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A Problem Course in
Mathematical Logic
Version 1.5

Volume I
Propositional and
First-Order Logic

Stefan Bilaniuk
Author address:
Department of Mathematics
Trent University
Peterborough, Ontario
Canada K9J 7B8
E-mail address: sbilaniuk@trentu.ca
1991 Mathematics Subject Classi¬cation. 03.

A Problem Course In Mathematical Logic
Volume I: Propositional and First-Order Logic
Version 1.5
Copyright c 1994“1999 by Stefan Bilaniuk.

Abstract. This is a text for a problem-oriented undergraduate
course in mathematical logic. It covers the basics of propositional
and ¬rst-order logic through the Soundness, Completeness, and
Compactness Theorems. Volume II, Computation, covers the ba-
sics of computability using Turing machines and recursive func-
tions, the Incompleteness Theorems, and complexity theory through
the P and NP.
Information on availabality and the conditions under which
this book may be used and reproduced are given in the preface.
This book was typeset using L TEX, using the AMS-L TEX and
AMSFonts packages of the American Mathematical Society.

Preface v
Introduction 1

Propositional Logic 5
Chapter 1. Language 7
Chapter 2. Truth Assignments 11
Chapter 3. Deductions 15
Chapter 4. Soundness and Completeness 19

First-Order Logic 21
Chapter 5. Languages 23
Chapter 6. Structures and Models 33
Chapter 7. Deductions 41
Chapter 8. Soundness and Completeness 47
Chapter 9. Applications of Compactness 51

Hints 57
Chapter 1. Hints 59
Chapter 2. Hints 61
Chapter 3. Hints 63
Chapter 4. Hints 65
Chapter 5. Hints 67
Chapter 6. Hints 69
Chapter 7. Hints 71

Chapter 8. Hints 73
Chapter 9. Hints 75

Appendices 77
Appendix A. A Little Set Theory 79
Appendix B. The Greek Alphabet 81
Appendix C. Logic Limericks 83
Bibliography 85
Index 87

This book is intended to be the basis for a problem-oriented full-year
course in mathematical logic for students with a modicum of mathe-
matical sophistication. Volume I covers the basics of propositional and
¬rst-order logic through the Soundness, Completeness, and Compact-
ness Theorems, plus some material on applications of the Compactness
Theorem. It could easily be used for a one-semester course on these
topics. Volume II covers the basics of computability using Turing ma-
chines and recursive functions, the Incompleteness Theorem, and basic
complexity theory; it could also be used as for a one-semester course
on these topics.
In keeping with the modi¬ed Moore-method, this book supplies
de¬nitions, problems, and statements of results, along with some ex-
planations, examples, and hints. The intent is for the students, indi-
vidually or in groups, to learn the material by solving the problems
and proving the results for themselves. Besides constructive criticism,
it will probably be necessary for the instructor to supply further hints
or direct the students to other sources from time to time. Just how
this text is used will, of course, depend on the instructor and students
in question. However, it is probably not appropriate for a conventional
lecture-based course nor for a really large class.
The material presented in this volume is somewhat stripped-down.
Various concepts and topics that are often covered in introductory
mathematical logic courses are given very short shrift or omitted en-
tirely, among them normal forms, de¬nability, and model theory.1 In-
structors might consider having students do projects on additional ma-
terial if they wish to to cover it. A diagram giving the dependence of
the chapters in this volume can be found in the Introduction.

Acknowledgements. Various people and institutions deserve the
credit for this text: All the people who developed the subject. My

Future versions of both volumes may include more “ or less! “ material. Feel
free to send suggestions, corrections, criticisms, and the like ” I™ll feel free to ignore
them or use them.

teachers and colleagues, especially Gregory H. Moore, whose math-
ematical logic course convinced me that I wanted to do the stu¬.
The students at Trent University who su¬ered, su¬er, and will suf-
fer through assorted versions of this text. Trent University and the
taxpayers of Ontario, who paid my salary. Ohio University, where I
spent my sabbatical in 1995“96. All the people and organizations who
developed the software and hardware with which this book was pre-
pared. Anyone else I™ve missed.
Any blame properly accrues to the author.
Conditions. This book may be freely transmitted, stored, copied,
and used until 31 December, 2000, subject to the following restrictions:2
1. It may not be modi¬ed in any way without the express written
permission of the author.
2. It may not be sold at a pro¬t, but only to recover the cost of
3. After 31 December, 2000, it may no longer be reproduced, stored,
or transmitted in any form without the express written permis-
sion of the author, except that printed copies existing as of 31
December, 2000, may be retained and used after this date.
The reason for the time-limit is that I hope to improve this book
and make a new version available.3
Availability. The URL of the home page for A Problem Course
In Mathematical Logic, with links to L TEX, PostScript, and Portable
Document Format (pdf) ¬les of the latest available releases of both
volumes, is:
• http://www.trentu.ca/mathematics/sb/misc/pcml.html
A text-only information ¬le and LTEX, PostScript, and Portable Doc-

ument Format (pdf) ¬les of the latest release of both volumes can be
accessed by anonymous ftp in the directory:
• ftp://ftp.trentu.ca/pub/sbilaniuk/pcml/
Please note that in addition to LTEX you will need the AMS-L TEX

and AMSFonts packages to typeset and print either volume.
If you have any problems, feel free to contact the author at the
addresses given on the title page, preferably by e-mail, for assistance
or even to ask for a paper copy.
Author™s Opinion. It™s not great, but the price is right!
If you violate these restrictions, I will be ¬‚attered, but you will still be in the
Who knows, maybe even ¬nd a publisher . . .

Mathematical Logic. What sets mathematics aside from other
disciplines is its reliance on proof as the principal technique for de-
termining truth, where science, for example, relies on (carefully ana-
lyzed) experience. So what is a proof? Practically speaking, it is any
reasoned argument accepted as a proof by other mathematicians.4 A
more precise de¬nition is needed, however, if one wishes to discover
what mathematical reasoning can accomplish in principle. This is one
of the reasons for studying mathematical logic, which is also pursued
for its own sake and ¬nding new tools to use in the rest of mathematics
and in related ¬elds.
In any case, mathematical logic is concerned with formalizing and
analyzing the kinds of reasoning used in the rest of mathematics. The
point of mathematical logic is not to try to do mathematics per se
completely formally ” the practical problems involved in doing so are
usually such as to make this an exercise in frustration ” but to study
formal logical systems as mathematical objects in their own right in
order to (informally!) prove things about them. For this reason, the
formal systems developed in this book are optimized to be easy to prove
things about, rather than to be easy to use. Natural deductive systems
such as those developed by philosophers to formalize logical reasoning
are equally capable in principle and much easier to actually use, but
harder to prove things about.
Part of the problem with formalizing mathematical reasoning is the
necessity of precisely specifying the language(s) in which it is to be
done. The natural languages spoken by humans won™t do: they are
so complex and continually changing as to be impossible to pin down
completely. By contrast, the languages which underly formal logical
systems are, like programming languages, much simpler and less ¬‚exible
than natural languages but rigidly de¬ned. A formal logical system also
requires the careful speci¬cation of the allowable rules of reasoning, plus
some notion of how to interpret statements in the underlying language

If you are not a mathematician, gentle reader, you are hereby temporarily

and determine their truth. The real fun lies in the relationship between
interpretation of statements, truth, and reasoning.
This volume develops the basics of two kinds of formal logical sys-
tems, propositional logic and ¬rst-order logic. Propositional logic at-
tempts to make precise the relationships that certain connectives like
not, and , or , and if . . . then are used to express in English. While it
has uses, propositional logic is not powerful enough to formalize most
mathematical discourse. For one thing, it cannot handle the concepts
expressed by all and there is. First-order logic adds all and there is to
those which propositional logic could handle, and su¬ces, in principle,
to formalize most mathematical reasoning. To be sure, it will not han-
dle concepts which arise outside of mathematics, such as possible and
relevant, among many others. (Trying to incorporate such concepts
into systems extending ¬rst-order logic is a substantial industry in phi-
losophy, but of marginal interest in mathematics.) Propositional logic,
which is much simpler, will be dealt with ¬rst in order to gain some
experience in dealing with formal systems before tackling ¬rst-order
logic. Besides, some of the results about propositional logic carry over
to ¬rst-order logic with little change.

Approach. This book supplies de¬nitions and statements of re-
sults, plus some explanations and a number of problems and examples,
but no proofs of the results. The hope is that you, gentle reader, will
learn the material presented here by solving the problems and proving
the results for yourself. Brief hints are supplied for almost all of the
problems and results, but if these do not su¬ce, you should consult
your peers, your instructor, or other texts.

Prerequisites. In principle, not much is needed by way of prior
mathematical knowledge to de¬ne and prove the basic facts about
propositional and ¬rst-order logic. Some knowledge of the natural
numbers and a little set theory su¬ces; the former will be assumed
and the latter is very brie¬‚y summarized in Appendix A. What re-
ally is needed to get anywhere with the material developed here is
competence in handling abstraction and proofs, especially proofs by
induction. The experience provided by a rigorous introductory course
in algebra, analysis, or discrete mathematics ought to be su¬cient.
Some problems and examples draw on concepts from other parts of
mathematics; students who are not already familiar with these should
consult texts in the appropriate subjects for the necessary de¬nitions.

Other Sources and Further Reading. [4], [8], and [9] are texts
which go over similar ground (and much more), while [1] and [3] are

good references for more advanced material. Entertaining accounts of
some related topics may be found in [7] and [10]. Those interested
in natural deductive systems might try [2], which has a very clean
Credit. Almost no attempt has been made to give due credit to
those who developed and re¬ned the ideas, results, and proofs men-
tioned in this work. In mitigation, it would often be di¬cult to assign
credit fairly because many people were involved, frequently having in-
teracted in complicated ways. (Which really means that I™m too lazy
to do it. I apologize to those who have been hurt by this.) Those
interested in who did what should start by consulting other texts or
reference works covering similar material.
Chapter Dependencies. The following diagram indicates how
the chapters in this volume depend on one another, with the excep-
tion of a few isolated problems or results.

¨¨ rr

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