# Teach yourself logic, #1: Basic first-order logic

[Added later: if you find yourself at this page, you might be (more) interested by the full Teach Yourself Logic Guide which is now available.]

It is an odd phenomenon. Serious logic is taught less and less, at least in UK philosophy departments. Yet the amount of formally-informed work in philosophy is ever greater. It seems then that many beginning graduate students (if they are not to cut themselves off from working in some of the most exciting areas) will need to teach themselves, solo and by organising reading groups. But what to read?

Here then is the first of a planned series of posts covering different areas of logic of interest to philosophers. This instalment covers the basics, up to a good grasp of the elements of classical first-order logic. I’ll assume that you’ve already done a smidgin of logic of some kind (utter beginners innocent of all logic and worried by symbols might find Guttenplan’s book a useful preliminary).

Two general points I’ve made before. (a) Mathematics (and that’s what we are talking about here, to be honest!) is not a spectator sport: you should try some of the exercises in the books as you read along to check and reinforce comprehension. (b) It is much the best to proceed by reading a series of books which overlap in level, with the next one covering some of the same ground and then pushing on from the previous one, rather than to try to proceed by big leaps. Again this will help reinforce and deepen understanding as you re-encounter the same material from different angles.

OK, with that preamble off we go (and though I don’t for a moment expect nearly fifty sets of comments as with the post on fun reads in philosophy, do please add comments — either on what you’ve found works in teaching first-order logic, or on what you’ve found particularly helpful as a student yourself). All the in-print books should be in any decent university library: order them if they aren’t in yours!

1. MyIntroduction to Formal Logic (2003, 2009) is intended for beginners (and has been the first year text in Cambridge): but it in fact already goes further than seems to be covered in whole undergraduate courses in some good UK universities. It was written as a teach-yourself book. It covers proposition and predicate logic ‘by trees’. It even has a completeness proof for predicate logic, though for a beginning book that’s very much an optional extra!
2. Paul Teller’s A Modern Formal Logic Primer (1989) predates my book, is now out of print, but is freely available online. It is excellent, and had I known about it at the time (or listened to Paul’s good advice when I got to know him), I’m not sure that I’d have written my own book. Unlike my book, as well as introducing trees this also covers a (user-friendly) version of natural deduction. It is notably user-friendly.
3. David Bostock’s Intermediate Logic (1997) goes only slightly further than either Paul’s book or my own, and is rather discursive, but it is very well done (and touches on some issues such as free logic which are not dealt with in our book).
4. A notch up in sophistication of approach we find the excellent Ian Chiswell and Wilfrid Hodges, Mathematical Logic (2007). This is now getting a little more ‘mathematical’ in flavour, but should be manageable at this stage. It deals nicely with natural deduction.
5. Neil Tennant’s Natural Logic (1978, 1990) is also now out of print, but freely available from the author’s website. I still like this a lot as a text on natural deduction (Tennant thinks that this approach to logic is philosophically significant, and it shows); but it isn’t always an easy read which is why I list it at this point rather than earlier.
6. You should now be able to cope, by way of wonderful revision summary, with Wilfrid Hodges’s ‘Elementary Predicate Logic’, in Handbook of Philosophical Logic, Vol. 1, (ed. by D. Gabbay and F. Guenthner: Reidel, 1984-89). An expanded version of this appears in the 2nd edition of the Handbook.
7. Chiswell and Hodges’s book, they say, started life as teaching notes for a course based on the classic book by Dirk van Dalen, Logic and Structure (4th ed, 2004 — at this stage you can omit the last chapter). If you can cope with this book, you are doing just fine!
8. Finally, for desert, look at Raymond Smullyan, First-Order Logic (first published 1968) which is an utter classic: you should certainly now be able to read Parts I and II.
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### 24 Responses to Teach yourself logic, #1: Basic first-order logic

1. Daniel says:

I haven’t read any on your list besides the Bostock (which is very good, though certainly not for a complete beginner); I’d suggest Guttenplan’s ‘Languages of Logic’, which I worked through by myself at sixth-form. Halbach’s ‘Logic Manual’ is the current Oxford text and is fairly easy to work through alone also (with exercises online).

• Peter Smith says:

Thanks, Daniel. I should have made it clearer I was thinking of recommendations for people who had done the smidgin of logic (e.g. Guttenplan’s book) that is all that some places seem to offer. I’ll edit to clarify. As to Volker Halbach’s book, I still — surprise, surprise — prefer mine, but it would indeed make a suitable entry point.

2. Nick Smith says:

My new book Logic: The Laws of Truth, just out from Princeton University Press, is intended to be suitable for independent study. It goes from the very beginning up to the early stages of metalogic (e.g. proving some soundness and completeness results, and discussing in some detail, but not proving, the undecidability of first order logic). It also includes a healthy dose of philosophy of logic, as well as the formal material — and it covers all the major forms of proof (trees, axiomatic proofs, natural deduction, sequent calculus) in all their major varieties. It’s a long book, but the Preface sets out which sections cover the core logic that everyone should know, and which sections cover material of particular interest to certain audiences (e.g. those wishing to prepare themselves to study mathematical logic, or nonclassical logics, or formal semantics, etc.). One thing that will be helpful to independent readers is that the answers to the exercises are online (or at least will be very soon, once the final formatting is completed).

3. Great list. I really like Teller’s Primer, too. Its great advantage (besides being free) is that it is short. Many authors think that they can make the material more accessible by being prolix, whereas the only result they achieve is boring the reader. I use the Primer regularly when teaching intro courses.

4. Matthew says:

Warren Goldfarb’s Deductive Logic is superb; I used to recommend it to students who wanted something more than Guttenplan and they all loved it. It’s crystal clear, compact without being terse, and makes it all seem so easy. It’s a real joy to read.

5. Rowsity Moid says:

There’s a book I first saw mentioned on one of your blog posts (iirc the same one that first mentioned Chiswell and Hodge): Richard Kaye’s The Mathematics of Logic: A Guide to Completeness Theorems and their Applications.

I thought it was one of the more interesting into books I’ve seen.

• Peter Smith says:

I think Kaye’s intended audience is people with a certain mathematical background. For example, the book starts with chapters on König’s Lemma and on posets; the philosophy graduate student starting out learning some logic might find that a bit alarming. But I certainly agree that this is a very good book, and readers coming with enough “mathematical maturity” could love it. And even non-mathmos who having got as far as coping with e.g. Hodges’s overview article should now be able to read it with profit.

6. Clark says:

Herb Enderton’s A Mathematical Introduction to Logic is still worthy of honorable mention. It informs the beginner without condescension.

One of the virtues of this subject is that each work on the list adds some unique contribution to its reader.

• Peter Smith says:

Agreed on both counts. A bit arbitrarily, Enderton’s book, like Kaye’s, is on my draft list for a later instalment, ‘Further into Mathematical Logic’, because of the later chapters. But perhaps the first half of the book should indeed be on this introductory list too.

7. AZ says:

I don’t know if I am doing right in interpreting «serious logic» as meaning metalogic. If yes, then the only book that I can think of for self-teaching serious first-order logic is «Mathematical logic» by Stephen C. Kleene.

It was love at first reading:
«It will be very important as we proceed to keep in mind this distinction between the logic we are studying (the object logic) and our use of logic in studying it (the observer’s logic). To any student who is not ready to do so, we suggest that he close the book now, and pick some other subject instead, such as acrostics or beekeeping.»

8. Rowsity Moid says:

I looked at Jeffrey’s Formal Logic: Its Scope and Limits in a bookshop the other day, and it seemed interesting. It looked like its approach to incompleteness was to do it for 2nd-order logic, something I don’t think I’ve seen before.

There’s also Machover’s Set Theory, Logic and their Limitations, but it’s been quite a while since I’ve opened a copy.

• Peter Smith says:

I love Jeffrey’s book. In fact, my book is pretty much the first-order logic part of Jeffrey’s, done more slowly. If you can manage the fast version on self-study, great! But I wouldn’t have written my book if I hadn’t found that many readers need to go my slowly at the beginning.

Machover’s book is very nice too. But this belongs on a later ‘Further into Mathematical Logic’ list.

9. Edward says:

Dear Peter,

What do you think of J C Beall’s “Logic The Basics” and of Geoffrey Hunter’s “Metalogic: An Introduction to the metatheory of Standard First Order Logic”?

• Peter Smith says:

Beall’s book — which is discussing paracomplete and paraconsistent logics within sixty pages — is hardly the place to start. It will be on my reading list for non-classical logics, but doesn’t belong here.

Hunter’s book by contrast is mostly bang on topic (though goes further towards into e.g. Gödelian incompleteness at the end) and I used to like this book a lot. It is perhaps showing its age in some ways (which is why it wouldn’t make my current shortlist), but it certainly has its virtues.

10. Luke says:

Another intermediate-level text worth putting on this list is “Logical Options: An Introduction to Classical and Alternative Logics” by John L. Bell, David DeVidi, and Graham Solomon. It covers all sorts of interesting material, including modal, intuitionistic, three-valued, many-sorted, fuzzy, and second-order logics.

• Peter Smith says:

An intermediate text, yes. And it has some virtues (though when I taught a seminar based on it, I found myself having to write very extensive accompanying handouts). Still, I don’t think this belongs to the list “Getting your head around basic first-order logic”, so much as to another list “Going beyond the first-order”.

11. ChrisE says:

I have for some years been using Bergmann, Moor, and Nelson’s The Logic Book because I like how fully they spell things out. But as I’m choosing a book for next year, and reading the above post and also Aldo Antonelli’s comment, I’m thinking that that might not be the greatest virtue. I’m not sure how many students actually work through the fairly dense text, and I’m not sure it justifies the \$USD 130 price tag against, e.g., Teller which I enjoyed as an undergraduate. (It also has frustratingly many errors, though an errata sheet is published.) I wonder if anyone else has views on The Logic Book.

• Peter Smith says:

I find The Logic Book rather ‘dense’ (in your word) and stodgy, and I’ve never tried using it. But how did your students find it — what was the feedback in course questionnaires? I can’t imagine they found the book very enticing.

The price tag is just outrageous.

• ChrisE says:

You’re right to deflect the question towards my students’ evaluations, but while my students’ responses to The Logic Book have never been glowing, suggestions that it was overly challenging have come roughly at the same rate as they do for the likes of Hume, Mill, Ayer, Hempel, Blackburn … and so I’ve discounted them, but perhaps prematurely. And they haven’t experienced other books. For my part, TLB’s stodginess was initially a welcome change from having had to teach from Posposel when I was a teaching assistant.

It seems it’s time to revisit whether there’s a happy balance between lightness and rigor in Teller’s book, Nick Smith’s, yours, and others.

• I have used The Logic Book to teach intro courses, and know many people who have likewise used it for the same purpose. I think the book is not bad. The meta-theory chapters are irrelevant for intro classes, and the real strength of the book is the wealth of exercises to choose from. I agree with Peter that the price is outrageous, but I have also heard that the publisher is willing to put together customized version leaving out any unwanted chapters (for a reduced price).

Having said that, I find most introductory logic textbooks to be about as good, with minor differences in style of presentation etc. The subject matter has not changed in decades, for heaven’s sake, and there is no reason to push new and “improved” editions on the students.

That is why I like Teller’s Primer. It’s about as good as any other, and you can’t beat the price. (I do regret that Paul has posted answers to the exercises, which forces me to make up my own, but, hey, that’s why I am being paid the big bucks.)

• Peter Smith says:

Here’s one reason for have at least a lot of answers to exercises provided on the web. You can now set class work in the following format. (i) Do such and such exercises. (ii) Then check your answers against the model answers from the web, and self-correct your answers, noting anything you still don’t understand. (iii) Hand in your self-marked work, along with the answers to these further exercises [now you are on your own!].

Two-part Worksheets in this format (example here) can very significantly cut the amount of work needed from the poor grad students you are employing as markers!

12. Rowsity Moid says:

Colin Howson’s Logic with Trees: An Introduction to Symbolic Logic?

• Peter Smith says:

Well, surprise, surprise, I prefer my version of logic-with-trees — if only because mine has more worked examples, and we are thinking here particularly about books that work for solo study (as against a classroom text backing up lectures).

13. Chris S. says:

Stumbled across this website by accident. What about Rosen’s Discrete Mathematics and its Applications?