Teach Yourself Logic

Note: Beginning Mathematical Logic, Part  I is currently being updated — you can see draft revised chapters being posted on the blog!

A Study Guide (and other Book Notes)

About the Study Guide

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 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 went through a lot of iterations over the years, ending up in a somewhat untidy and uneven state. It’s time for an end-to-end rewrite. So I’m working on the (re-titled) Beginning Mathematical Logic: A Study Guide. Chapters 1 to 9, forming Part I of the Guide, are newly (re)written, and are in one PDF. The remaining chapters are as yet unrevised, taken from the mid-2020 version of TYL, form a separate PDF. (I’m leaving these later parts  online with some reluctance, as they very much need improving). As the weeks and months go by, the first PDF should slowly grow as more new chapters are added, and the second PDF will correspondingly shrink. Or at least, that’s the plan!

Mathematical logic is indeed a big subject, and different people have different backgrounds/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 TYL was so long: its replacement is even longer as I’m adding some sections giving a quick overview or checklist of topics in various areas.

About the Appendix and Book Notes

Most of the recommendations are in the Guide are for books which focus on particular areas.  But I have added an Appendix reviewing some of the big multi-area textbooks on mathematical logic. The same reviews can also be found in the suite of Book Notes, which also comment on various other books on logic and the philosophy of mathematics.


It goes without saying, of course, that all constructive comments and suggestions continue to be most warmly welcomed. Many thanks, in particular, to all those who have earlier sent comments which are now deleted because I’ve taken up (or plan to take up) the suggestions in newer versions of the Guide.

90 thoughts on “Teach Yourself Logic”

  1. Rodrigo de Almeida

    Dear 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

  2. Samuel T. X. Khoo

    I 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?

      1. Peter — 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

  3. I’ve noticed a new category theory book that takes a different sort of approach: Category Theory for the Sciences by 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.

  4. I’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.

    1. There 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.

  5. I 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.

      1. Yeah, 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.

  6. As 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.

    1. Well, 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.

  7. I 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.

  8. First 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?

    1. I 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!

  9. Thank 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!!

    1. I 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.

  10. Christofer Koch

    Thank 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.

    1. Church’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!

  11. What 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’.

  12. Shealton George

    Thank 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?

  13. Hey 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.

  14. I 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?

  15. This 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?

    1. Oooh 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!

  16. I 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.

  17. I 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?

  18. I 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 :)

  19. Hi,
    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

  20. Hello, 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.

  21. I 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.

  22. Are 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?

    1. In 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!

  23. There 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’.

  24. Mr 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!

  25. hey 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!

  26. I’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?

  27. Maximilian Stöckel

    Hi!

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

    Thanks for this wonderful resource!

    1. I 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.

    1. Thanks 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.

      1. Dear 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.

    1. Yes … 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.

      1. For 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.

  28. Jaikrishnan Janardhanan

    Dear 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.

    1. That’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! ;)

  29. Hello 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!

    1. On p. 28 I say

      Smullyan’s … A Beginner’s Guide to Mathematical Logic (Dover Publications, 2014) is more conventional [than his puzzle books]. The first 170 pages are relevant to FOL. A rather uneven read, it seems to me, but … perhaps an illuminating supplement to the texts recommended above.

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

  30. Remedial Logician

    Judith 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!

  31. Hi
    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

    1. Fuzzy logic isn’t my thing at all. I do recall Fuzzy Sets, Fuzzy Logic, Fuzzy Methods by 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.

  32. Hi 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

  33. Dear 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.

  34. Thanks 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 !

    1. I 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!

  35. Dear 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!

  36. Hi 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.

    1. This 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.

  37. Dear 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

    1. Short 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?

  38. This 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!

    1. Thanks, 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!

  39. Thank 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.

      1. I’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.

      1. Also, 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’.

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