
7.5.2 Contradictions
7.5.3 Contingent Sentences
7.6 Semantic Properties of Multiple Sentences in PL
7.6.1 Equivalence
7.6.2 Contradictory Pairs
7.6.3 Consistency
7.7 Validity in PL

Chapter 8: Natural Deduction in Predicate Logic
8.1 Using the SL Rules in PL Proofs
8.2 ∃I Rule
8.3 ∀E Rule
8.4 ∀I Rule
8.5 ∃E Rule
8.6 Quantifier Negation Rules
8.7 Theorems and Zero-Premise Proofs in PL
8.8 Rules for “=”
8.9 Soundness and Completeness of PL

Part IV: Beyond Elementary Symbolic Logic
Chapter 9: Beyond Elementary Symbolic Logic
9.1 Logicism and Russell’s Paradox
9.2 Trivalent Logic
9.3 Modal Logic

Answers to Select Problems
Works Referenced
Index