# The Very Short Teach Yourself Logic Guide

The full TYL Study Guide is a 96 page PDF. That’s full of good things, including detailed descriptions of lots of books, including alternative suggestions to the top recommendations. But some readers may just want the unvarnished headline news. So here, excerpted from the current version of the full Guide, is a one-webpage mini-guide to some particularly good entry-level reading on the standard core mathematical logic curriculum (a step up from baby logic). NB:

- This is a short Guide to getting started on teaching yourself mathematical logic (a serious and time-consuming business) not a short introduction to mathematical logic!
- It’s assumed you have access to a university library (so cost/availability in print is not a factor in the choosing what to recommend).
- There’s nothing here about extensions of classical logic like modal logic or variants like relevance logic. They are treated in the full Guide. The only extras that get mentioned are second order and intuitionistic logic.
- These are, to repeat, entry-level readings. Advice on more advanced topics and more sophisticated texts is also in the full Guide.
- You can tackle Sections 2, 3, 4 in any order.

## 1. Basic first-order logic

- Read Ian Chiswell and Wilfrid Hodges,
*Mathematical Logic*(OUP 2007) for a natural deduction approach, followed by - Christopher Leary and Lars Kristiansen,
*A Friendly Introduction to Mathematical Logic*(Milne Library, 2015 – first edn. by Leary alone, Prentice Hall 2000) up to §3.2, or Herbert Enderton,*A Mathematical Introduction to Logic*(Academic Press 1972, 2002) up to §2.5, for an axiomatic approach.

- If you struggle slightly, or just want a comfortingly manageable additional text: Derek Goldrei’s
*Propositional and Predicate Calculus: A Model of Argument*(Springer, 2005) is explicitly designed for self-study. Read Chs. 1 to 5 (except for §§4.4, 4.5). - Then for overview/revision/sideways looks at other ways of doing things: Wilfrid Hodges, ‘Elementary Predicate Logic’, in the
*Handbook of Philosophical Logic*, Vol. 1, ed. by D. Gabbay and F. Guenthner, (Kluwer 2nd edition 2001, but little changed from 1st edition). Read first twenty short sections.

## 2. Beginning model theory

- Derek Goldrei,
*Propositional and Predicate Calculus: A Model of Argument*(Springer, 2005) §§4.4, 4.5 and then Ch. 6 on ‘Some uses of compactness’ — gives a very clear introduction to some model theoretic ideas. That should smooth the path to reading the main text: - Maria Manzano,
*Model Theory*, Oxford Logic Guides 37 (OUP, 1999).

## 3. Beginning computability theory/Gödel’s theorems

- Peter Smith,
*An Introduction to Gödel’s Theorems*(CUP 2007, 2nd edition 2013), highlights incompleteness but has something on the general theory of computability. - The particularly clear Richard Epstein and Walter Carnielli,
*Computability: Computable Functions, Logic, and the Foundations of Mathematics*(Wadsworth 2nd edn. 2000) tackles computability first. - Tougher but classic: George Boolos, John Burgess, Richard Jeffrey,
*Computability and Logic*(CUP 5th edn. 2007)

## 4. Beginning set theory

- Start with Herbert B. Enderton,
*The Elements of Set Theory*(Academic Press, 1977): or alternatively with - Derek Goldrei,
*Classic Set Theory*(Chapman & Hall/CRC 1996) (subtitle ‘For guided independent study’).

Additional introductory reading:

- Karel Hrbacek and Thomas Jech,
*Introduction to Set Theory*(Marcel Dekker, 3rd edition 1999). - Yiannis Moschovakis,
*Notes on Set Theory*(Springer, 2nd edition 2006). - Michael Potter,
*Set Theory and Its Philosophy*(OUP, 2004).

## 5. Extras

On second-order logic:

- Herbert Enderton,
*A Mathematical Introduction to Logic*(Academic Press 1972, 2002), Ch. 4. And/or Dirk van Dalen,*Logic and Structure*, Ch. 4. - For a fuller story: Stewart Shapiro,
*Foundations without Foundationalism: A Case for Second-Order Logic*, Oxford Logic Guides 17 (Clarendon Press, 1991), Chs. 3–5.

On intuitionistic logic:

- John L. Bell, David DeVidi and Graham Solomon,
*Logical Options: An Introduction to Classical and Alternative Logics*(Broadview Press 2001). §§5.2, 5.3 give an elementary explanation of the constructivist motivation for intuitionist logic. - See also Dirk van Dalen,
*Logic and Structure*(Springer, 4th edition 2004), §§5.1–5.3.

RodrigoHello,

Thank you very much for this list and the guide – it has been very helpful. I haven’t seen any of W.V. Quine’s mentioned – do you reccomend any of them for self study? (in particular “Mathematical Logic” and “Set Theory and Logic”). I ask because I’m very interested in Principia Mathematica and the history of Logicism and Quine’s work seems to relate directly to the work of Whitehead and Russell.

Thank you

Peter SmithQuine’s ML is 1940/1951; STL is 1963. Both are very dated, and (I suppose) rather idiosyncratic. So that’s why I wouldn’t particularly recommend either in the Guide which is, after all, for those looking for initial reading in various areas/at various levels. But yes, if you are interested in the history of logic post Principia, ML in particular is an important text. And Quine puts a great deal of effort into writing as clearly as possible, so both are at least in that respect suitable for self-study.

Adam HowardDear Prof Smith

Thank you for taking the time to write these recommendations

I am curious if you have any recommendations for getting started with computer proof assistants as a basis for self-study, to supplement these works. I am interested in that to make the self-study route more interactive and help cement the concepts.

Best

Peter SmithIf you are asking about software to accompany textbooks, software that can be used for elementary proof-checking, I know there are options out there, but I don’t have any hands-on experience with them so I can’t make any useful suggestions, sorry. Perhaps I should do some investigation (for the reason you suggest!).