Preface

1 Classical Propositional Logic

1. Introductory Remarks
1. Some Basic Concepts
2. Formal Logics
2. Propositional Logic
1. Preliminaries
2. Truth Values, Valuations, and validity
3. Implication, Tautology, and Other Important Concepts
4. Equivalence and Expressive Completeness
5. Arithmetical Representation of Statements and Logical Operations
3. Trees for Classical Propositional Logic
1. Tree Rules for Classical Propositional Logic
2. Trees as a Test for Validity
3. Further Applications of the Tree Method
4. Metatheorems
5. Other Proof Methods
1. Classical Propositional Calculus
2. Natural Deduction
3. Sequent Calculus

2 Classical Predicate Logic

1. Introductory Remarks
2. Tree Rules for Classical Predicate Logic
1. Rules for Quantifiers
2. Identity
3. Functions
3. Predicate Languages and Their Interpretations
1. The Languages
2. Interpretations: Preliminary Remarks
4. Set Theory
1. Sets
2. Relations
3. Equivalence Relations
4. Orderings
5. Functions
5. Interpretations of Languages for Predicate Logic
6. Validity, Satisfiability, and Models
1. Some Difficulties
2. Dealing with Difficulties
3. The Proofs

3 Using and Extending
Predicate Logic: Postulates, Sorts and Second-Order Logic

1. Postulate Systems
1. Postulate Systems for Arithmetic
2. Noncategoricity of First-Order Peano Arithmetic
2. Many Sorted Logic
1. Introductory Remarks
2. Many Sorted Languages and Interpretations
3. Reducing Many Sorted to Unsorted Logic
3. Second-Order Logic
1. Languages and Interpretations
2. Second-Order Trees
3. The Strength of Second-Order Logic
4. Metatheory of Second-Order Logic

4 Introducing Contextual Operators: Modal Logics

1. The Propositional Modal Language and Models
1. The Language
2. Interpretations
3. Classes of Frames and Different Logics
2. Trees for Contextual Logics
1. Proving Correctness for ▪
2. Counterexamples
3. Other Systems of Contextual (Modal) Logic
1. Correctness and Adequacy for Trees
4. Provability Logic
1. Arithmetic Provability and Contextual Logic
2. Frames and Provability Logic
3. Trees for Provability Logic
5. Multi-Modal Logic
6. Quantificational Contextual Logic
1. The Languages
2. Introduction
3. Semantics for Contextual Predicate Logic
4. Concluding Remarks

5 Getting Away From Bivalence: Three-Valued and Intuitionistic Logic

1. Three-Valued Logics
1. Trees for Three-Valued Logic
2. Intuitionistic Logic
1. Introduction – Constructivism
2. A More General Account
3. Semantics and Countermodels
4. Metatheorems for Intuitionistic Propositional Logic
5. Comparing Intutionistic Logic to Other Logics
3. Intuitionistic Predicate Logic
1. Interpretations
2. Trees for Intuitionistic Predicate Logic
3. Intuitionistic Identity

6 A Sampling of Other Logics

1. Fuzzy Logic
2. Algebraic Logic
3. Term Forming Operators and Free Logics
1. Term Forming Operators
2. Free Logics

7 Solutions to *-ed Exercises

1. Solutions for Chapter 1
2. Solutions for Chapter 2
3. Solutions for Chapter 3
4. Solutions for Chapter 4
5. Solutions for Chapter 5
6. Solutions for Chapter 6

Index

Posted on October 29, 2015