About the TYL Guide

Teach Yourself Logic: A Study Guide is a hundred page PDF. Who is it for? What’s in it? Is it for you?

Why this Guide for philosophers?

It is an odd phenomenon, and a rather depressing one too. Serious logic is seemingly taught less and less, at least in UK philosophy departments, even at graduate level. Fewer and fewer serious logicians get appointed to teaching posts. Yet logic itself is, of course, no less exciting and rewarding a subject than it ever was, and the amount of good formally-informed work in philosophy is ever greater as time goes on. Moreover, logic is far too important to be left entirely to the mercies of technicians from maths or computer science departments with different agendas (who often reveal an insouciant casualness about conceptual details that will matter to the philosophical reader).

So how is a competence in logic to be passed on if there are not enough courses, or are none at all?

It seems that many beginning graduate students in philosophy – if they are not to be quite dismally uneducated in logic and therefore cut off from working in some of the most exciting areas of their discipline – will need to teach themselves from books, either solo or (much better) by organizing their own study groups.

In a way, that’s perhaps no real hardship, as there are some wonderful books written by great expositors out there. But what to read? Logic books can have a very long shelf life, and one shouldn’t at all dismiss older texts when starting out on some topic area: so there’s more than a fifty year span of publications to select from. Without having tried very hard, I seem to have accumulated on my own shelves some three hundred formal logic books at roughly the level that might feature in a Guide such as this – and of course these are still only a selection of what’s available.

Philosophy students evidently need a study guide if they are to find their way around the available literature old and new: the Guide is my (on-going, still developing) attempt to provide one.

Why this Guide for mathematicians too?

The situation of logic teaching in mathematics departments can also be pretty dire. Indeed there are full university maths courses in good UK universities with precisely zero courses on logic or set theory (maybe a few beginning ideas are touched on in a discrete maths course, but that is all). And I believe that the situation is equally patchy in many other places.

So again, if you want to teach yourself some logic, where should you start? What are the topics you might want to cover? What textbooks are likely to prove accessible and tolerably enjoyable and rewarding to work through? Again, this Guide – or at least, the sections on the core mathematical logic curriculum – will give you some pointers.

True, this is written by someone who has – apart from a few guest mini-courses – taught in a philosophy department, and who is no research mathematician. Which probably gives a distinctive tone to the Guide (and certainly explains why it ranges into areas of logic of more special interest to philosophers). Still, mathematics remains my first love, and these days it is mathematicians whom I mostly get to hang out with. A large number of the books I recommend are very definitely paradigm mathematics texts. So I shouldn’t be leading you too astray.

And I didn’t want to try to write two overlapping Guides, one primarily for philosophers and one aimed primarily at mathematicians. This was not just to avoid multiplying work. Areas of likely interest don’t so neatly categorize. Anyway, a number of philosophers develop serious interests in more mathematical corners of the broad field of logic, and a number of mathematicians find themselves interested in more foundational/conceptual issues. So there’s a single but wide-ranging menu here for everyone to choose from as their interests dictate.

What does the Guide cover?

It’s assumed (if you are a philosopher) that you’ve already done a ‘baby logic’ course, and we continue at an upper undergraduate/graduate level. Mathematicians can probably dive straight in, with the first books starting at a mid-to-upper undergraduate level. We look, first, at the basic mathematical logic curriculum at entry level:

  • Basic First-Order Logic (up to and including the completeness theorem)
  • From First-Order Logic to Model Theory
  • Beginning computability theory/Gödel’s theorems
  • Beginning set theory
  • Extras: Second-order logic, intuitionistic logic

(After the initial segment on FOL, you can pick and choose your way through the other segments, in any order.) Then glancing sideways at other topics in logic — perhaps mostly of interest to philosophers — we briefly look at

  • Modal Logic
  • Other classical variants and extensions
  • Non-classical logics

(Again, you can pick and choose, or ignore, any of these.) Then, pushing on to rather more advanced work on topics in and around the main  mathematical logic curriculum, there are sections on

  • Proof theory
  • Beyond the model-theoretic basics
  • Computability
  • Gödelian incompleteness again
  • Theories of arithmetic
  • Serious Set Theory
  • There is also a supplement on Category Theory

Why is the Guide so long?

We cover a lot in the Guide and I often say quite a bit about the recommended choices, which is one reason for its length (don’t be daunted — as just indicated, you can be very selective, depending on your interests).

But there is another reason the Guide is long. I very strongly recommend tackling an area of logic (or indeed any new area of mathematics) 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 trying to proceed by big leaps.

In fact, I probably can’t stress this advice too much, which is why I am highlighting it here. For this approach will really help to reinforce and deepen understanding as you re-encounter the same material from different angles, with different emphases.

The multiple overlaps in coverage in the reading lists are therefore fully intended, and this explains why the lists are always longer rather than shorter (and also means that you should more often be able to find options that suit your degree of mathematical competence). You will certainly miss a lot if you concentrate on just one text in a given area, especially at the outset. Yes, very carefully read one or two central texts, at a level that appeals to you. But do cultivate the additional habit of judiciously skipping and skimming through a number of other works so that you can build up a good overall picture of an area seen from various somewhat different angles of approach.

What guides the choices in the Guide?

So what has guided the choices of what to recommend in the main Guide?

Different people find different expository styles congenial. For example, what is agreeably discursive for one reader is irritatingly verbose and slow-moving for another. For myself, I do particularly like books that are good on conceptual details and good at explaining the motivation for the technicalities while avoiding needless complications or misplaced ‘rigour’, though I do like elegance too. Given the choice, I tend to prefer a treatment that doesn’t rush too fast to become too general, too abstract, and thereby obscures intuitive motivation. (There’s a certain tradition of masochism in maths writing, of going for brusque formal abstraction from the outset: that is unnecessary in other areas, and just because logic is all about formal theories, that doesn’t make it any more necessary here.)

The selection of books in the Guide no doubt reflects these tastes. But overall, I don’t think that I have been downright idiosyncratic. Nearly all the books I recommend will very widely be agreed to have significant virtues (even if some logicians would have different preference-orderings).

Want to explore further? Then get the main Guide here.

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