# Teach yourself logic, #2: Modal logic

An expanded, improved, version of the full Guide to teaching yourself various areas of logic is now here.

So let’s move on to looking at books to read on modal logic.

The ordering of some of the instalments here is necessarily going to be a little bit arbitrary. But I’m putting this one next for two reasons. First, the basics of modal logic don’t involve anything mathematically more sophisticated than the elementary first-order logic covered in the first instalment. Second, and more much importantly, philosophers working in many areas surely ought to know a little modal logic, even if they can stop their logical education and manage without knowing too much about some of the fancier areas of logic we are going to be looking at later.

Again, the plan is to offer a list of books of increasing range and difficulty, choosing those which look most promising for do-it-yourself study. The place to start is clear, I think:

1. Rod Girle, Modal Logics and Philosophy (Acumen 2000, 2009). Girle’s logic courses in Auckland, his enthusiasm and abilities as a teacher, are justly famous. Part I of this book provides a particularly lucid introduction, which in 136 pages explains the basics, covering both trees and natural deduction for some propositional modal logics, and extending to the beginnings of quantified modal logic.

Also pretty introductory, though perhaps rather brisker than Girle at the outset, is

1. Graham Priest, An Introduction to Non-Classical Logic (CUP, much expanded 2nd edition 2008): read Chs 2–4, 14–18. This book — which is a terrific achievement and enviably clear and well-organized — systematically explores logics of a wide variety of kinds, always using trees in a way that can be very illuminating.

If you start with Priest’s book, then  at some point you will need to supplement it by looking at a treatment of natural deduction proof systems for modal logics. A possible way in would be via the opening chapters of

1. James Garson, Modal Logic for Philosophers (CUP, 2006). This again is intended as a gentle introductory book: it deals with both ND and semantic tableaux (trees). But — on an admittedly rather more superficial acquaintance —  this doesn’t strike me as being as approachable or as successful.

We now go a step up in sophistication:

1. Melvin Fitting and Richard L. Mendelsohn, First-Order Modal Logic (Kluwer 1998): also starts from scratch. But — while it should be accessible to anyone who can manage e.g. the Hodges overview article on first-order logics that I mentioned before — this goes quite a bit more snappily, with mathematical elegance. But it still also includes a good amount of philosophically interesting material. Recommended.

Getting as far as Fitting and Mendelsohn will give most philosophers a good enough grounding. Where, if anywhere, you go next in modal logic, broadly construed, would depend on your own further concerns (e.g. you might want to investigate provability logics, or temporal logics).  But if  you want to learn more about mainstream modal logic, here are some suggestions (skipping past older texts like the estimable Hughes and Cresswell, which now rather too much show their age). Though note, further technical developments do tend to take you rather quickly away from what is likely to be philosophically interesting territory.

1. Sally Popkorn, First Steps in Modal Logic (CUP, 1994). The author is, at least in this possible world, identical with the mathematician Harold Simmons. This book, entirely on propositional modal logics, is written for computer scientists. The Introduction rather boldly says ‘There are few books on this subject and even fewer books worth looking at. None of these give an acceptable mathematically correct account of the subject. This book is a first attempt to fill that gap.’ This perhaps oversells the case: but the result is still  illuminating and readable — though its concerns are not especially those of philosophers.
2. Going further is Patrick Blackburn, Maarten de Ricke and Yde Venema, Modal Logic (CUP, 2001). One of the Cambridge Tracts in Theoretical Computer Science. But don’t let that put you off. This text on propositional modal logics is (relatively) accessibly and agreeably written, with a lot of signposting to the reader of possible routes through the book, and interesting historical notes. I think it works pretty well. However, again this isn’t directed to philosophers.
3. Alexander Chagrov and Michael Zakharyaschev’s Modal Logic (OUP, 1997) is a volume in the Oxford Logic Guides series and also concentrates on propositional modal logics. This one is probably for real enthusiasts: it tackles things in an unusual order, starting with a discussion of intuitionistic logic, and is pretty demanding of the reader.  Still, a philosopher who already knows just a little about intuitionism might well find the opening chapters illuminating.
4. Nino B. Cocchiarella and Max A. Freund, Modal Logic: An Introduction to its Syntax and Semantics (OUP, 2008). The blurb announces that ‘a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight’. That sounds hopeful, and the authors are right about the unusually wide range. As noted, the previous three books only deal with propositional logics, while many of the more challenging philosophical issues about modality tangle with quantified modal logic. So the promised coverage makes the book potentially of particular interest to philosophers. However, when I looked at this book with an eye to using it for a graduate seminar, I didn’t find it appealing: I suspect that many readers will indeed find the treatments in this book uncomfortably terse and rather relentlessly hard going.
5. Finally, in the pretty unlikely event that you want to follow up even more, there’s the giant Handbook of Modal Logic, ed van Bentham et al., (Elsevier, 2005). You can get an idea of what’s in the volume by looking at the opening pages of entries available online here.

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### 21 Responses to Teach yourself logic, #2: Modal logic

1. wo says:

Great series! I would add Johan van Benthem’s Modal Logic for Open Minds early on in the list, perhaps after 1 and 2. It provides a gentle but illuminating introduction to contemporary (Amsterdam-style) modal logic, highlighting many features that are still often neglected by philosophers.

• Peter Smith says:

I didn’t know of this book, so many thanks for the recommendation: I’ve had a quick look, and indeed it looks interesting. Though I’d say it should feature a bit further down a revised list than #3: diving in and talking about bisimulations so early would make it a bit conceptually tough for some readers, I’d have thought. But again, thanks for this.

2. Marco says:

The web page related to the handbook of modal logic contains a preview of the essays, but not the full texts.

• Peter Smith says:

Oops — you are right, of course! I’ve changed the post to reflect that!

3. Rowsity Moid says:

Robert Goldblatt, Logics of Time and Computation, which I think is now available as downloadable pdf.

4. Rowsity Moid says:

BTW, would you want to mention Boolos’s The Logic of Provability in this section?

• Marco says:

Boolos wrote a milestone in the history of contemporary logic: a perfect bridge between modal logic and proof-theoretic analysis.

• Peter Smith says:

Boolos will most certainly feature somewhere! (Maybe I should put a reference forward in this section.)

• Rowsity Moid says:

It occurs to me that if you’re considering books that connect one area with another, then Smullyan and Fitting’s Set Theory and the Continuum Problem would connect modal logic and set theory, because they use modal logic for their version of forcing. (It may also be the most accessible treatment of forcing that’s appeared in a book.)

5. Daniel says:

Thank you very much for these lists. The terrain of different areas and systems of logic is a little difficult to map for a complete outsider. The lists are much more helpful – and up to date – than the reading lists to courses in logic(predicate logic) and modal logic in my university. I shall most certainly take the recommendations in consideration when(more or less) diving into this field.

Best regards from Norway

• Peter Smith says:

Thanks for this: always nice to know when people are finding this kind of post potentially useful!

6. Rowsity Moid says:

Another book it might be worth mentioning, either positively or negatively (it’s been a while since I’ve looked at it), is Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic by JC Beall and Bas van Fraassen. It is reasonably short, and reasonably affordable, so that someone looking for a modal logic introduction might consider it.

• Peter Smith says:

Thanks! I’d clean forgotten about this book, even though it is on my shelves a couple of yards away … I’ll have another look at it before I write the revised version of the modal logic list.

7. KT says:

As a beginner I’ve more than once heard the remark, “Hughes & Cresswell’s NML 1996 is outdated because it employs axiom systems.” But the remark often stops there. More substantiation would be helpful (so that, e.g. a beginner like me could as well avoid certain approach, not just this or that book). I can think of — just suggestions: I’m neither defending nor opposing NML — 1) terrible typesetting; 2) pedagogically superseded (but how?); 3) tableaux bring you more directly to completeness etc.; 4) it’s written before some so-and-so crucial notion was fully understood; 5) certain of its philosophical stances no longer tenable or 5) (simply) it just looks too old fashioned. Any idea?

• Peter Smith says:

The first thing to say is that I used to love the 1968 version of Hughes and Cresswell. That book taught modal logic to generations of logic students: in it’s time it was a notable and very admirable achievement.

Now, the first part of the 1996 version sticks pretty closely, I think, to the 1968 original (I no longer have my copy of the earlier book so I can’t do a close comparison — but I’m going on memory, and what H&C say in their introduction to the later version). As you note, both in 1968 and 1966, the introduced modal proof systems are built on a very old-fashioned axiomatic presentation of non-modal logic. That basis will look both very unfamiliar and very clumsy to nearly every modern reader. So that’s a major strike against the book, when there are now introductions based on the kind of trees-systems or natural deduction systems familiar to the beginning logician. For example, if you are used to my tree-based intro logic book, then you can smoothly progress to reading Graham Priest’s book without skipping a beat. By contrast, you’ll have to work a lot harder to get into H&C. In that sense, indeed, the book is “pedagogically superseded” and “just looks too old fashioned”. Of course, between 1968 and 1996, the book is updated in all kinds of ways — but not so distinctively, I think, for it to trump more recently written books.

• KT says:

Thanks Peter. That’s very helpful indeed.

8. Tony Roy says:

For what it is worth, my (longish), “Natural Derivations for Priest” AJL (2006)develops natural derivation systems for logics in the first edition of Priest — and so for sentential versions of the systems in the longer edition of his book. Online at, http://philosophy.unimelb.edu.au/ajl/2006/

9. Rick Taylor says:

As a mathematician with no background in modal logic, I found Fitting and Mendelsohn’s First Order Modal Logic just about perfect for me. I was wondering what to read next, and was considering Kenneth Konyndyk’s “Introductory Modal Logic” as the reviews on amazon said it was especially good for discussing the philosophical issues involved. Would you recommend that, or another book that emphasizes more philosophical issues.

• Peter Smith says:

I don’t know Konyndyk’s book at all, so I can’t comment on that. For discussions of philosophical issues, Rod Girle’s book has some introductory discussions. For more detailed discussions there are some good online articles at the Stanford Encyclopedia of Philosophy e.g. on modal fictionalism, actualism, etc.