PHILOSOPHY, MATHEMATICS AND FORMAL LOGICS

Philosophy, Mathematics
and Formal Logics

Craig Harrison

Department of Philosophy
San Francisco State University

and

William M. Pezzaglia Jr.
Department of Physics
Santa Clara University

May 15, 2004

ii

Acknowledgements

Draft prepared in LATEX format.

iii

iv ACKNOWLEDGEMENTS

About the Authors

Craig Harrison

born 1933, London, England

B.A. History (1959), Ph.D. Philosophy (1967), Stanford University

Associate Professor of Philosophy

San Francisco State University
1600 Holloway Avenue
San Francisco, CA 94132

William Marvyn Pezzaglia Jr.

born 1953, Sacramento, California
B.S. Physics (1975), Ph.D. (1983) Physics, University of California, Davis

Adjunct Professor of Physics

Santa Clara University
500 El Camino Real
Santa Clara, CA 95053

v

vi ABOUT THE AUTHORS

Preface

What we Hope to Accomplish in this Work

An introductory course in formal or symbolic logic is a subject widely required
for philosophy majors at both the graduate and undergraduate levels, and is
often considered essential to the pursuit of philosophy. Why this should be so

is not entirely clear, especially to the students who take the course.

A typical text first teaches the student to “symbolize” certain artificial state-
ments. Here is a random sample quoted from a few highly regarded works which
happen to be on my bookshelf:

• Either Adam is blond and loves Eve, or he is not blond and Eve loves him.
–Teller (1989)[94].

• If Holmes has bungled or Watson’s on the job, then Moriarty will make a
mistake. –Jeffrey (1991)[38].

• Either Sam will come to the party and Max will not, or Sam will not come
to the party and Max will enjoy himself. –Mendelson (1987)[59].

• If Jones is ill or Smith is away, then neither will the Argus deal be con-
cluded nor will the directors will declare a dividend unless Robinson comes
to his senses and takes matters into his own hands. – Quine (1972)[68].

These are to be symbolized into formulas in two-valued “propositional” or
“sentence” logic, depending on whether the author believes in propositions or
not. Having done this, the student is to learn to solve problems, typically to
determine whether a given argument containing assumptions and a conclusion
consisting of statements of this sort, is valid or not, by determining whether the
resulting formal argument is valid by means of truth tables, or a formal proof
of the conclusion from the assumptions (or a combination of both). A similar
though more general method is used to solve such problems in first order logic
or “predicate logic”. Truth trees are also gradually increasing in popularity.

Having been drilled by constant repetition in these methods, one of two

outcomes in, in or experience, all too common:

1. The student satisfies the other requirements, but forgets or rarely applies
the methods learned in the introductory course.

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viii PREFACE

2. The student then continues the study of logic and finds herself unprepared
and disoriented by the material next presented, which is more or less a
formal study of meta-logic or meta-mathematics.

Now some students manage to survive the “just do it and don’t ask ques-
tions” approach, which is arguably the most efficient, and is also not uncommon
in the teaching of high school algebra. But students at the college level, partic-
ularly students of philosophy, want to know and understand what the subject
is about, what it has to do with philosophy, and what the motivations are for
the ideas being introduced, and are uncomfortable in proceeding further until
they have got some answers with which they are satisfied.

We believe that they deserve answers, and in this book, we try to provide
them. To this end, we discuss the basics in each chapter, and reserve more
extended discussion of the motivations and philosophical significance of the ideas
introduced, as well as more detailed and exact explanations of the nature of the
concepts and the abstract objects which represent them, to optional sections.
This gives the reader (or the instructor, as the case may be) some flexibility
when it comes to deciding which additional material to study.

We also try to be forthright and candid about the scope and applicability

of the material introduced, and not to pretend by carefully selected examples,
that it is wider than it is.

Some Philosophical Presuppositions Concerning
the Nature of Formal Logic

Formal systems of logic, especially two-valued propositional or Boolean logics,
have a close affinity to algebra, in particular to Boolean algebras. Experience
has shown that Boolean logics, and their extension to first and higher order
logics, and other related systems, are helpful when it comes to formalizing and
clarifying mathematical concepts and proofs. We also try, especially in the
concluding chapter, to explain how all these ideas came about, as well as their
range of application, and their limitations. Besides that, we try to show how
these ideas developed historically, and their connection with philosophy, which
since Plato have been deep and numerous.

Introduction

Logic is a branch of mathematics, and the mathematical structures that arise
from it are extensively studied, within the framework of such mathematical
disciplines as set theory, topology, category theory and lattice theory. It is also
a traditional part of the philosophy curriculum, especially at the graduate level,
and it receives some attention in mathematical studies and in computer science
as well.

It was long the province of metaphysics and of general philosophy, until as
a result of the work of George Boole, it became inseparable from mathematics.
The study of mathematics has in turn been important in Western philosophy
since its inception, at the time of Pythagoras and Plato, and has remained so
to this day.

Yet the way in which formal logic is introduced to the beginner is all to often
devoid of context and entirely unmotivated. The emphasis is exclusively on the
development of “skills”, such as symbolizing statements nobody in their right
mind would ever make, or on endless drills in performing algorithms. And this
does a disservice to the student and to the subject, and makes further progress
all the more difficult.

Philosophers do not take things on faith. When they are pursuing a subject,
they want to understand what they are doing and why they are doing it. We
believe that a basic knowledge of science and of mathematics in particular,
is important to philosophy, and to the understanding of reality. You can’t
philosophize about science without knowing what you’re talking about.

We have tried to introduce the subject of logic with candor and without
evasion, to explain the underlying motivations and the context which made
them important, and hopefully, to give some idea of the manifold possibilities
and the diversity of the subject, which like many other fields, has increased
dramatically in the last century or so.

In the first introductory chapter, we provide an account of the concepts on
which formal logic is based. In the second chapter, we introduce the simple
arithmetical operations which define what is meant in symbolic logic by ‘not’,
‘and’, and ‘or’. In the next chapter, we take a closer look at what this all
means than is customary in regular algebra, particularly when it concerns the
substitution of actual numbers for the variables in algebraic formulas.

In Chapter 4, we discuss various methods of determining whether a sentence
follows from given assumptions, and we also show that these methods really

ix

x INTRODUCTION

work. In the next chapter, we explore various alternatives to the regular two-
valued logic which was heretofore our main concern. In the next three chapters,
we extend the discussion to cover generalities and statements of existence, which
cannot be done in the logic of “not’s, and’s and or’s” alone. In the final chapter,
we provide an account of the philosophical issues which lend importance to the
most dramatic discoveries in the past century.

We may add that we take formulas to denote the Boolean truth values 0 and
1, much as in ordinary algebra, formulas denote numbers. We also define formu-
las involving quantifiers so as not to allow colliding quantifiers, an unnecessary
and avoidable subtlety. For similar reasons, we permit only closed formulas in
proofs or proof trees. Moreover, we take every element in the domain of a struc-
ture to have a name, even if there are uncountably many of them, instead of
taking free variables to have variable denotations. If this precludes a physicalist
interpretation of proper names, so be it.

We have added optional sections to supplement the core material, hopefully
to provide further insight into it, or else alternative approaches. Chapters 5 and
8 are also largely optional. This will, we hope, provide the reader (or instructor)
with more flexibility in what subjects to address.

Contents

Acknowledgements iii

About the Authors v

Preface vii

Introduction ix

I Propositional Logics 1

1 Introductory Concepts: Sets, Functions,. . . 3

1.1 Logic: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 What is Logic For? . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Naming Sets . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 The Principle of Extensionality: When are Two Sets Equal? 6

1.2.3 Unit Sets, the Empty Set and Finite and Infinite Sets . . 7

1.2.4 Subsets, and Extensionality Again . . . . . . . . . . . . . 7

1.2.5 Basic Operations on Sets . . . . . . . . . . . . . . . . . . 8

1.2.6 Ordered Pairs, Triples, Quadruples, etc. . . . . . . . . . . 12

1.2.7 What Is an Ordered Pair . . . . . . . . . . . . . . . . . . 13

1.3 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.1 On Functions . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.2 Functions with Several Inputs . . . . . . . . . . . . . . . . 20

1.4 Truth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4.1 Truth Valued Functions and Truth Functions . . . . . . . 21

2 Binary Boolean Arithmetic 25

2.1 Reasoning by Calculation from Leibniz to Boole . . . . . . . . . . 25

2.1.1 Boolean Algebra and Modern Logic: Introductory Remarks 25

2.2 Boole’s Arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.1 Symbolizing Boolean Statements and Calculating their

Truth Values . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Calculating Truth Values: Some Examples . . . . . . . . . 33

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xii CONTENTS

2.3 Material Implication, . . . Exclusive ‘Or’ . . . . . . . . . . . . . . . 36
2.3.1 Truth Functional or Boolean Implication . . . . . . . . . . 36
2.3.2 Material Equivalence and Exclusive Disjunction . . . . . . 39

2.4 About The Truth Functional Conditional . . . . . . . . . . . . . 40
2.4.1 Why the conditional of Boolean . . . . . . . . . . . . . . . 40
2.4.2 The Material Conditional in Ordinary Reasoning. . . . . . 42

2.5 Boolean Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5.1 What is a Boolean Language? . . . . . . . . . . . . . . . . 45
2.5.2 Boolean Formation Rules . . . . . . . . . . . . . . . . . . 47
2.5.3 Trees and their Uses . . . . . . . . . . . . . . . . . . . . . 48
2.5.4 The Unique Readability Theorem. . . . . . . . . . . . . . . 52

2.6 Game-Theoretical Semantics . . . . . . . . . . . . . . . . . . . . 56
2.6.1 Boolean Games . . . . . . . . . . . . . . . . . . . . . . . . 56
2.6.2 How To Win a Boolean Game. . . . . . . . . . . . . . . . . 59

2.7 How to Believe that you will Believe a Falsehood . . . . . . . . . 62
2.7.1 On the Island of Knights and Knaves . . . . . . . . . . . 62

3 Truth Functional Logic: Determining Validity and Satisfiability

by Calculation 37

3.1 Boolean Paralogisms as “Errors in Calculation” . . . . . . . . . . 37

3.1.1 Boolean Word Problems . . . . . . . . . . . . . . . . . . . 37

3.2 TRUTH FUNCTIONAL ENTAILMENT . . . . . . . . . . . . . 39

3.2.1 Boolean Formulas and Truth Functional Entailment Be-

tween Them . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 PROPOSITIONAL LANGUAGES . . . . . . . . . . . . . . . . . 41

3.3.1 What is a Boolean Language? . . . . . . . . . . . . . . . . 41

3.4 BOOLEAN SEMANTICS: TRUTH AND SATISFIABILITY . . 42

3.4.1 Truth Value Assignments and Boolean Valuations . . . . 42

3.4.2 Satisfying Formulas and Sets of Formulas . . . . . . . . . 43

3.5 The Basic Problems of Boolean Logic . . . . . . . . . . . . . . . 43

3.5.1 The Basic Problems of Boolean Logic, and How to Solve

Them by Calculating with Vectors . . . . . . . . . . . . . 43

3.5.2 Solving the Basic Problems by Truth Value Analysis . . . 43

3.6 The Syntax of Boolean Formulas II . . . . . . . . . . . . . . . . . 43

3.6.1 Subformula Trees and Valuation Trees . . . . . . . . . . . 43

3.6.2 An Algorithm for Finding the Main Connective and the

Immediate Subformulas of a Given Formula . . . . . . . . 43

3.7 Game-Theoretical Semantics for Boolean Logic: An Alternative

to Direct Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7.1 Boolean Games . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8 About the Truth Functional or Material Conditional . . . . . . . 43

3.8.1 Why the Conditional of Boolean Reasoning is the Material

Conditional . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.8.2 The Material Conditional in Ordinary Reasoning: Some

Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.9 Spatializing Logical Relations: The Geometry of Entailment . . . 43

CONTENTS xiii

3.9.1 Formal Propositions a Points in Space . . . . . . . . . . . 43

4 Boolean Reasoning 45

4.1 Methods of Boolean Logic: Boolean Equations . . . . . . . . . . 46

4.1.1 Boolean Equations . . . . . . . . . . . . . . . . . . . . . . 46

4.1.2 Finding Solutions to Sets of Boolean Equations, or Show-

ing that None Exist . . . . . . . . . . . . . . . . . . . . . 46

4.1.3 Solving the Basic Problems of Boolean Logic with Equa-

tion Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Methods of Boolean Logic: Formal Proofs . . . . . . . . . . . . . 46

4.2.1 Boolean Natural Deduction . . . . . . . . . . . . . . . . . 46

4.2.2 Summarizing the Rules . . . . . . . . . . . . . . . . . . . 46

4.3 Methods of Boolean Logic: Truth Trees . . . . . . . . . . . . . . 46

4.3.1 Truth Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Solving the Satisfiability, Validity and Tautology Prob-

lems using Truth Trees . . . . . . . . . . . . . . . . . . . . 46

4.4 Reasoning about Boolean Reasoning . . . . . . . . . . . . . . . . 46

4.4.1 Basis Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.2 Vectors and Duality . . . . . . . . . . . . . . . . . . . . . 46

4.4.3 The Proof of the Pudding: the soundness and Complete-

ness Theorems for the Natural Deduction System . . . . . 46

5 Non-Classical Logics 47

5.1 Some Alternative Propositional Logics . . . . . . . . . . . . . . . 47

5.1.1 Philosophy and Logic . . . . . . . . . . . . . . . . . . . . 47

5.1.2 On Modal Logics . . . . . . . . . . . . . . . . . . . . . . . 48

5.1.3 Intuitionist Logic . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.4 Many-Values Logics and Fuzzy Logics . . . . . . . . . . . 51

5.1.5 Quantum Logic . . . . . . . . . . . . . . . . . . . . . . . . 53

5.1.6 Philosophy and Logic Again: The Struggle Between Op-

posites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1.7 Embracing Contradictions: Formal Systems which are Non-

Trivial but Inconsistent . . . . . . . . . . . . . . . . . . . 57

II Quantifier Logics 63

6 Zero Order Reasoning and . . . 65

6.1 Zero Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1.1 Zero Order vs. Boolean Languages . . . . . . . . . . . . . 66

6.1.2 Doing Arithmetic in a Zero Order Language . . . . . . . . 69

6.1.3 The Completeness of the Axiom System . . . . . . . . . . 72

6.1.4 Structures and Valuations of Zero Order Languages . . . 72

6.2 Interpretations and Validity in . . . . . . . . . . . . . . . . . . . . 74

6.2.1 Transition from Monadic Zero Order Languages to Uni-

form Monadic First Order Languages . . . . . . . . . . . 75

xiv CONTENTS

6.2.2 Categorical Syllogisms in Uniform First Order Languages 77

7 First Order Reasoning 81

7.1 First Order Semantics . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.1 First Order Languages and Multiple Quantification: Sen-

tences, Interpretations and Truth . . . . . . . . . . . . . . 82

7.1.2 Elementary and Higher Order Structures . . . . . . . . . 84

7.1.3 Models of Sets of Sentences: Some Examples . . . . . . . 86

7.1.4 First Order Games . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Methods of First Order Logic I: Truth Trees . . . . . . . . . . . . 90

7.2.1 First Order Truth Trees . . . . . . . . . . . . . . . . . . . 91

7.3 Methods of First Order Logic II. . . . . . . . . . . . . . . . . . . . 97

7.3.1 Natural Deduction in First Order Logic: Formal Proofs . 98

7.3.2 The Basic Problems of First Order Logic . . . . . . . . . 103

7.4 First Order Theories . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.4.1 First Order Theories . . . . . . . . . . . . . . . . . . . . . 106

7.4.2 Introducing Functions and Equality: First Order Theories

with Equality . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4.3 Some First Order Theories with Functions and Equality . 111

7.5 Reasoning about First Order Reasoning . . . . . . . . . . . . . . 118

7.5.1 The First Order Soundness Theorem . . . . . . . . . . . . 118

7.5.2 The Completeness Theorem for First Order Natural De-

duction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.5.3 Deriving the First Order Completeness Theorem. . . . . . 127

8 Questions of Consistency, . . . 135

8.1 First Crisis: Arithmetizing Geometry . . . . . . . . . . . . . . . . 135

8.1.1 Crises in Mathematical Foundations I: Facing the Irrational136

8.2 Taming the Infinite: The analysis of Change . . . . . . . . . . . . 141

8.2.1 Concepts of the Calculus from Newton and Leibniz Through

the Seventeenth Century: the Analysis of Change and its

Cumulative Effects in Space and Time . . . . . . . . . . . 141

8.3 Second Crisis: Paradoxical Series . . . . . . . . . . . . . . . . . . 151

8.3.1 The Problem of the Convergence of Power Series: Cauchy’s

Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.3.2 The Problem of the convergence of Fourier Series: Can-

tor’s Introduction of Sets and Completed Infinities . . . . 155

8.3.3 Reducing the Concept of Real Numbers to the Concept

of Natural Numbers, with a Little Help from Sets . . . . . 156

8.3.4 Defining Natural Numbers as Sets . . . . . . . . . . . . . 161

8.4 Third Crisis, Set Theory Paradoxes . . . . . . . . . . . . . . . . . 161

8.4.1 Trouble in Cantor’s Paradise: Hilbert to the Rescue . . . 161

8.4.2 Which Sets are Admissible? Logicism and the Formaliza-

tion of Mathematics within Type Theory . . . . . . . . . 161

8.4.3 The Consistency of Mathematics: Hilbert’s Program . . . 161

8.4.4 Set-Theoretical Foundations: First Order Set Theories . . 161

CONTENTS xv

8.5 Classical Mathematics and Loss of Certainty. . . . . . . . . . . . . 161
8.5.1 Prelude to Găodel I: Exact Denitions of Decidability . . . 161
8.5.2 Prelude to Găodel II: Găodel Numbering: the Code for Talk-
ing about Arithmetic in the Language of Arithmetic . . . 161
8.5.3 Prelude to Găodel III: Putting all the Ducks in a Row and
How to Make Sentences Talk about Themselves . . . . . . 161
8.5.4 The Heart of the Matter: the Basic Arguments . . . . . . 161
8.5.5 Formalism and the “Loss of Certainty”: Implications for
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.5.6 The Demise of Metamathematical Certainty: Some Con-
sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Appendix 1: Operator Precedence 63

xvi CONTENTS

List of Figures

1.1 Euler Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Intersecting Circles . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Euler Diagram with 3 Regions . . . . . . . . . . . . . . . . . . . . 10
1.4 Tree Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Internal Diagram for Five Functions . . . . . . . . . . . . . . . . 15
1.6 Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Natural Number Equinumerous with Even Natural Numbers . . 17
1.8 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 The Two Element Boolean Lattice B2 . . . . . . . . . . . . . . . 33
2.2 Subformula Tree for U . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3 Valuation Tree for U . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4 Game Tree for 1 ∨ (1 ∧ 0) (Example 1) . . . . . . . . . . . . . . . 57
2.5 Game Tree for (1 ∨ 1) ∧ 0 (Example 2) . . . . . . . . . . . . . . . 58
2.6 Example 3 Valuation and Game Trees . . . . . . . . . . . . . . . 61
2.7 Example 4 Valuation and Game Trees . . . . . . . . . . . . . . . 61

5.1 Deducing Disjunctive Syllogism from the Spread Rule . . . . . . 58
5.2 Proof of What using Minimal Logic . . . . . . . . . . . . . . . . . 59

6.1 Formalization of Argument . . . . . . . . . . . . . . . . . . . . . 71

7.1 Directed Graph of Strict Partial Order on A = {1, 2, 3, 6} . . . . 88
7.2 Game Tree for ‘ ∀x∃y xF y ’ . . . . . . . . . . . . . . . . . . . . . 90
7.3 Valuation Tree for ‘ ∀x∃y xF y ’ . . . . . . . . . . . . . . . . . . . 90
7.4 Game Tree for ‘ ∃y∀x xGy ’ . . . . . . . . . . . . . . . . . . . . . 90
7.5 Valuation Tree for ‘ ∃y∀x xGy ’ . . . . . . . . . . . . . . . . . . . 91
7.6 Closed tree constructed from sentence . . . . . . . . . . . . . . . 94

8.1 Plot of Falling Speed versus Time . . . . . . . . . . . . . . . . . . 143

8.2 Plot of Falling Distance versus Time . . . . . . . . . . . . . . . . 144
8.3 Something About Limits . . . . . . . . . . . . . . . . . . . . . . . 147

xvii

xviii LIST OF FIGURES

List of Tables

2.1 Logical Multiplication for B2 . . . . . . . . . . . . . . . . . . . . 30
2.2 Logical Addition for B2 . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Logical Complementation for B2 . . . . . . . . . . . . . . . . . . 30
2.4 Truth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5 Matrix for Material Conditional . . . . . . . . . . . . . . . . . . . 36
6.1 Categorical Statements . . . . . . . . . . . . . . . . . . . . . . . . 77

xix