# Mathematics

The Cali Garmo does Math

Tag: logic

## A Mathematical Introduction to Logic by Herbert B. Enderton

Feb 10

A Mathematical Introduction to Logic

I think Enderton does a really nice job introducing logic to those who have never studied it before. Although not always the easiest to follow Enderton lays out all the necessary topics in a nice organised fashion so it’s fairly simple to follow everything. I found his explanation of the pumping lemma to be lacking which made it difficult to follow, but his proof of the compactness theorem more than makes up for it. He does a good introduction of not only proposition and predicate logic, but also goes into second-order logic as well and tackles it the same way as his previous material so it’s easy to follow it all. Although this wasn’t my favourite logic book, it is a nice book to peruse.

## Introduction to Mathematical Logic by Elliott Mendelson

Jan 27

Introduction to Mathematical Logic

This week we’re doing another book on Logic! Mendelson does an ok job with his book in introductory logic. He sets up a nice introduction to logic concepts, but then fails to deliver much needed exposition. Although the book is very descriptive and helps to get your hands very dirty in logic, it can sometimes be a little to follow what Mendelson is trying to prove and whether a certain problem is an exercise or a proof. Although at times it’s a little hard to follow this text does give an amazing introduction to the topic of logic. It doesn’t assume any presupposed knowledge and actually goes over almost every topic that a new student would be expected to know in the topic. Not only that, but he also breaks down Number Theory to prove Gödel’s incompleteness theorem and also does a fairly good job of showing the different axiomatizations of set theory. Not only that, but his introduction to second-order logic and modal theory are also fairly nicely laid out. I chose not to look at his computability section as it seemed that it would be lacking since it only seemed to barely touch on the subject and there are tons of books out there that go more in depth on the topic and do a phenomenal job. I’d definitely recommend this book with a secondary companion text to help guide you along if you have trouble understanding what Mendelson is trying to state.

## Beginning Logic by E.J. Lemmon

Jan 13

Beginning Logic

Edward John Lemmon was a logician whose main area of expertise was modal logic. His book ‘Beginning Logic‘ is likely one of the better ones out there for those just getting into logic in order to understand the different rules in modal logic. Lemmon begins with an introduction on logic and why it is necessary to talk about the subject. After giving a very well laid out overview of why logic is required he begins by talking about the rules of logic and why each one is necessary. Although the book on it’s own is likely difficult to understand, it is a good reference book to see why certain rules are the way they are. (By rules I refer to the rules of derivation from one set of formulae to another). Not only does he give a detailed explanation of each rule he also goes into the concepts of completion and why each set of rules are complete in their respective areas.

He covers 2 separate areas of logic: propositional and predicate. Propositional logic is logic that only uses ‘operators’ such as . (Here he uses  instead of the now traditional ). Propositional logic is also sometimes referred to as sentential logic or propositional calculus. In predicate logic Lemmon adds the symbol  for ‘there exists an x’, and  for ‘for all x’. This format is not necessarily traditional, but Lemmon is working before standards were fully developed. His syntax can be slightly difficult to follow, but after working on it, it is not the worst syntax out there.

Who this book is for: This book should be used in conjunction with some other books and is good for someone who is looking for an english language description of logic and the different rules associated with proving results.