### A Study Guide

A re-titled, expanded version of the old *Teach Yourself Logic* study guide. This is a book length guide to the main topics and some suitable texts either for teaching yourself logic by individual self-study, or to supplement a university course. You only need to read just the first half-dozen pages to see if this is for you.

*Beginning Mathematical Logic: A Study Guide*[corrected reprint 5 Jan 2024]

The *Guide *is also available as a *very* inexpensive paperback, about the price of a couple of coffees, but only direct from Amazon (sorry!) as this minimizes the price for you: ISBN 1916906338.

In addition there is an Appendix to the Guide saying more about some Big Books on mathematical logic:

- Appendix to the Guide (a 60 page PDF, discussing 23 books).
- Book Notes (a page of links to separate webpages on some 63 books, including those covered in the Appendix)

### A little more about the Study Guide and its Appendix

Most philosophy departments, and many maths departments too, teach little or no serious logic, despite the centrality of the subject. Many students will therefore need to teach themselves, either solo or by organizing study groups. But what to read? Students need annotated reading lists for self-study, giving advice about the available texts. In around 2012, I started the *Teach Yourself Logic *Study Guide, which aimed to provide the needed advice by suggesting some stand-out books on various areas of mathematical logic. NB: it covered *mathematical* logic — so we are working a step up from what’s rudely called “baby logic” (that philosophers may encounter in their first year courses).

The Guide has now been radically rewritten. After a couple of preliminary chapters, Chapters 3 to 9 introduce core topics from the mathematical logic curriculum, and suggest suitable entry-level reading. Chapters 10 and 11 broaden the scope just a bit. And then there is a final Chapter 12 pointing to more advanced texts on the core topics.

Mathematical logic is indeed a big subject, and different people have different backgrounds and/or requirements. So you’ll want detailed advice from which you can work out which books on which areas might be suitable for you. That’s why the Guide is so long. But there are a lot of pointers to help you find your way around.

The recommendations in the *Guide* are organised topic by topic. The *Appendix* then reviews some of the big multi-topic textbooks on mathematical logic book by book.

### The Book Notes

The same reviews in the *Appendix* can also be found as separate webpages in linked to this page of Book Notes, which also links to comments/reviews about some forty(!) more books on logic and the philosophy of mathematics. These range from half-pages to (in one case) a 52 page essay. Most are a useful two or three pages!

Rodrigo de AlmeidaDear Prof. Smith,

firstly, thank you for assembling all this information about Logic and writing the Guide. I not only bought your books (willing to have the proper time to finally get through IGT) but also rely on your inputs – for instance: it was a very refreshing and rewarding experience reading the first 4 or 5 chapters of Chiswell&Hodges after I had Van Dalen as a course book on introductory logic (ouch!), so your comments on the Guide were surely very useful.

I’d like to ask then if you have any thoughts about Haskell Curry’s Foundations of Mathematical Logic, I didn’t find nothing about it here. I got this book and I wonder if there is something interesting about it.

Best Regards,

Rodrigo de Almeida

Samuel T. X. KhooI would like to know what you think of Katalin Bimbó’s new book “Proof Theory: Sequent Calculi and Related Formalisms” (2014, Taylor and Francis). It’s a textbook aimed at advanced undergraduates focusing ‘on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic.’

Do you think it would be suitable for learning more about sequent calculi in general, and proof theory in specific?

Peter SmithThanks for alerting me to this book, which I didn’t know of before. I can’t give you a view, then, though preview pages look pretty encouraging.

Andrew Lewis-SmithPeter — In your Teach Yourself Logic p 63 you mention Katalin Bimbo’s text, in which you say you’re “not inclined to recommend it.” I’m working thru it now, and I must say, it’s chock full of information you won’t find anywhere else. In fact, I’d say it fills a major gap in the community and is a vastly more detailed approach to the proof theory of substructural logics — and maybe even proof theory generally — than Restall’s text (which I keep coming back to, it is an amazing book).

Just curious why you wouldn’t recommend it, what faults you see in it, etc. It seems very thorough to me. :)

Thanks!

Andrew

Rowsety MoidI’ve noticed a new category theory book that takes a different sort of approach:

Category Theory for the Sciencesby David I. Spivak (MIT Press). It’s not quite out in the UK but is available from US Amazon. It focuses on ideas and examples, rather than proofs for theorems, and it looks like it aims to show how category theory can be useful outside mathematics.Peter SmithThere is a version online at the author’s website, here: http://math.mit.edu/~dspivak/teaching/sp13/CT4S–static.pdf

I’ll take a look, and thanks for the info!

Mige KrokbenI’d be interested in hearing what you think of Johan van Benthem’s “Modal logic for open minds”. I just got it in the mail today and I like what I see on a quick flip-trough. Of the other books I’ve used (Hughes & Cresswell, Sider, Girle…) this seems by far most similar to Girle’s book—not just in content but also in being written in a readable and engaging style. However, it’s more than 100 pages bigger than Girle’s, and I believe a bit wider in scope.

Rowsety MoidThere was a brief comment in version 10 of the Teach Yourself Logic Guide. It said:

Some would say that Johan van Benthem’s Modal Logic for Open Minds (CSLI 2010) belongs much earlier in this Guide. But, though developed from a course intended to give ‘a modern introduction to modal logic’, it is not really routine enough in coverage and approach to serve at an elementary level. It takes up some themes relevant to computer science: worth having a look at to get an idea of how modal logic fares in the wider world.

Andrew CrawshawI would like to know what you think of Paul Tomassi’s ‘Logic’? One difficulty I found with this book, is that there are no solutions therein, and the webpage for access to the solutions has, since Paul Tomassi’s passing, taken them offline.

Peter SmithTomassi’s book is OK — but I’d say counts as baby logic, which isn’t really the topic of the Guide, and there are better books at that level.

Andrew CrawshawYeah, after I reread the introduction to your book, I realized that you might not include it for that reason. Thanks for the great resource, I am especcially pleased that you introduced the books that deal,with mathematical topics that might be missing from an introductory Logic course like More Precisley by Steinhart, very useful.

Gustavo BandeiraAs the guide is made towards people studying logics for the purposes of both mathematics and philosophy, why not suggest Susan Haack’s Philosophy of Logics? I am a mathematics student with interest in logics and lately bought this book. It’s an amazing read, and it talks a lot about why we need logic and how to build a logic.

Peter SmithWell, I’m remember Haack’s old book as indeed being good of its kind, and I’m glad that you found it helpful. I’ll have to take another look at it and consider whether this (and some similar books) might be mentioned in what is, basically, a guide to mathematical logic.

Jan NoordhofI used J L Bell & M Machover’s ‘A Course in Mathematical Logic” (1977) when it first appeared as a friendlier alternative to Schoenfield. At the time this was in conjunction with Bell & Slomson’s “Models and Ultraproducts”. Bell & Machover’s book is still in print and not particularly expensive.

Peter SmithBell and Machover is indeed pretty good — it’s on my list of books to comment on one day!

Michael SloanFirst off, thank you for providing this great resource. Having a guide is great for allowing more time to admire the scenery, rather than focusing wholly on not falling over cliffs, so to speak. Second, what are your thoughts on “The development of Logic”, by W. & M. Kneale?

Peter SmithI think the Kneale’s book was a remarkable achievement in its time, and it does stand up remarkably well 50 years on. But obviously a lot more, some very good indeed, has been written on the history of logic since then!

Troy LargeThank you for your work in laying out a path to follow for self-study.

I am a little sad there isn’t more in the way of free books as the price can be very much like a closed door to so many of us who do not have access to large universities, and getting worse these days with the crunch in fund to public libraries. All the same, it look like you have done a very good service to people – I hope to prove that in coming days!!

Peter SmithI very much appreciate the point about the expense of logic books (even libraries in not-so-rich universities have problems keeping up). I do mark in the Guide some particularly good-value books and even a few freely available resources: but I realise that isn’t enough for those who have no access to major libraries (though you might find that such public libraries which you do have access to have an interlibrary load system.

Obviously, I can’t link to the well-known PDF repositories which break copyright (even when the copied books are old ones and even out-of-print).

I do think there are various good reasons for maintaining traditional book publishing (though I’m open to persuasion on the point). But I do think that it should be default that academic publishers — especially those that are university presses — put a significant amount of their back catalogue into the public domain e.g. a decade after publication. By that point sales will usually be low, so neither press nor author will lose much, but the book can gain a new lease of life.

Christofer KochThank you very much for your guide. I have found it very useful in preparing for graduate school. I was wondering what you thought about “Introduction to Mathematical Logic” by Alonzo Church.

Peter SmithChurch’s book was, in its time, a wonderful achievement and an immensely influential classic. It is, however, ages since I have looked at it. I ought to do so again one day!

Marco LoffaWhat do you think about Schaum’s Outline of Logic, Second Edition of Nolt, Rohatyn and Varzi. I think it is the best for ‘baby logic’.

Peter SmithI don’t know the book, so can’t comment, sorry!

Shealton GeorgeThank you for putting this together. I stopped the guide where it says its not for elementary logic. I don’t have any experience with logic. Are there any free resources you recommend to learn elementary logic?

Peter SmithWell, in the Guide I do recommend Paul Teller’s introductory book which is freely available online.

Charlie SpiraHey Dr. Smith! I’m a baby logic student reading your introductory text and the TYL Guide and I thought you might want to know that there’s a typo on page twenty-seven of the TYL Guide: “Now, I recommended A Friendly Introduction as a follow-up to C&H: but Leary’s book might not in every library”. I think the word ‘be’ was intended to be in there?

Also, I’m wondering what you think of Smullyan’s A Beginner’s Guide to Mathematical Logic. It might help me gauge what I’ll think of other texts.

Nathan GlassI found your Guide really helpful for up-skilling in logic, sufficient to TA a class in Intermediate Logic—thank you.

I’m wondering if you’re aware of anything comparable in other areas of mathematics, particularly probability and statistics?

Peter SmithThanks for the nice words about the Guide. But no, I don’t know of anything comparable in the probability area — I’d be reduced to googling, like you!

Jia EnThis looks very interesting and I want to start but I didn’t study philosophy at an undergraduate level at all. Could you recommend me any texts I could read to familiarise myself with Baby Logic?

Jia EnOooh sorry, I saw your answer to Shealton George. Will try Paul Teller’s book. Would be grateful for any other suggestions you can throw out.

Thank you so much for putting this guide together. It looks like a very helpful map!

Souvik DasI think that in the latest version of Appendix there are references to other sections which has been removed (see page 6,8, 33 and 36). Please take a look at them.

CharlieI love this guide! It’s very helpful.

I was wondering if you’ve read John Burgess’ “Philosophical Logic”. If so, what are your thoughts on it?

GarrettI am not a philosopher with no academic prospects whatsoever but I am interested in formal logic and this exactly what I have been looking for! There’s always a mountain of books to choose from when looking into any academic field and it is positively dizzying to choose among them, for a neophyte it often feels like being beset by a swarm of locusts. As soon as the uni opens up I’m heading to the library and scouting out your beginners recommendations! Thank you again :)

Pierre FranckxHi,

Thank you so much for putting this online! I have spent some time reading your guide, and have concluded that my knowledge of mathematical logic is restricted to some baby logic: I have finished reading (and doing all the exercises of) Patrick Suppes’ and Shirley Hill’s ‘First course in Mathematical Logic’ and have started reading G.T. Kneebones’ ‘Mathematical Logic and the Foundations of Mathematics’ (with S.T. Kleene’s Mathematical Logic waiting to be read – I know it’s in your recommendations). I’m still a bit confused at what the next step should be: reading Modern Formal Logic Primer by Paul Teller, or can I start with a book on FOL? Or is this too early too? My budget is limited, so I’m restricted to (legally) free online resources or the Dover publications. Thanks in advance.

Pierre

ChristianHello, do you have any references about these 2 books?

1- Introduction to Logic by Harry J. Gensler

2- Introduction to Logic by Irving M. Copi.

Thank you very much.

Peter SmithI don’t know Gentler. But Copi is at a more elementary level than the Guide is dealing with.

E. ShengI have found many books recommended in your guide encouraging so far as I could preview them online. But, equally, many become quite unencouraging to a poor student trying to teach himself logic from scratch when he sees their price.

P. D. Magnus’s text «forall x» is freely available online, and, I believe, is currently used for the Part IA logic paper. What is your opinion of this text?

The other more affordable books I have found are

Volker Halbach’s «The logic manual»,

Raymond Smullyan’s «A beginner’s guide to mathematical logic»,

Joel W. Robbins’ «Mathematical logic: a first course»,

Patrick Suppes’ «Introduction to logic»,

Suppes and Shirley Hill’s «First course in mathematical logic»,

Wilfrid Hodges’ «Logic», and

Alice Ambrose and Morris Lazerowitz’s «Logic: the theory of formal inference».

I would appreciate your comments on as many of these books as you have encountered.

Peter SmithI’ll hope to comment on a few of these in the next edition of TYL

Rowsety MoidThere’s a brief (1-paragraph) comment on Smullyan’s

A Beginner’s Guide to Mathematical Logicin the current version of TYL and a proper discussion of Robbins,Mathematical Logic: A First Course, in the TYL book notes, here:https://logicmatters.net/tyl/booknotes/robbin/

AlonAre you familiar with Gamut’s Logic, Language, and Meaning, Volume 1: Introduction to Logic as an intro?

I’m hearing about this quite a bit.

Do you have any opinion on his?

Peter SmithIn the past, I did take a quick look at this, but obviously wasn’t enthused enough to recommend it in TYL. But I have had other recommendations, so perhaps I should take another look!

Henk SchotelSome 50 years ago I learned a lot of (the Dutch translation of) J.E. Lemmon’s “Beginning Logic”: https://www.goodreads.com/book/show/606295.Beginning_Logic

DanielThere seem to be a small mistake in the TYL, in section 4.4.1.: Leila Haaparanta’s book is called ‘The Development of Modern Logic’ instead of ‘The History of Modern Logic’.

FelixMr Smith,

Thank you so much for writing and posting this guide! I study philosophy in a continental philosophy-oriented college and, desperate as I was for advanced courses in logic, I almost cried when I found this jewel on the web (ok, maybe not, but I was very happy). Keep up the good work, you’re actually helping people out there!

Peter SmithAlways good to hear the Guide is helping someone!

roberthey i was wondering your thoughts on one of the other books in the second oxford texts in logic series: proof and disproof by bornat.

i realize that it is much more germane to computer science, but all the same, if youre familiar with it im curious to know where it might be placed in the guide in terms of difficulty and coverage and if you might know a better way of familiarizing oneself with the concepts covered: constructive proof, disproof, hoare triples, etc.

in any case,

thank you for the guide it has helped immensely!

Peter SmithMaybe in the next edition! But at the moment I don’t know the book well enough to say anything helpful about it.

SeamusI’m actually new to philosophy and wanting to delve into philosophy of logic. Is this guide recommended for a complete amateur? And is the book: A Concise Introduction to Logic by Patrick J. Hurley a good elementary book?

Peter SmithThe Guide isn’t really for beginners. Hurley’s book is ok as an elementary text; but I would rather recommend Nick Smith’s Logic, The Laws of Truth.

Maximilian StöckelHi!

I am just interested in your thoughts on Daniel Cunninghams “Set Theory: A First Course” (Cambridge Mathematical Textbooks).

Thanks for this wonderful resource!

Peter SmithI did take a look at Cunningham’s book when it came out, but wasn’t immediately excited. It is at quite an elementary level, and I’m not sure why one would choose it over other classic elementary texts. But I’d need to spend more time on the book to make a fair comparison.

KarlDear Prof. Smith,

thank you for your helpful guide!

I wonder what is your opinion on one of the newer set theory textbooks, namely, the one by Abhijit Dasgupta, please see the link:

https://www.amazon.com/Set-Theory-Introduction-Real-Point/dp/1461488532/

Should one prefer it to Goldrei/Enderton as a first course?

Thank you very much!

Yours sincerely,

Karl.

Peter SmithThanks for bringing Dasgupta’s book to my attention — I hadn’t come across it before. Looking at the detailed table of contents, it looks rather different in style and approach to Goldrei/Enderton. The latter two both introduce axioms early and don’t develop a lot of what you might call the ordinary mathematics of sets; in Dasgupta you get much more set-theory-for-mathematicians and the axiomatic approach comes much later. Which approach is appropriate to a first course would depend very much on the aims and objectives of the course.

KarlDear Prof. Smith,

thank you very much for precisely pointing the difference between the books. It is interesting to note that recently there appeared another book on the set theory that seems to belong to the opposite pole in the sense that the strong accent is put on the axioms from the very beginning. This is the book of Daniel Cunningham,

https://www.amazon.com/Set-Theory-Cambridge-Mathematical-Textbooks/dp/1107120322

They might complement each other well.

It is important to choose the right textbook from the start, that is why I am so captious as to finding the most appropriate one for me. I was heavily traumatized in my school years by the Gindikin’s book on the algebraic logic, which I was trying to learn the logic from. I would avoid repeating the experience.

JonasHi Peter.

Will the TYL study guide be updated for 2018? When in that case? I plan to start reading it soon.

Peter SmithYes … eventually. But mostly in minor, presentational ways. I’m not planning a major overhaul of the recommendations this time around. Though it is worth highlighting that The Friendly Introduction to Mathematical Logic which I warmly recommend anyway is now freely downloadable.

LouisGreat read! Have you every considered putting this study guide on Github in Markdown?

Peter SmithWell, I haven’t. Is there any good reason (as someone who isn’t at the moment a Github user) why I should do?

Jason HemannFor things like changes of URLs, releases of new editions, small typographical issues and the like, I imagine it would be helpful to you the author for your readers to submit fixes you can adopt, rather than leaving comments registering the issue. These submissions would still be yours to accept/reject/discuss.

Jaikrishnan JanardhananDear Prof. Smith,

Thanks for the wonderful guide! What is your opinion of Yuri Manin’s “A Course in Mathematical Logic for Mathematicians”? The book offers the perspective of a first-rate mathematician and is quite different in both content and style from other logic books.

Peter SmithThat’s a good question. But it is far too long since I looked at Manin’s book for me to really give an opinion off the top of my head. On my list, perhaps, as something else I should look at again before the TYL Guide 2020! ;)

AndrewHello Kind Sir,

Thank you for this wonderful guide. I’m curious about Smullyan’s book, A Beginner’s Guide to Mathematical Logic, published just a few years back. Any take on whether it too may be a good place to start before leaping into the TYL guide?

Cheers!

Peter SmithOn p. 28 I say

That strikes me still as about right: a good supplement, but not the best starting point.

Remedial LogicianJudith Roitman’s Introduction to Modern Set Theory book is no longer available at link provided in the guide but I found it by googling her name + set theory + pdf. I think the source is an University in Israel but I cannot read Hebrew so beats me. Thanks for your help!

JawadHi

First of all appreciate the effort by you professor Smith for the guide.specially for students like me who were contacting all the professors around the best philosophy departments.but they accomplish nothing except empty answers.i wanted to say the be sure the guide won’t be pointless and students like me (which live in countries without any mathmatical logic fields presented in the universities)will try enlighten and broaden their understanding by the use of it . At last i wanted a bit help with fuzzy logic subjects if it is possible .maybe my request will be irrelevant but this is the only place i found my answers in it.

Big thanks Professor

Peter SmithFuzzy logic isn’t my thing at all. I do recall

Fuzzy Sets, Fuzzy Logic, Fuzzy Methodsby Hans Bandemer and Siegfried Gottwald seemed helpful at its level. But that book is about 20 years old, and is a couple of steps up from really introductory. I don’t know what’s a good modern intro.JawadHi again professor

I moved according to your list in the logic study program and almost i read all books mentioned till i got to chapter 4.1

2 books Jane Bridge’s very compact Beginning Model Theory and Maria Manzano, Model Theory are realy hard for me to obtain specially no amazon and such markets available in my country and even if amazon was available due to differences in my country’s currency and dollar it will be ultra expensive if i want to order them from eu or us markets( I didn’t find any pdf versions of mentioned books) .i will be really grateful if u suggest some other alternatives As the same level as these two or some other cheaper ways to get access to these books so i can keep going according the program.

Thanks for your help

Francis LeeDear Prof. Smith,

Daniel Cohen’s ‘Computability and Logic’ is quite interesting. (actually you metioned it in a footnote in IGT2)

I once read a book by Kfoury, Moll, Arbib, using the so called ‘while-language’ as the model of computation. And they gave a proof that ‘every GOTO program can be mimicked by a WHILE program’ (as an exercise).

But unfortunately, in the other versions of while-language, instructions are even more restricted, so we can’t immediately get the result above.

But in Cohen’s book, if we restrict the total register used in a computation then using exercise 5.7 we can prove ‘every GOTO program can be mimicked by a WHILE program given an upper-bound of the input length’.

This book lacks depth, but I found it’s better than other books on elementary computability for CS students. And his concise treatment of Hilbert’s tenth problem is also impressive.

Xavier NortonThanks a lot for doing this! I have a few questions ..

1 – I like that graph/ illustration for the cover page – is there any hidden meanings /logical truths or allusions behind it? And, may I use it for art purposes (with attribution, of course, but whose?)

2 – I don’t really have the time to study this in depth .. but as a former mathematician, my one and only semi serious encounter with logic was reading the verbose first book of Bourbaki’s Book 1 (maybe called introduction to set theory). If you are aware of it, how would this be classified in the overall scheme of things logic?

3 – reading some of Einstein’s early papers, I was struck by how much it all was relying on thought experiments. How would this sort of logic be classified?

4 – and finally, in part re Einstein and the advent of (modern) computing, software languages and AI (generalized or not, but especially Alpha Go Zero) – what do (modern) logicians have to say?

Thanks !

Michael RoseDear Professor Smith,

Do you have an opinion on the online introduction to logic course offered by Stanford?

The link is http://intrologic.stanford.edu/homepage/index.html.

Peter SmithI don’t know it well — but from what I’ve seen of it, I can’t say I like it. (For example, its Fitch-style proof system is horrible!) Not a “best buy”, certainly!

DanielDear Professor Smith,

I am a graduate student in mathematics, looking to learn some computability theory.

What I am looking for is a book which is on the one hand very exhaustive, and on other hand can be read as be read as a textbook.

I’m vary flexible on the requirement that it should be possible to read it as a textbook (“possible” is the key word here). Namely, I’m o.k. if the exposition is a bit rough. What is important to me is that it at least in principle, chapters in the book only rely on those that came before them. Also, it should have exercises.

The best comparison is probably Jech’s 2003 “Set theory”, which I used (two thirds of) to learn some set theory. I’d be happy to find a book on computability theory that has a similar macro-level structure.

Do you have any recommendations?

Thankyou very much!

Peter SmithS. Barry Cooper,

Computability Theory(2004) is very well thought of and should match your requirements!PouriaHi everyone

Does anyone know how I can possibly get my hands on the solutions for all exercises in Ian Chiswell and Wilfrid Hodges? Or any other equivalent book from the guide? Because I’m almost finished with An Introduction to Formal Logic now and the solutions on the website were very very helpful with the self-study. Thanks.

Peter SmithThis is a very good question to raise. And it would be a good addition to the Guide to add indications of which books have (some) answers to exercises. I’ll try to add some indications to the next version of the Guide.

Peter SmithOne thing I should add to TYL is more explicit detail on when and where books have answers to exercises: good point.

SteveDear Prof. Smith,

I wanted to ask if you could recommend any book that takes a top-down approach to logic, starting with the greatest generality possible and only once those foundations are rigorously established moving on to more specific and applicable material.

The paper “General Logics” from Prof. Meseguer, freely available at “https://courses.engr.illinois.edu/cs522/sp2016/GeneralLogics.pdf”, gave me the hope that similar approaches might have been taken elsewhere as well, and that over the 30 years since the paper’s publication they might have been reconciled in a comprehensive treatise.

The paper itself takes a category-theoretical view and introduces institutions, entailment systems, and of course logics, in awe-inspiring generality. It was, even if very hard to read for a relatively inexperienced undergraduate like myself, immensely satisfying, to get a birds-eye view on the topics that were being introduced in my Intro to Gödel’s class with little context and great specificity.

I tried my luck with a few classics (Kleene’s, Shoenfield’s, Barwise’s Handbook, and basically all other books from your TYL “big books” guide that I could get my hands on), never going past skimming the first few chapters, realising that they just weren’t what I’m looking for.

With deep gratitude for your time,

Steve

Peter SmithShort answer is that I really don’t know anything much about Meseguer’s sort of project! (I notice, though, that there are logical systems that don’t have at least one of the monotonicity and transitivity conditions which Meseguer requires. So does that mean his idea of general logic is not general enough? Or does it mean that the appropriate level of generality we go for is going to be interest-relative: and going for the “greatest generality possible” we may end up with something too general to be very interesting?

Stephen HarrisThis is such a valuable resource, Dr. Smith. I’m curious: are you aware of any mathematicians out there who have put together a study guide similar to your own that helps those interested in mathematics as well to build their knowledge, step-by-step, from more basic mathematics to some of the most advanced?

I’m using your guide to learn with a group right now, and while I have stumbled into a few reading lists for mathematics, none of them are quite like the richly annotated guide that you have provided with several carefully curated volumes to assist the learner at each stage. Have any of your colleagues put such a thing together for interested laymen, like me, who are interested in studying mathematics and logic on our own?

Thanks for your help!

Peter SmithThanks, glad you are finding TYL helpful. I don’t know of comparably detailed resources on other areas of mathematics — but to be honest, I haven’t really looked around much!

SjruujThere are a few, here’s one https://m.imgur.com/a/JGWZqVp.

4chan’s sci has a list of materials but they’re more disparately organized https://4chan-science.fandom.com/wiki/Mathematics.

JeanGreat list of resources on many topics: https://github.com/rossant/awesome-math

Peter SmithThanks — I didn’t know about this list and there are indeed some good-looking resources here.

J. David VellemanThe interactive online textbook BLOGIC is moving to http://www.learn-logic.org. Graded online quizzes are also available, on request.

PetrThank you very much for the guide, Dr. Smith!

I was wondering whether you have anything to say about Alexander Zinoviev and his contributions to logic, in particular his “Complex Logic” and “Logical Physics”.

I don’t know if all his ideas are considered outdated at the moment, or perhaps too crazy, or maybe both.

Are you aware of any of his work that might merit inclusion in the guide?

Thank you very much once again.

Peter SmithI really don’t know anything about this work. Is there a pointer to short piece that would give the headline news?

PetrI’m afraid there are only 2 (maybe 3) works translated into English (Zinoviev was a Soviet logician who was active in the West in the 60s-80s), and neither of them are articles or in any sense short pieces.

One of them is the “Foundations of the logical theory of scientific knowledge (complex logic)” (1973, Reidel) and the other is “Logical Physics”, published under the auspices of the “Boston Studies in the Philosophy and History of Science” in 1983. A short abstract, however, may be found on the Springer website with regards to Logical Physics.

Previously the author had published a review of almost all extant multivalued (polyvalential) logics (a translated version, “Philosophical Problems of Multivalued Logic” was published in 1963), on which further work was based.

The gist is that the author was attempting a general semantics of science, to bridge the gap between logic and the empirical sciences, with his own concept of logic.

Zinoviev was somewhat of a celebrity in the West back in the day, on many accounts (fiction author, political dissident, logician), although I don’t know how much of his legacy remains today.

Ultimately, one of the main reasons I myself have chosen to undertake the study of logic (and thus have come across your Guide), is to one day be able to absorb his work, but I have a long way to go yet.

I’d be interested to hear your thoughts.

Thank you very much.

HMHi Dr. Smith, not sure if you have already known it but Halbeisen and Krapf have published a new book (https://www.springer.com/gp/book/9783030522780) covering a number of areas that interest most readers of your blog. Maybe you want to have a look and comment on it?

Peter SmithOh heavens, yet another book! :)) I didn’t know about this one, thanks for the pointer! I’ll take a look ….

HMAlso, if it is not already known to you, the name of the author of Real Analysis through Modern Infinitesimals was misspelt. It should be ‘Vakil’, not ‘Vakin’.

HellerDear Dr. Smith, could you tell me what you think about Mathematical Logic: A Course with Exercises by Ren´e Cori and Daniel Lascar? I’m looking for a book that introduces some sort of concatenation theory in order to buttress some common syntactic claims. The book I’ve mentioned provides it but I would like to know whether there are other books which do this (maybe better than that one).

Peter SmithI think back in the day, in some earlier version of the Guide, I did mention Cori and Lascar. I’m not sure now why they have dropped out of sight! Thanks for the reminder, and I’ll refresh my memory of their book(s) before the next revision of the Guide.

ButteredBaconI think I found a book which seems to cover a wide range(therefore brisk), named (Philosophical and Mathematical Logic) by Harrie De Swart.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://core.ac.uk/download/pdf/326762898.pdf&ved=2ahUKEwj0yMyw5qv0AhU7zIsBHZFZCEIQFnoECCYQAQ&usg=AOvVaw3jgYFvcutKicHgE8yHnK2p

Can I ask what your impressions might be when skimming this book?

Peter SmithI do positively mention De Swart’s chapter on intuitionism. Skimming other chapters, they looked OK, maybe a bit idiosyncratic. But not “best buys”!

BUTTEREDBACONOops, there was! Thank you!

Jonathan RadaDr. Smith, do you have an opinion on the books Deduction by Daniel Bonevac and Logic, Sets and Functions by Bonevac, Asher, and Koons? Bonevac also has a book called Simple Logic, but as the name implies, it’s probably too rudimentary for this site.

Peter SmithI think both books are really for beginning philosophy students, so a level or two below the purview of the Guide. And, to be honest, I wouldn’t particularly recommend either of them even for their intended audience. For a start, I really don’t like their treatment of natural deduction.

JM ZThe latest version of the study guide is simply brilliant! Thank you so much for keeping it updated!

Peter SmithMany thanks! It has been quite fun to put together, but is is always good to hear when someone finds it useful!

Cheng Yui ToHi, your guide is amazing. My graduate recursion theory class last semester used Soare’s textbook ‘Turing computability: Theory and Application”. I suppose this one should be pretty well-known, would you include it in your guide as well?

Peter SmithMany thanks for this — I confess I didn’t know about Soare’s new book. I’ll try to take a look at it before while I finish revising the Guide.

Raphael Farias.Perfect guide.

I am so grateful that there is such a good, kind and competent human being as to build and publish for FREE such a comprehensive, consistent and complete guide as this one. My faith in humanity is restored.

Thank you, Peter. I wish to delve into our dear vast ocean of logic, seeking to find its secret gardens and, surely, your guide will be the light that illuminates my path.

These few words are my humble form of thanks. I will buy all your books as a second way of gratitude. Know that you helped a soul of our world a lot.

THANK YOU!

ZebedeeI would echo so many of the comments posted here.

It is so wonderful in this age of ‘monetise everything’ to have such quality so generously given to all who wish to enrich their lives by examining this important topic.

thank you very, very much!

JustinHey Peter thank you very much for your guide!

What do you think of The two books by Patrick Suppes

Introduction to logic and Axiomatic set theory.

Personally I really like them clearly that’s all that matters, but still just thought I’d ask for your opinion on them!

Peter SmithSuppes’s

Intro to Logic(1957) doesn’t get a mention inBeginning Mathematical Logicas it is really at quite an elementary level. It was a decent book in its day, but there a quite a few better elementary books to choose from now. And I’d certainly recommend supplementing it, even while staying at that level, with a more modern treatment of a natural deduction system.As for his

Axiomatic Set Theory, I do very briefly mention it inBMLat the start of §7.7. I say “Clear and straightforward as far as it goes: but there are better alternatives now.” — and I don’t have anything very useful that I can quickly add to that!JustinThank you sir! I am definitely buying C&I that you talk a lot about in your guide! Excited to study it out of interest and as a hobby!

GetahunProf Smith, thank you very much for your guide.I just finished ur IFL which was a pleasure as result of core concepts being explained gradually and clearly. After this what logic book shall I dive into as philosophy student??

Peter SmithIt is very difficult to give general advice for

philosophystudents about this, because it depends so much on (i) the direction your philosophical interests are taking you, and (ii) how mathematically minded you are. Of the topic areas covered in the Guide, I think the computability/Gödel’s Theorems theme is a particularly fun one for phil. students to meet sooner rather than later, because it is technically relatively accessible and conceptually interesting.autoI think chapter 4 of van Dalen’s book is the one covering model theory, not chapter 3, at least in the 5th edition.

CameronCan you recommend a study guide for “baby logic”? All the resources I’ve found so far start at an advanced level, and I’d like a basic introduction.

Peter SmithIn §1.5 of the Study Guide, I recommend a couple of elementary books: there’s my own

Introduction to Formal Logic(2nd edition, CUP, 2020; now freely downloadable from logicmatters.net/ifl), or Nicholas Smith’s excellentLogic: The Laws of Truth(Princeton UP 2012).”MehmoodI am a student of Analysis and I really want to understand Gödel’s Theorems rigorously but I did not take any course about Formal logic and Philosophy of mathematics. Can I understand Gödel’s Theorems formally without taking the courses of formal logic and Philosophy or I must have to take these courses?

And also suggest me books about Gödel’s Theorems which deals formally.

Peter SmithYou can understand Gödel’s Theorems without taking a full course on formal logic, so long as you have a little basic background knowledge. So, for example,

Gödel Without (Too Many) Tearsrequires minimal prior logical knowledge.EyobHi Peter,

Thanks so much for the work you’ve done to put together your red books, and especially “Beginning Mathematical Logic”, they’ve been a huge help to me in my grad program.

Just wanted to follow up and ask about your thoughts regarding a recommendation I got for a modal logic: Brian Chellas’ Modal Logic. I haven’t seen that you’ve discussed it anywhere. I’d love to hear what you think it’s strengths and weaknesses are, and with what you’d supplement those weakness with.

Thanks again!

Peter SmithI confess I have hardly glanced at Chellas’ book since it came out half-a-lifetime ago. As far as I recall, it is perfectly respectable but not

especiallyinviting (perhaps a bit old-school in style). But pulling my copy of the shelves and blowing off the dust, I guess on a quick glance that I should revisit it properly before next updating BML. I can’t offer a “strengths and weaknesses” assessment straight off the bat, though!NedThanks for this fantastic guide!

Would you consider including some material on linear logic in the next iteration? Aside from its inherent logical interest, the topic has become important in computer science and linguistics. It gets some attention, of course, in the books by Troelstra & Schwichtenberg and Restall that the guide already mentions – but I think more tailored suggestions would make an improvement.

Some resources which I wish I’d known about when first approaching the topic: Troelstra’s

Lectures on Linear Logic, Wadler’s “A Taste of Linear Logic,” and Frank Pfenning’s online course notes. They make for a friendlier introduction than Girard’s original papers.You may think the topic is too advanced, or too marginal, to earn space in a guide about

beginningmathematical logic that’s already close to 200 pages. If that’s so, fair enough. But I bet there’s a number of readers out there who would appreciate some guidance.SDHave you ever looked into Classical Mathematical Logic by Richard L. Epstein? I happened to stumble on this book in the library, and it seems like it’d be worth reviewing. This is one volume in a series called The Semantic Foundations of Logic. The other two volumes are titled Propositional Logics and Predicate Logic, respectively.

https://books.google.com/books?id=-7HpkQRvQhoC&printsec=frontcover#v=onepage&q&f=false

https://www.advancedreasoningforum.org/publications

SDI found a book called Computability: A Mathematical Sketchbook by Douglas S. Bridges. It’s a very compact book. It might be a bit too dense for the beginner, but it has lots of solutions to exercises in the book. Maybe it’s a good way to review or get another perspective on the material after working through a book like Cutland? I would love to hear your thoughts.

SDThere is a new edition of “First-Order Modal Logic” by Fitting and Mendelsohn that was very recently released in 2023. It is available through Springer at this link: https://link.springer.com/book/10.1007/978-3-031-40714-7.

Peter SmithThanks — I’m on the case :). I’ve already read some with enjoyment and it will certainly be featuring in the next version of the Guide.

Joe PedersenProfessor Smith,

I am enjoying “An Introduction to Formal Logic” very much. I think it’s the clearest book I’ve ever read on a mathematical topic (if I’m allowed to call logic that). Although I’m only half way through it, I’m already wanting to go on to Second Order Logic next. However, your guide seems to recommend a chapter from one book, or a chapter from another, possibly ignoring certain sections, and the ones I’ve looked at seem to cover other material: model theory, proof theory, 3rd and higher order logic. Also, some seem to mention presenting non-standard views on the topic.

Is there any book focused only on Second Order Logic which describes the subject almost as clearly and completely as your book does First Order Logic? I’m hoping that there is one book that you can recommend I use after finishing “An Introduction to Formal Logic.” If not, can you please write one ;-)

Peter SmithThanks for the friendly remarks about

IFL!I’ll be updating the

Guidelater in the year, all being well, and it is on my “to do” list to look around to see if there is now something better that I can recommend on second-order logic. But in the meantime, if you really do want a book “focused only on Second Order Logic”, then Shapiro’s remains the go-to option. It is very good.SDHave you taken a look at Handbook of Practical Logic and Automated Reasoning by John Harrison? It’s an introduction to automated theorem proving but done in conjunction with many of the usual topics for a first course in mathematical logic.

P.S. How do you italicize or bold text in the comments?

Peter SmithThanks for the pointer to the book, I’ll take a look in due course.

I’ll investigate whether I can add basic Markdown functionality — *italics* for

italics, for example — to comment boxes.Html code works, e.g. <em > .. </em> for italics, but is clunky

Peter SmithBasic Markdown

nowenabled …as here. Thanks for the prompt!