Suppose you want to learn more logic after working through *An Introduction to Formal Logic* (or a similar book). What should you look at next? Three general points:

- It is much the best to read a series of books, at increasingly more sophisticated levels, but with a
*lot*of overlap of content along the way. Having things explained by different authors, with different emphases, and at different levels of abstraction, can be a great aid to real understanding, and is almost never time wasted. - It can also help a lot to read every chapter twice: once quickly, skipping over proof details, to get the main ideas, the big picture; then again more carefully, joining up all the dots.
- Remember, mathematics is not a spectator sport. Do some of the exercises as you go along to check understanding and help fix ideas.

For revision, read

- Richard Jeffrey,
*Formal Logic: Its Scope and Limits*(McGraw Hill, 2nd end 1981). [I love this book: in fact you can think of*IFL*as Jeffrey spoilt by being made plodding and with everything spelt out more laboriously! In my experience, however, many beginners do find Jeffrey a bit too brisk as a self-study book. But now you are no longer a beginner, you should be able to read him with pleasure and instruction.]

The following isn’t exactly revision, but can be warmly recommended as a way of hammering home a range of ideas that any logician should be happy to work with:

- Daniel J. Velleman,
*How to Prove It*(CUP, 1994). [This is written for beginning maths students, but count yourself in. Various ideas and notation from logic and elementary set theory are explained and put to work. Nicely done.]

Then the first books that come to mind as ‘next steps’ are

- David Bostock,
*Intermediate Logic*(Clarendon Press, Oxford: 1997). [As the title suggests, written as a second-level book, though in fact much of it doesn’t go much further than*IFL*. However it does cover sorts of proof system other than trees, and is nicely accessible.] - Ian Chiswell and Wilfrid Hodges,
*Mathematical Logic*(OUP, 2007). [Don’t be scared off by the title: this is beautifully accessible and doesn’t go a*lot*further than*IFL.*] - Neil Tennant,
*Natural Logic*(Edinburgh University Press, second edn 1990). [Seemingly out of print but should be in your university library. Notionally for beginners but tougher than than*IFL*. But a classic presentation of logic via natural deduction.]

I’d then mention, as just a bit more advanced/terse

- Wilfrid Hodges, ‘Elementary Predicate Logic’, in
*Handbook of Philosophical Logic, Vol. 1*, ed. by D. Gabbay and F. Guenthner (Reidel, 1984-89). Also an expanded version of this appears in the 2nd edition of the Handbook. [This is a beautifully done presentation of basic first-order logic.] - Dirk von Dalen,
*Logic and Structure*, (Springer, 4th edn. 2004) [Something of a modern classic. A rich treatment of natural deduction, though this book shouldn’t be your first ever encounter with that approach.] - Herbert B. Enderton,
*A Mathematical Introduction to Logic*(Academic Press, 2nd edn. 2002) [Another modern classic.] - Christopher Leary,
*A Friendly Introduction to Mathematical Logic*(Prentice Hall, 2000). [Chs. 1–3 are indeed a very friendly and helpful introduction to first-order logic.]

By the time you get to these last items, you’ll perhaps need a bit of ‘mathematical maturity‘, as they say. There is no magic route to this desirable state! But it should come from enough immersion in some maths (in our case, in some serious logic books) and with sufficient hard work at details and exercises. And if you’ve been managing some of those last books, you should also find the following beautiful classic accessible and illuminating

- Raymond Smullyan, First-order Logic (Springer-Verlag, 1968: Dover, 1995).

After getting on top of first-order logic, then wide fields open up that traditionally belong to mathematical logic broadly understood. There’s modal logic, and other non-classical logics. The theory of computability and recursive functions. Set theory. But all that really is another, further story — though there are some pointers here in this reading list which I recently edited, and I’ll finish here by mentioning one nice book that introduces elements of both set theory and recursion theory that might make a very good stepping stone into those topics:

- Moshe Machover,
*Set Theory, Logic and Their Limitations*(CUP, 1996).

For a longer and much more detailed Guide to teaching yourself some serious logic, see here.

I used Jeffrey for the baby logic for many years. Then I switched to Hodges’ little Penguin book. (The Logic Book, Bergmann et al, is another reasonable intro book, with the advantage of approximately 1 ton of exercises.) When I’m despairing of what to use for the intermediate course, I fall back to Suppes (Dover reissued it)–I like it because it covers some first-order theories, not just pure logic, and also the notion of definition. I’ve tried Sider’s book, which is odd (works well if you’re willing to expand on the too brief parts and skip a lot; not good for self-study, I think.) Smullyan’s book (Logical Labyrinths) is his wonderful First-Order Logic book with a pedagogical overlay. Hastily done, with typos. Not sure that the first-order logic book wouldn’t actually work better.

I have fondish memories of Mates, but haven’t looked at it lately. Also in the “showing my age” category is Kalish and Montague, but students may find that notationally off-putting.

I seem to remember you recommending Christopher Leary’s

A Friendly Introduction to Mathematical Logicas well. (I don’t remember where though.) If I’m correct in this, where would you put that text on this list? Would it be supplanted by something else you mention here? Just curious.I like Suppes too. I would like to mention 2 wonderfull books: (i) Machover’s «Set theory and their limitations» and (ii) Smullyan’s «First order logic» (Dover series).

L. T. F. Gamut (a collective pseudonym for five Dutch professors, led by Johan van Benthem), Logic, Language, and Meaning (vol. 1 Intro. to Logic; vol. 2 Intensional Logic & Logical Grammar) The University of Chicago Press (1991) [translated from the original (1982) Dutch version: Logica, Taal en Betekenis]

I’ve the paperback version. They are very good. Don’t know whether they’re still around (in print).

Thanks for all these. I’ve added the indeed wonderful Smullyan

FOLto the list. I didn’t know about hisLogical Labyrinths— but Christopher Leary’s review isn’t encouraging, and others too comment on the terrible number of misprints.I was wavering about keeping Leary’s own book on the list now there’s Chiswell and Hodges. But I’ve put it back! I still admire its style and approach.

I don’t know the Gamut volumes at all: but they

arestill available e.g. from amazon.co.uk (and amazon’s “look inside” gives some rather promising glimpses).Finally, I agree about Machover’s book too, and added it.

Just checked the University of Chicago Press website. Pleasantly surprised to see that both books are available, as follows:

L. T. F. Gamut, Logic, Language, and Meaning, Volume 1: Introduction to Logic, Paperback $35.00, ISBN: 9780226280851

L. T. F. Gamut, Logic, Language, and Meaning, Volume 2: Intensional Logic and Logical Grammar, Paperback $37.50, ISBN: 9780226280882

As it is available in a softcover reprint from Dover, I wonder whether Robbin’s

Mathematical Logic: A First Coursewould be a useful addition to the list, slotted in as a bit more sophisticated and faster-paced than the Enderton text.