Mathematical Logic: A First Course (Dover Books on Mathematics) by Joel W. Robbin (PDF)

Description

Suitable for advanced undergraduates and graduate students, this self-contained text will appeal to readers from diverse fields and varying backgrounds — including mathematics, philosophy, linguistics, computer science, and engineering. It features numerous exercises of varying levels of difficulty, many with solutions.A survey of the propositional calculus is followed by chapters on first-order logic and first-order recursive arithmetic. An examination of the arithmetization of syntax follows, along with a review of the incompleteness theorems and other applications of the Liar Paradox. The text concludes with a study of second-order logic and an appendix on set theory that will prove valuable to students with little or no mathematical background.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This 1969 book by Joel William Robbin (a student of Alonzo Church) is more advanced and modern than most of the “first course” logic textbooks which appeared in the 1960s and 1970s. It seems to me like a good first introduction to model theory, whereas most of the “first courses” in that time were more about logic and set theory axioms and theorems. Model theory is, after all, a “different kettle of fish” to the more practical kind of mathematical logic.This book is well within the modern mainstream of mathematical logic and model theory. It seems to me like a relatively gentle introduction to model theory concepts which can be painfully brain-twisting in some of the more modern literature. The model theory presented here (starting on pages 38-40) is quite concrete compared to some excessively abstract modern textbooks.Chapter 1 presents a semi-formal propositional calculus language which interestingly is based on the implication operator and the “falsum” (always-false) nullary operator. (The only other textbook in my collection of 45 logic textbooks which starts from these two operators is Church’s ”

⭐”, page 72.) This language is given truth-table semantics, and a “deduction theorem” is proved for it. (It’s really a metatheorem.) Another metatheorem given for this language is completeness.Chapter 2 introduces a form of first-order logic which has individual variables and multi-variable predicates. Then things get more “interesting” with model theory starting on page 38. A semantic (double-barred) assertion symbol is introduced for validity of sentences in the model. A sentence is defined to be “valid” is it is true in every model. This is really the core concept of model theory. (Everything else is just definitions and theorems!) Then there is a deduction theorem for FOL, a completeness theorem, and a FOL with equality. (Very useful for set theory!!)Chapter 3 is on recursive arithmetic, which is de rigeur in every proof theory and model theory book.Chapter 4 presents the “arithmetization of syntax”, which is coincidentally what Gödel was famous for. And prime numbers are used for this, just as Gödel did in the 1930s. So it’s clear where this is all heading….Chapter 5 is about incompleteness theorems by…. Gödel. And there’s a Tarski theorem and a Church’s theorem too.The final Chapter 6 is about second-order logic, which is not really necessary for the daily work of mathematicians, but it allows yet another Gödel theorem to be given. The part I like about Chapter 6 is section 55 (pages 164-165), which presents Skolem’s paradox. This is very fundamental, I think, because it exposes some important fundamental differences between languages and models. Maybe model theory doesn’t really prove all the things people think it proves about ZF set theory, for example, because things which appear in models aren’t necessarily accessible from the language.Some small negatives.1. The use of dots (defined on page 5) instead of parentheses. Thankfully these are only single dots, not the crazy multiple dots of Whitehead/Russell.2. The use of archaic Fraktur font, which is thankfully kept to a minimum.All in all, this is about the friendliest introduction to model theory I’ve seen in a long time. I think this is a great place to start before delving into the more modern austere abstract textbooks. It’s also a really good explanation of Gödel’s theorems and Skolem’s paradox.

⭐DESCRIPTION AND INITIAL READING IN 2009I read all of the first three chapters, most of chapter 4 and about half of chapter 6 in 2009 and this concise and for me clear text is one of my two favorite logic books. First, I like Robbin’s full formation rules and axiomatics for each level of logic he describes and that he both fully explained and didn’t overdo dot notation for elimination of parentheses. Then also, he used implication as his only logical connective, and primarily used universal quantification in his first and second order logics, both of which were nicely simplifying. Interestingly, a falsity sign ‘f’ was used instead of a ‘not’ unary connective, and this worked naturally with that use of implication only. Plus there were a number of helpful insights stated that added a nice dimension to this book. Chapter 2 on First-Order Logic and chapter 3 on First-Order Recursive Arithmetic were my favorites in this book.CHAPTER TITLES / PAGE NUMBERSThe six chapters are: 1) The Propositional Calculus-1 / 2) First-Order Logic-32 / 3) First-Order Recursive Arithmetic-66 / 4) Arithmetization of Syntax-90 / 5) The Incompleteness Theorems and Other Applications of the Liar Paradox-111 / 6) Second-Order Logic-132 + four appendices-171 on related subjects and answers to selected exercises-183–224 pages total.COMPARISON OF MY TWO FAVORITE LOGIC BOOKSMy other favorite logic book, fully read in Nov11 is at the following link. This other book by Uwe Schoning is much different than the currently reviewed book by Joel Robbin but I like both books equally:

⭐While Schoning focuses on computational logic, including both propositional and first-order resolution in detail, Robbin covers more territory, and gets well into outright mathematical foundations in chapters 3-5.THE LARGER 2012 READINGOn Thu 8Mar12, I did start reading this great book again, starting with excellent chapter 2, as I also have other books in process. This second time thru the Robbin, I am really liking chapter 4 on arithmetization of syntax in sections 26-31. Last sections 32-36 though are great subjects, but are very difficult, with many huge expressions in proofs that go on and on and so are difficult to understand well. Finished chapter 4 on Tue 10Apr12. Next day, I started unplanned chapter 5 on incompleteness and the liar paradox, and it has been quite interesting, and easier than late chapter 4. Started planned chapter 6 on second order logic after finishing chapter 5, both on Fri 13Apr12. Chapter 6 is billed as mostly an optional chapter, and starting with section 50 / p.148 the writing mostly transitions to a bare bones definition/theorem/proof type of operation, and this reader skipped the only 3.5 page proof in the book in that section 50. On Thu and Fri 19-20Apr12, I did push thru several final sections of excessively proofy late chapter 6 to finish reading and extending my reading of this book since the 2009 read, finishing at 2:36pm on Fri 20Apr12. Still a terrific book!

⭐One of the best introductions to mathematical logic.A concise, elegant and deep text, despite being introductory.It is worth reading the book!

⭐Awful in my opinion. Bizarre non-standard notation and little to no explanation, things seem to just pop out of nowhere. I returned it and bought Hodel. Much better book if you want to teach yourself.

⭐This book is suitable for advanced undergradutes and graduate students for learning mathematical logic.It contains a wide selection of exercises. In the back of the book, the author gave answers to selected exercises. But I think the notational conventions is a little of old, not stylish. I don’t like this kind of notational convensions in the book.

⭐There are 5 chapters in total, of which I’ve worked through the first two on my own. These take you clearly through the metatheory of propositional and predicate calculus. I’m not a mathematician, so I was looking for something clear. I got this from the library along with several others including Enderton and Mendelsohn, and this was the one I ended up working with because it struck me as the clearest. Robbin does a good job of getting to the essentials without getting bogged down in a mass of theorem proving, but at the same time doesn’t skip anything that you really need to know. This is a good, fast intro to the subject.

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