Logical Options

John L. Bell and David DeVidi are members of the Departments of Philosophy at the University of Western Ontario and the University of Waterloo, respectively. The late Graham Solomon was a member of the Department of Philosophy at Wilfrid Laurier University.

Preface

1 Classical Propositional Logic



Introductory Remarks


Some Basic Concepts
Formal Logics


Propositional Logic


Preliminaries
Truth Values, Valuations, and validity
Implication, Tautology, and Other Important Concepts
Equivalence and Expressive Completeness
Arithmetical Representation of Statements and Logical Operations


Trees for Classical Propositional Logic


Tree Rules for Classical Propositional Logic
Trees as a Test for Validity
Further Applications of the Tree Method


Metatheorems
Other Proof Methods


Classical Propositional Calculus
Natural Deduction
Sequent Calculus



2 Classical Predicate Logic



Introductory Remarks
Tree Rules for Classical Predicate Logic


Rules for Quantifiers
Identity
Functions


Predicate Languages and Their Interpretations


The Languages
Interpretations: Preliminary Remarks


Set Theory


Sets
Relations
Equivalence Relations
Orderings
Functions


Interpretations of Languages for Predicate Logic
Validity, Satisfiability, and Models
Correctness and Adequacy


Some Difficulties
Dealing with Difficulties
The Proofs



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



Postulate Systems


Postulate Systems for Arithmetic
Noncategoricity of First-Order Peano Arithmetic


Many Sorted Logic


Introductory Remarks
Many Sorted Languages and Interpretations
Reducing Many Sorted to Unsorted Logic


Second-Order Logic


Languages and Interpretations
Second-Order Trees
The Strength of Second-Order Logic
Metatheory of Second-Order Logic



4 Introducing Contextual Operators: Modal Logics



The Propositional Modal Language and Models


The Language
Interpretations
Classes of Frames and Different Logics


Trees for Contextual Logics


Proving Correctness for ▪
Counterexamples
Proving Adequacy for ▪


Other Systems of Contextual (Modal) Logic


Correctness and Adequacy for Trees


Provability Logic


Arithmetic Provability and Contextual Logic
Frames and Provability Logic
Trees for Provability Logic


Multi-Modal Logic
Quantificational Contextual Logic


The Languages
Introduction
Semantics for Contextual Predicate Logic
Concluding Remarks



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



Three-Valued Logics


Trees for Three-Valued Logic


Intuitionistic Logic


Introduction – Constructivism
A More General Account
Semantics and Countermodels
Metatheorems for Intuitionistic Propositional Logic
Comparing Intutionistic Logic to Other Logics


Intuitionistic Predicate Logic


Interpretations
Trees for Intuitionistic Predicate Logic
Intuitionistic Identity



6 A Sampling of Other Logics



Fuzzy Logic
Algebraic Logic
Term Forming Operators and Free Logics


Term Forming Operators
Free Logics



7 Solutions to *-ed Exercises



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

Index