A Beginner's Guide to Mathematical Logic

by Raymond M. Smullyan

Written by a creative master of mathematical logic, this introductory text combines stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic. Author Raymond Smullyan offers clear, incremental presentations of difficult logic concepts. He highlights each subject with inventive explanations and unique problems.
Smullyan's accessible narrative provides memorable examples of concepts related to proofs, propositional logic and first-order logic, incompleteness theorems, and incompleteness proofs. Additional topics include undecidability, combinatoric logic, and recursion theory. Suitable for undergraduate and graduate courses, this book will also amuse and enlighten mathematically minded readers. 2014 edition.

Genres: Mathematics, Logic, Nonfiction, Philosophy, Science, Reference, Stem

284 pages, Paperback

First published January 1, 2014

About the author

Raymond M. Smullyan was a logician, musician, Zen master, puzzle master, and writer.

Reviews

[Richard Thompson · Rating 4 out of 5]

I first encountered Raymond Smullyan through his popular logic puzzle books. They are lighthearted, funny and just hard enough to make you think without getting frustrated. And the guy looks like Gandalf The Gray! So it seemed reasonable for me to try Smullyan for the first real math book that I have read in a few years. I took a mathematical logic class in high school, and I have read a simplified proof of Godel's theorem demonstrated through the formal system of a Turing Machine, but all of that stuff was pretty simple. Notwithstanding the title and the fact that no prior knowledge of logic is strictly speaking necessary to read this book, this book is challenging for someone like me who has some mathematical ability but is very rusty. I did great with the first half of the book understanding nearly everything, but my comprehension level dropped to around 70%-80% in the later chapters. I still got enough to stay interested and keep reading, and if it hadn’t been a bit challenging, I probably would not of liked it as much. If I went back and reread it and reworked the problems, I'd probably be able to absorb all of it. I doubt that I will do that, but I'd like to read more mathematical logic books so that I can soak it in better.

[William Schram · Rating 5 out of 5]

A Beginner’s Guide to Mathematical Logic introduces a reader to the subject of Mathematical Logic. The author, Raymond M Smullyan, wrote several books. This is one of the many books he wrote for the layman to get them to understand Logic.

From the cover of this book, one can see a statue of Aristotle, a portrait of George Boole, and a photograph of Kurt Godel. The book is organized somewhat like the cover, where it opens with basic Aristotelian Logic. The syllogism is used and Smullyan attempts in many cases to demonstrate the truth or falsity of statements. When we get to Boole the kid gloves come off and we are introduced to Boolean Algebra and Propositional Logic. This introduced rigor to the field of Logic and was an attempt to axiomatize all statements in mathematics. The main proponents of such an idea were Alfred North Whitehead and Bertrand Russell. Well, not really; they invented a language and system that could be used to attempt such a thing, but many mathematicians wanted to be able to “complete” mathematics. Of course, fans of the field know what happened next; Godel came along and presented a proof showing that such an action was impossible.

Smullyan explains how that happened in an easy to understand manner. All Godel did was create a sentence out of a mathematical phrase that said something like “This Sentence Is False.” The important part is that he used the rigor of the Principia Mathematica to do so, turning this logical machine into a double-bladed sword.

The book also talks about First-Order Logic, which has the little upside-down capital ‘A’ or the universal quantifier and the backward capital ‘E’ or the existential quantifier. It goes into Tableaux as well, so that you can verify truth statements and truth tables.

So the book covers all of that information. It does so in a manner that is easy to understand if you are paying attention. It presents practice problems for each chapter with solutions at the end explaining the answer. For example, chapter 14 contains a statement called Proposition 1. It says “If S is an extension of R, then all recursive sets and relations are definable in S.” The first problem of the chapter asks you to prove this Proposition.

I enjoyed this book and it introduced me to some ideas in mathematics that I am not completely familiar with.

[Dmk · Rating 2 out of 5]

Not very interesting book. It gives you only basic knowledge about math. logic, so if you have some knowledge, you'll be bored.

[Neeraj Adhikari · Rating 3 out of 5]

DNF. Stopped studying after Chapter 4 of 14.

I picked up this text for self-study, because at a first glance it looked very attractive; it contained the main topics I was looking for, was small (~250 pages) and was written by a famous author.

When I started working through the book, I started getting frustrated due to two reasons, which eventually made me decide to DNF. First, the book uses terminology that is different from the mainstream for no reason at all (for example, the well-ordering principle is called the least number principle) which makes searching for more details on the web difficult. Second, and this is the much bigger problem for me, the book is quite terse. It packs a lot of content in a few pages. Most of the major proofs are exercises for the reader, and although there are solutions to these exercises at the end of chapters, the proofs given feel very informal and non-rigorous, which is at the same time weird for a book on logic, and also impedes understanding and digestion of the material.

Lesson learnt: don't go for a textbook that packs in a lot of content in a small number of pages. Choose books that have plenty of exercises that warm you up and help build understanding and intuition before asking you to prove major results.

I will be picking up Set Theory and Logic by Stoll as a replacement. The topics covered don't overlap completely, but my hope is that it will be much less frustrating than this book.

[Rin · Rating 3 out of 5]

For me, the book isn't really for beginner's. It started off pretty simple but quickly exponentially, recursively scaled the difficulty of the subject. I think it would serve as a good beginner's guide for people looking to do coding, programming or going into computer science/ information technology. Or just pretty smart math students in general might appreciate and grasp these topics quickly.

[Amasi Godgift · Rating 2 out of 5]

It is a good to help children learn

[Kaidi Pan · Rating 3 out of 5]

Witty, lighthearted, interactive, yet incredibly clear.

[Mills College Library]

511.3 S6669 2014

[Vader · Rating 4 out of 5]

5 star - Perfect
4 star - i would recommend
3 star - good
2 star - struggled to complete
1 star - could not finish