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Teach Yourself Logic: a Study Guide

Peter SmithUniversity of Cambridge

September 4, 2014

Version 12.0a
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Pass it on, .... Thats the game I want you to learn. Pass it on.

Alan Bennett,The History Boys

cPeter Smith 2014

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Contents

Whats new? iii

A very quick introduction v

I Preliminaries, and some logical geography 1

1 Introduction 2

1.1 Why this Guide for philosophers? . . . . . . . . . . . . . . . . . 21.2 Why this Guide for mathematicians too? . . . . . . . . . . . . . 31.3 Choices, choices . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 A strategy for reading logic books (and why this Guide is so long) 41.5 Other preliminary points . . . . . . . . . . . . . . . . . . . . . . 5

2 Logical geography 8

2.1 Mapping the field . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The shape of the Guide . . . . . . . . . . . . . . . . . . . . . . . 112.3 Assumed background: baby logic . . . . . . . . . . . . . . . . . 142.4 Do youreallyneed more logic? . . . . . . . . . . . . . . . . . . . 162.5 How to prove it . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

II Basic first-order logic 20

3 FOL: a checklist 21

4 FOL: what to read 28

4.1 The main recommendations . . . . . . . . . . . . . . . . . . . . 284.2 Some other texts for first-order logic. . . . . . . . . . . . . . . . 31

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III Going beyond the basics 40

5 Elementary Mathematical Logic 41

5.1 From first-order logic to elementary model theory . . . . . . . . 425.2 Second-order logic . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3 Computability and Godelian incompleteness . . . . . . . . . . . 485.4 Beginning set theory . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Variant Logics 57

6.1 Getting started with modal logic . . . . . . . . . . . . . . . . . . 576.2 Other classical extensions and variants . . . . . . . . . . . . . . 61

6.2.1 Free Logic . . . . . . . . . . . . . . . . . . . . . . . . . . 616.2.2 Plural logic . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3 Non-classical variants . . . . . . . . . . . . . . . . . . . . . . . . 646.3.1 Intuitionist logic . . . . . . . . . . . . . . . . . . . . . . 646.3.2 Relevance logics (and wilder logics too) . . . . . . . . . 66

IV Exploring further 69

7 More advanced reading 70

7.1 Proof theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.2 Beyond the model-theoretic basics . . . . . . . . . . . . . . . . . 757.3 Computability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.3.1 Computable functions . . . . . . . . . . . . . . . . . . . 797.3.2 Computational complexity . . . . . . . . . . . . . . . . . 81

7.4 Godelian incompleteness again . . . . . . . . . . . . . . . . . . . 827.5 Theories of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 847.6 Serious set theory . . . . . . . . . . . . . . . . . . . . . . . . . . 86

7.6.1 ZFC, with all the bells and whistles . . . . . . . . . . . 877.6.2 The Axiom of Choice. . . . . . . . . . . . . . . . . . . . 90

7.6.3 Other set theories? . . . . . . . . . . . . . . . . . . . . . 92

8 What else? 97

Index of authors 99

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Whats new?

Version 12.0: September 2014 This has ended up a more substantialrevision than originally intended, hence another jump in versionnumber. The division into Parts has changed, with some reorgani-zation between the first two chapters: Part I now makes for a more

coherent whole. The recommendations in 4.2 (on FOL) and 5.1and 7.2 (on model theory) have been significantly updated, alsothe entries on modal logic and intuitionism have been revised. Thematerial on serious set theory has been reworked into the Guide(though the stand-alone web-page remains). There has been a lot ofother minor tinkering. And there is now an Index of authors.

Version 11.0: July 2014 This Guide has been getting longer and longer,

and consequently more and more daunting in appearance (which israther unfortunate, given its intended role). Things have been get-ting out of hand!

So to cut things back down to size, the material that was pre-viously all in one document is now distributed between this newupdate of the Guide (still in PDF form, but under 60% the lengthof Version 10), an Appendix (also in PDF form), and further web-pages.

The core of the old Guide is still here. What has been made intoa separate PDF is the notes on some of the Big (and not so big)Books on Mathematical Logic. I have also exported the pages onSerious Set Theory. Entries on constructivism and category theoryhave been deleted, but will reappear shortly as further supplemen-tary webpages. Otherwise there has been little change in content.

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Version 10.1: mid-April 2014 Up until now, the Guide has taken theform of a PDF in standard A4 format. But I cant imagine thatmany people actually print it out (especially now the document hasgrown so much). Hence some will think very belatedly this ver-

sion is now designed primarily for reading onscreen.In particular, it should look good on an iPad (download and read

full-screen, e.g. in iBooks) and also on a laptop (again, downloadand e.g. read in Adobe Reader, two pages side-by-side, in full-screenmode). I did wonder about converting the whole thing to an iBook,but thats a hassle to produce, embedding symbols or diagramsdoesnt produce a reliably good effect, and reading on non-Applekit is a problem. So Ill stick with PDF output from LATEX.

There are only minor changes of content, though the Guide is nowdivided into three Parts, which signals its structure more clearly.

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A very quick introduction

I retired from the University of Cambridge in 2011. Previously, it wasmy greatest good fortune to have secure, decently paid, university postsfor forty years in leisurely times, with almost total freedom to follow myinterests wherever they led. Like many of my generation, I am sure I didnt

at the time really appreciate just how lucky I and my contemporarieswere. This now rather long and multi-layered Guide to logic textbooksfor students is one very small but heartfelt effort to give just somethingback by way of thanks.

Dont be scared off by the Guides length! This is due to its startingjust one step above the level of baby logic and then going quite a longway down the road towards pretty advanced stuff. Different readers willtherefore want to jump on and off the bus at different stops.

If you are impatient and want a quick start guide to the Guide,then just read 2.1, 2.2 from from Part I. Then glance throughChapter 3 to see whether you already know enough about first-order logic to skip the rest of Part II. If you dont, continue fromChapter4.Otherwise, start in earnest with Part III.

For aslower introduction, read through PartIover a cup of coffee

to get a sense of the purpose, shape and coverage of the Guide.That should enable you to pick and choose your way through theremaining chapters.

However, if you are hoping for help with very elementary logic(e.g. as typically encountered by philosophers in their first-yearcourses), then I am afraid that this Guide isnt designed for you.

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The only section that directly pertains to baby logic is 2.3; allthe rest is about rather more advanced and eventually verymuchmore advanced material.

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Part I

Preliminaries, and somelogical geography

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Chapter 1

Introduction

This chapter and the next are for new readers, and include a gentle guide

to the Guide, explaining its aims and how it is structured.

1.1 Why this Guide for philosophers?

It is an odd phenomenon, and a rather depressing one too. Serious logicis 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 rewardinga subject than it ever was, and the amount of good formally-informedwork in philosophy is ever greater as time goes on. Moreoever, logic is fartoo important to be left entirely to the mercies of technicians from mathsor computer science departments with different agendas (who often revealan insouciant casualness about conceptual details that will matter to thephilosophical 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 cutoff from working in some of the most exciting areas of their discipline will need to teach themselves from books, either solo or (much better) byorganizing their own study groups.

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In a way, thats perhaps no real hardship, as there are some wonderfulbooks written by great expositors out there. But what to read? Logicbooks can have a very long shelf life, and one shouldnt at all dismissolder texts when starting out on some topic area: so theres more than a

fifty year span of publications to select from. Without having tried veryhard, I seem to have accumulated on my own shelves some three hundredformal logic books at roughly the level that might feature in a Guide suchas this and of course these are still only a selection of whats available.

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

1.2 Why this Guide for mathematicians too?

The situation of logic teaching in mathematics departments can also bepretty dire. Indeed there are full university maths courses in good UKuniversities with precisely zero courses on logic or set theory (maybe afew beginning ideas are touched on in a discrete maths course, but thatis 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 textbooksare likely to prove accessible and tolerably enjoyable and rewarding towork through? Again, this Guide or at least, the sections on the coremathematical logic curriculum will give you some pointers.

True, this is written by someone who has apart from a few guestmini-courses taught in a philosophy department, and who is no research

mathematician. Which probably gives a distinctive tone to the Guide (andcertainly explains why it ranges into areas of logic of more special interestto philosophers). Still, mathematics remains my first love, and these daysit is mathematicians whom I mostly get to hang out with. A large numberof the books I recommend are very definitely paradigmmathematicstexts.So I shouldnt be leading you too astray.

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And I didnt want to try to write two overlapping Guides, one pri-marily for philosophers and one aimed primarily at mathematicians. Thiswas not just to avoid multiplying work. Areas of likely interest dontso neatly categorize. Anyway, a number of philosophers develop serious

interests in more mathematical corners of the broad field of logic, anda number of mathematicians find themselves interested in more founda-tional/conceptual issues. So theres a single but wide-ranging menu herefor everyone to choose from as their interests dictate.

1.3 Choices, choices

So what has guided my choices of what to recommend in this Guide?Different people find different expository styles congenial. For exam-ple, what is agreeably discursive for one reader is irritatingly verbose andslow-moving for another. For myself, I do particularly like books thatare good on conceptual details and good at explaining the motivationfor the technicalities while avoiding needless complications or misplacedrigour, though I do like elegance too. Given the choice, I tend to prefer atreatment that doesnt rush too fast to become too general, too abstract,

and thereby obscures intuitive motivation. (Theres a certain tradition ofmasochism in maths writing, of going for brusque formal abstraction fromthe outset: that is unnecessary in other areas, and just because logic is allabout formal theories, that doesnt make it any more necessary here.)

The selection of books in later chapters no doubt reflects these tastes.But overall, I dont think that I have been downright idiosyncratic. Nearlyall the books I recommend will very widely be agreed to have significantvirtues (even if some logicians would have different preference-orderings).

1.4 A strategy for reading logic books (and whythis Guide is so long)

We cover a lot in this Guide, which is one reason for its size. But there isanother reason.

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I very strongly recommend tackling an area of logic (or indeed any newarea of mathematics) by reading a series of books which overlap in level(with the next one covering some of the same ground and then pushingon from the previous one), rather than trying to proceed by big leaps.

In fact, I probably cant stress this advice too much, which is why Iam highlighting it here in a separate section. For this approach will reallyhelp to reinforce and deepen understanding as you re-encounter the samematerial from different angles, with different emphases.

The multiple overlaps in coverage in the reading lists below are there-fore fully intended, and this explains why the lists are always longer ratherthan shorter (and also means that you should more often be able to findoptions that suit your degree of mathematical competence). You will cer-

tainly 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 sothat you can build up a good overall picture of an area seen from varioussomewhat different angles of approach.

1.5 Other preliminary points

(a) Mathematics is not merely a spectator sport: you should try someof the exercises in the books as you read along to check and reinforcecomprehension. On the other hand, dont obsess about doing exercises ifyou are a philosopher understanding proof ideas is very much the crucialthing, not the ability to roll-your-own proofs. And even mathematiciansshouldnt get too hung up on routine exercises (unless you have specific

exams to prepare for!): concentrate on the exercises that look interestingand/or might deepen understanding.

Do note however that some authors have the irritating(?) habit ofburying important results in the exercises, mixed in with routine home-work, so it is often worth skimming through the exercises even if you dontplan to tackle very many of them.

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(b) Nearly all the books mentioned here should be held by any universitylibrary which has been paying reasonable attention to maintaining a corecollection in the area of logic (and any book should be borrowable throughyour interlibrary loans system). Certainly, if some recommended books

from Chapters4,5and6are not on the shelves, do make sure your libraryorders them(in my experience, university librarians overwhelmed by thenumber of publications, strapped for cash, and too familiar with the never-borrowed purchase are rather happy to get informed recommendationsof books which are reliably warranted actually to be good of their kind).

Since Im not assuming you will be buying personal copies, I havenot made cost or even being currently in print a significant considera-tion: indeed it has to be said that the list price of some of the books

is just ridiculous. (Though second-hand copies of some books at betterprices might well be available via Amazon sellers or from abebooks.com.)However, I have marked with one star* books that are available new ata reasonable price (or at least are unusually good value for the lengthand/or importance of the book). And Ive marked with two stars** thosebooks for which e-copies are freely (and legally!) available and links areprovided.1 Most articles, encyclopaedia entries, etc., can also be down-

loaded, again with links supplied.(c) And yes, the references here are very largely to published booksrather than to on-line lecture notes etc. Many such notes are excellent,but they do tend to be a bit terse (as befits material intended to supporta lecture course) and so perhaps not as helpful as fully-worked-out book-length treatments for students needing to teach themselves. But Im surethat there is an increasing number of excellent e-resources out there whichdo amount, more or less, to free stand-alone textbooks. Id be very happyindeed to get recommendations about the best.

(d) Finally, the earliest versions of this Guide kept largely to positiverecommendations: I didnt pause to explain the reasons for the then ab-

1We will have to pass over in silence the issue of illegal file-sharing of PDFs of e.g.

out-of-print books: most students will know the possibilities here.

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sence of some well-known books. This was partly due to considerations oflength which have now quite gone by the wayside; but also I wanted tokeep the tone enthusiastic, rather than to start criticizing or carping.

However, enough people have asked what I think about the classic

X, or asked why the old warhorse Y wasnt mentioned, to change mymind. So I have very occasionally added some reasons why I dont par-ticularly recommended certain books. I should perhaps emphasize thatthese relatively negative assessments are probably more contentious thanthe positive ones.

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Chapter 2

Logical geography

This chapter explains how the field of logic is being carved up in this

Guide; I add something about the (limited!) background that you aregoing to be assumed to have.

2.1 Mapping the field

Logic is a big field. Its technical development is of concern to philoso-phers, mathematicians and computer scientists. Different constituencies

will be particularly interested in different areas and give different em-phases. For example, modal logic is of considerable interest to some philoso-phers, but also parts of this sub-discipline are of concern to computerscientists too. Set theory (which falls within the purview of mathematicallogic, broadly understood) is an active area of research interest in math-ematics, but because of its (supposed) foundational status even quiteadvanced results can be of interest to philosophers too. Finite model the-ory is of interest to mathematicians and computer scientists, but perhaps

not so much to philosophers. The incompleteness theorems are relativelyelementary results of the theory of computable functions, but are of par-ticular conceptual interest to philosophers. Type theory started off as adevice of philosophy-minded logicians looking to avoid the paradoxes: ithas become primarily the playground of computer scientists. And so itgoes.

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(Assumed background: Baby logic)

1st order logic

More model theory

Computability, incompleteness

Beginning set theory

2nd order logic Modal logic

Classical variations

Non-classical variations

In this Guide, Im going to have to let the computer scientists largelylook after themselves. Our focus is going to be the areas of logic of mostinterest to philosophers and mathematicians, if only because thats what Iknow a little about. So in the map above, we have a very rough-and-readyinitial indication of the ground surveyed in PartsIIandIII of the Guide.

We start with fundamentals of first-order logic at the top. The bold-

face left-hand branch will be the crucial one for most readers, covering thetraditional Mathematical Logic curriculum in advanced (graduate level?)philosophy courses and not-so-advanced mathematics courses: PartIVofthe Guide then mostly goes deeper into these same topics. The right-hand branch then highlights some areas that are likely to be of interestto philosophers and perhaps not so much to most mathematicians.

The dashed lines between the three blocks on the left indicate thatthey are roughly on the same level and with one caveat they caninitially be tackled in any order; likewise you can take the three blocks onthe right in any order.

In a bit more detail (though dont worryat allif at this stage you donot understand much of the jargon):

By Baby logic, which I say just a little more about later in thischapter in 2.3, I mean the sort of material that is covered in ele-

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mentary formal logic courses for philosophers or is perhaps coveredin passing in a discrete mathematics course: truth-functional logic(the truth-table test for validity, usually a first look at some formalproof system as well), a light-weight introduction to quantifiers and

identity (giving at least a familiarity with the use of the notation),and perhaps also an introduction to fragments of set theoretic nota-tion and such ideas as that of an equivalence relation or a one-to-onefunction, etc.

First-order logic (FOL) means a reasonably rigorous treatment ofquantification theory, studying both a proof-system (or two) for clas-sical logic, and the standard semantics, getting at least as far as a

soundness and completeness proof for your favourite proof system.

At some stage, we need to look briefly at second-order logic, butthe main task is to go down the right-hand column, to take in thethree substantial elements (after core first-order logic) which go in various proportions to make up the traditional MathematicalLogic curriculum and which form the principal topics of Chapter 5.The first of those elements, continuous with our initial study of

FOL, involves a little model theory, i.e. a little more exploration ofthe fit between theories cast framed in formal languages and thestructures they are supposed to be about. This will start withthe compactness theorem and Lowenheim-Skolem theorems (if thesearent already covered in your FOL reading), and then pushes on

just a little further. For some elements of this, you will need to knowa little, a very little, set theory (mainly ideas about cardinality), soyou might want to interweave starting out on model theory with the

beginnings of your work on set theory if those cardinality ideas arenew to you.

The second standard element of the curriculum is an introduction tothe ideas of mechanical computability and decidability, and proofsof epochal results such as Godels incompleteness theorems. (This isperhaps the most readily approachable area of mathematical logic.)

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Then we start systematic work on set theory.

Now we move on to consider the right-hand column of logic-mainly-for-philosophers. Even before encountering a full-on-treatment of

first-order logic, philosophers are often introduced to modal logic(the logic of necessity and possibility) and to its possible worldsemantics. You can indeed do an amount of propositional modallogic with no more than baby logic as background.

Further down that right-hand column we meet some variations andextensions of standard logic which are of conceptual interest butwhich are still classical in favour (e.g. free logic, plural logic).

Then there are non-classical variations, of which perhaps the mostimportant and the one of real interest to mathematicians too isintuitionist logic (which drops the law of excluded middle, motivatedby a non-classical understanding of the significance of the logicaloperators). We could also mention e.g. relevant logics which dropthe classical rule that a contradiction entails anything.

To repeat, dont be alarmed if (some of) those descriptions are at the

moment pretty opaque to you: we will be explaining things a little moreas we go through the Guide. And as I said, you probably shouldnt takethe way our map divides logic into its fields too seriously. Still, the mapshould be a helpful initial guide to understanding . . .

2.2 The shape of the Guide

(a) To help you navigate through and find the episodes that are mostrelevant to your needs/interests, the Guide is chunked into four mainParts, backed up a separate Appendix and some further web-pages:

The rest of this Part I of the Guide suggests some preliminary read-ing; this is aimed at those with little or no mathematical back-ground who might benefit from a gentle warm-up before tackling

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the recommendations in later chapters. The suggested reading in2.3,together perhaps with that mentioned in 2.4, may indeed al-ready provide some philosophers with enough for their purposes. Icertainly think that any philosopher should know this little amount,

but these modest preliminaries are not really the topic of this Guide.Our concern is with the serious stuff that lies beyond, of somewhatmore specialized interest.

PartII covers Basic first-order logic (at a level one step up fromwhat many/most philosophers will have met in their baby logiccourses, but at entry-level for mathematicians). Nearly everythinglater depends on this material, so it is absolutely imperative to re-

ally get on top of it if you want to go any distance in mathematicallogic. Thats why I have both given an outline of the topics thatneed to be covered and also suggested a number of different waysof covering them (Im not suggesting you read all the texts men-tioned!). This Part is therefore quite long even though it doesntcover a lot of ground.

Part III, Going beyond the basics, begins with Chapter 5 which

looks at more-or-less-introductory readings on each of the mainsubfields of the traditional mathematical logic curriculum, assum-ing a background in first-order logic. This is, roughly speaking, atan advanced undergraduate or beginning graduate student level forphilosophers, or equivalently at a middling-to-upper undergraduatelevel for mathematicians.

Much of Chapter6on Variant Logics will be primarily of interestto philosophers. Do note, though, that a number of the topics, likemodal logic, are in fact quite approachable even if you know littleother logic. So the fact that material in this chapter is mentionedafter the heavier-duty mathematical logic topics in the previous twochapters doesnt mean that there is always a jump in difficulty.

The introductory books already mentioned in the Chapters4,5and6already contain numerous pointers to further reading, more than

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enough to put you in a position to continue exploring solo. How-ever, Part IV, Exploring further, does give more suggestions fornarrower-focus forays deeper into various subfields, looking in Chap-ter7at more advanced work on core topics in mathematical logic.

This is mostly for specialist graduate students (among philosophers)and final year undergraduate/beginning graduate students (amongmathematicians). Finally, Chapter 8 very briefly gestures at sometopics outside the traditional basic menu of mathematical logic.

PartIVof the Guide used also to have a substantial Appendix con-sidering some of The Big Books on mathematical logic (meaning typ-ically broader-focus books that cover first-order logic together with one

or more subfields from the further menu of mathematical logic). Thesebooks vary a lot in level and coverage, but can provide very useful con-solidating/amplifying reading. This supplement to the main Guide is nowavailable online as anAppendix.Alternatively, for separate webpages onthose texts and a number of other reviews, visit the Logic Matters BookNotes.

(b) We are dividing up the broad field of logic into subfields in a fairly

conventional way. But of course, even the horizontal divisions into dif-ferent areas can in places be a little arbitrary. And the vertical divisionbetween the entry-level readings in earlier chapters and the further explo-rations in Chapter7is evidently going to be a lot more arbitrary. I thinkthat everyone will agree (at least in retrospect!) that e.g. the elemen-tary theory of ordinals and cardinals belongs to the basics of set theory,while explorations of large cardinals or independence proofs via forcingare decidedly advanced. But in most areas, there are far fewer natural

demarcation lines between the basics and more advanced work.Still, it is surely very much better to have somesuch structuring than

to heap everything together. And I hope the vertical divisions betweenParts will help to make the size of this Guide seem less daunting.

Within sections in Chapters 47, I have put the main recommenda-tions into what strikes me as a sensible reading order of increasing dif-ficulty (without of course supposing you will want to read everything

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those with stronger mathematical backgrounds might sometimes want totry starting in the middle of a list). Some further books are listed insmall-printasides orpostscripts.

2.3 Assumed background: baby logic

If you are a mathematician, you will have picked up a smattering of logicalideas and notations. But there is no specific assumed background youreally needbefore tackling the books recommended in the next chapter.So you can just skip on to 2.5.

If you are a philosopher without any mathematical background, how-

ever, you will almost certainly need to have already done some formallogic if you are going to cope with what are, in the end, mathematicsbooks on areas of mathematical logic.

So, if you have only done an informal logic or critical reasoningcourse, then youll surely need to read a good introductory formallogic text before tackling more advanced work. See below.

If you havetaken a logic course with a formal element, but it was

based on some really, really, elementary text book like Sam Gutten-plansThe Languages of Logic, Howard KahanesLogic and Philos-ophy, or Patrick Hurleys Concise Introduction to Logic, then youmightstill struggle with the initial suggestions in4.1(though thesethings are very hard to predict). If you do, one possibility would beto use Intermediate Logic by Bostock mentioned in 4.2to bridgethe gap between what you know and the beginnings of mathemat-ical logic. Or again, try one of the books below, skipping fast overwhat you already know.

If you have taken a formal logic course based on a substantial textlike those mentioned in a moment, then you should be well prepared.

Here then, for those that need them, are couple of initial suggestions offormal logic books that start from scratch and go far enough to provide

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a good foundation for further work the core chapters of these cover thebaby logic that it would be ideal to have under your belt:

1. My Introduction to Formal Logic* (CUP 2003, corrected reprint

2013): for more details see theIFL pages,where there are also an-swers to the exercises). This is intended for beginners, and was thefirst year text in Cambridge for a decade. It was written as an acces-sible teach-yourself book, covering propositional and predicate logicby trees. It in fact gets as far as a completeness proof for the treesystem of predicate logic without identity, though for a beginnersbook thats very much an optional extra.

2. Paul TellersA Modern Formal Logic Primer**(Prentice Hall 1989)predates my book, is now out of print, but is freely available onlineat the books website, which makes it unbeatable value! It is inmany ways quite excellent, and had I known about it at the time(or listened to Pauls good advice, when I got to know him, abouthow long it takes to write an intro book), Im not sure that Id havewritten my own book, full of good things though it is! As well asintroducing trees, Teller also covers a version of Fitch-style natural

deduction I didnt have the page allowance to do this, regrettably.(Like me, he also goes beyond the really elementary by getting as faras a completeness proof.) Notably user-friendly. Answers to exercisesare available at the website.

Of course, those are just two possibilities from very many. I have notlatterly kept up with the seemingly never-ending flow of books of entry-level introductory logic books, some of which are no doubt excellent too,

though there are also some rather poor books out there (and also thereare more books like Guttenplans that will probably not get readers to thestarting line as far as this Guide is concerned). Mathematicians shouldalso be warned that some of the books on discrete mathematics coverelementary logic pretty badly. This is not the place, however, to discusslots more options for elementary texts. Ill mention here just two otherbooks:

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3. I have been asked a number of times about Jon Barwise and JohnEtchemendysLanguage, Proof and Logic(CSLI Publications, 1999;2nd edition 2011 for more, see the books website). The uniqueselling point for this widely used book is that it comes with a CD

of programs, including a famous one by the authors called TarskisWorld in which you build model worlds and can query whethervarious first-order sentences are true of them. Some students reallylike it, and but at least equally many dont find this kind of thingparticularly useful. And I believe that the CD cant be registeredto a second owner, so you have to buy the book new to get the fulladvantage.

On the positive side, this is another book which is in many re-

spects user-friendly, goes slowly, and does Fitch-style natural deduc-tion. It is a respectable option. But Teller is rather snappier, and Ithink no less clear, and certainly wins on price!

4. Nicholas Smiths recent Logic: The Laws of Truth (Princeton UP2012) is very clearly written and seems to have many virtues. Thefirst two parts of the book overlap very largely with mine (it toointroduces logic by trees). But the third part ranges wider, including

a brisk foray into natural deduction indeed the logical coveragegoes almost as far as Bostocks book, mentioned below in 4.1, andthere some extras too. It seems a particularly readable addition tothe introductory literature. I havecommented further here. Answersto exercises can be found atthe books website.

2.4 Do you really need more logic?

This section is for philosophers. It is perhaps worth pausing to ask suchreaders if they are sure especially if they already worked through a booklike mine or Paul Tellers whether they reallydo want or need to pursuethings much further, at least in a detailed, formal, way. Far be it fromme to put people off doing more logic: perish the thought! But for manypurposes, you might well survive by just reading the likes of

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1. David Papineau, Philosophical Devices: Proofs, Probabilities, Pos-sibilities, and Sets (OUP 2012). From the publishers description:This book is designed to explain the technical ideas that are takenfor granted in much contemporary philosophical writing. . . . The

first section is about sets and numbers, starting with the member-ship relation and ending with the generalized continuum hypothesis.The second is about analyticity, a prioricity, and necessity. The thirdis about probability, outlining the difference between objective andsubjective probability and exploring aspects of conditionalizationand correlation. The fourth deals with metalogic, focusing on thecontrast between syntax and semantics, and finishing with a sketchof Godels theorem.

Or better since perhaps Papineau arguably gives rather too brisk anoverview you could rely on

2. Eric Steinhart, More Precisely: The Math You Need to Do Philos-ophy (Broadview 2009) The author writes: The topics presented. . . include: basic set theory; relations and functions; machines; prob-ability; formal semantics; utilitarianism; and infinity. The chapters

on sets, relations, and functions provide you with all you need toknow to apply set theory in any branch of philosophy. The chapter ofmachines includes finite state machines, networks of machines, thegame of life, and Turing machines. The chapter on formal semanticsincludes both extensional semantics, Kripkean possible worlds se-mantics, and Lewisian counterpart theory. The chapter on probabil-ity covers basic probability, conditional probability, Bayes theorem,and various applications of Bayes theorem in philosophy. The chap-

ter on utilitarianism covers act utilitarianism, applications involvingutility and probability (expected utility), and applications involvingpossible worlds and utility. The chapters on infinity cover recursivedefinitions, limits, countable infinity, Cantors diagonal and powerset arguments, uncountable infinities, the aleph and beth numbers,and definitions by transfinite recursion. More Precisely is designedboth as a text book and reference book to meet the needs of upper

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level undergraduates and graduate students. It is also useful as areference book for any philosopher working today.

Steinharts book is admirable, and will give many philosophers a handle

on some technical ideas going well beyond baby logic and which theyreally should know just a little about, without the hard work of doing areal logic course. Whats not to like? And if there indeed turns out tobe some particular area (modal logic, for example) that seems especiallygermane to your philosophical interests, then you can go to the relevantsection of this Guide for more.

2.5 How to prove itAssuming, however, that you do want to learn more serious logic, beforegetting down to business in the next chapter, let me mention one other rather different and often-recommended book:

Daniel J. Velleman,How to Prove It: A Structured Approach (CUP,2nd edition, 2006). From the Preface: Students of mathematics . . .often have trouble the first time that theyre asked to work seriously

with mathematical proofs, because they dont know the rules of thegame. What is expected of you if you are asked to prove something?What distinguishes a correct proof from an incorrect one? This bookis intended to help students learn the answers to these questions byspelling out the underlying principles involved in the constructionof proofs.

There are chapters on the propositional connectives and quantifiers, and

informalproof-strategies for using them, and chapters on relations andfunctions, a chapter on mathematical induction, and a final chapter oninfinite sets (countable vs. uncountable sets). This truly excellent studenttext could well be of use both to many philosophers and to some math-ematicians reading this Guide. By the way, if you want to check youranswers to exercises, heres along series of blog posts (in reverse order).

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True, if you are a mathematics student who has got to the pointof embarking on an upper level undergraduate course in some area ofmathematical logic, youshouldcertainly have already mastered nearly allthe content of Vellemans splendidly clear book. However, an afternoon or

two skimming through this text (except perhaps for the very final section),pausing over anything that doesnt look very comfortably familiar, couldstill be time extremely well spent.

What if you are a philosophy student who (as we are assuming) hasdone some baby logic? Well, experience shows that being able to handle

formal tree-proofs or natural deduction proofs doesnt always translateinto being able to construct good informal proofs. For example, one ofthe few meta-theoretic results usually met in a baby logic course is the

expressive completeness of the set of formal connectives t^,_,u. Theproof of this result is really easy, based on a simple proof-idea. But manystudents who can ace the part of the end-of-course exam asking for quitecomplex formal proofs inside a deductive system find themselves all atsea when asked to replicate this informalbookwork proofabouta formalsystem. Another example: it is only too familiar to find philosophy stu-dents introduced to set notation not even being able to make a start on a

good informal proof that ttau, ta, buu tta

1

u, ta

1

, b

1

uu if and only ifa a

1

and b b1.Well, if youareone of those students who jumped through the formal

hoops but were unclear about how to set out elementary mathematicalproofs (e.g. from the meta-theory theory of baby logic, or from baby settheory), then again working through Vellemans book from the beginningcould be just what you need to get you prepared for the serious study oflogic. And even if you were one of those comfortable with the informal

proofs, you will probably still profit from skipping and skimming through(perhaps paying especial attention to the chapter on mathematical induc-tion).

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Part II

Basic first-order logic

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Chapter 3

FOL: a checklist

With scene-setting preliminaries out of the way, we at last get down to

work. This chapter gives a checklist of the topics we will be treating asbelonging to the basics of first-order logic (FOL).

For those who already have some significant background in formallogic, this list will enable you to determine whether you can skip forwardin the Guide. If you find that you are indeed familiar with the highlightedtopics, then go straight to PartIII.

For those who know just (some fragments of) baby logic, then thislist should provide some preliminary orientation though dont worry, ofcourse, if at present you dont fully grasp the import of every point. Thenext chapter gives recommended reading to help you reach enlightenment!

So, without further ado, heres what you need to get to know about:

1. Starting with syntax, you obviously need to know how first-orderlanguages are constructed. And now you need to understand howto prove various things that seem obvious and that you perhaps

previously took for granted, e.g. that bracketing works to avoidambiguities, meaning that every well-formed formula has a uniqueparsing.

Note by the way that baby logic courses for philosophers veryoften ignore functions; but given that FOL is deployed to regimenteveryday mathematical reasoning, and that functions are of coursecrucial to mathematics, function expressions now become centrally

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important (even if there are tricks that make them in principle elim-inable).

2. On thesemanticside, you need to understand the idea of a structure

(a domain of objects equipped with some relations and/or functions,and perhaps having some particular objects especially picked out,e.g. as a zero or unit), and you need to grasp the idea of an in-terpretation of a language in such a structure. Youll also need tounderstand how an interpretation generates an assignment of truth-values to every sentence of the interpreted language this meansgrasping a proper formal semantic story with the bells and whistlesrequired to cope with quantifiers adequately. You now can define

the notion of semantic entailment (where semantically entails when no interpretation in any structure can make all the sentencesamong true without making true too.)

3. Back to syntax: you need to get to know one deductive proof-systemfor FOL reasonably well, and you ought ideally (and at quite anearly stage) acquire at least a glancing acquaintance with some al-ternatives. But where to concentrate your efforts first? It is surely

natural to give centre stage, at least at the outset, to so-callednat-ural deduction systems.

The key feature of natural deduction systems is that they allowus to make temporary assumptions for the sake of argument andthen later discharge the temporary assumptions. This is, of course,exactly what we do all the time in everyday reasoning, mathemati-cal or otherwise as, for example, when we temporarily suppose A,show it leads to absurdity, and then drop the supposition and con-

clude A. So surely it is indeed natural to seek to formalize thesekey informal ways of reasoning.

Different formal natural deduction systems will offer differentways of handling temporary assumptions, keeping track of themwhile they stay live, and then showing where in the argument theyget discharged. Suppose, for example, we want to show that frompP^ Qq we can infer pP Qq (where , ^ and are of

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course our symbols for, respectively,not,and, and [roughly]implies.)Then one way of laying out a natural deduction proof would be likethis (a so-called Fitch-style layout1) where we have added informalcommentary on the right:

1 pP^ Qq premiss

2 P supposition for the sake of argument

3 Q supposition for the sake of argument

4 pP^ Qq from 2, 3 by and-introduction

5 K 1 and 4 give us a contradiction

6 Q suppn 3 must be false, by reductio7 Q from 6, eliminating the double negation

8 pP Qq if the supposition 2 is true, so is 7.

Here, the basic layout principle is that, whenever we make a newtemporary assumption we indent the line of argument a column tothe right (vertical bars marking the indented columns), and when-ever we discharge the assumption at the top of an indented columnwe move back to the left.

An alternative layout (going back to Genzten) would display thesame proof like this, where the guiding idea is (roughly speaking)that a wff below an inference line follows from what is immediatelyabove it, or by discharge of some earlier assumption:

rPsp2q rQsp1q

pP^ Qq pP^ QqK

(1)QQ

(2)pP Qq

1See the introductory texts by Teller or by Barwise and Etchemendy mentioned in

2.3for more examples of natural deduction done in this style.

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Here numerical labels are used to indicate where a supposition getsdischarged, and the supposition to be discharged gets bracketed off(or struck through) and also given the corresponding label.

Fitch-style proofs are perhaps easier to use for beginners (indeed,

we might say, especially natural, by virtue of more closely followthe basically linear style of ordinary reasoning). But Gentzen-styleproofs are usually preferred for more advanced work, and thatswhat the natural deduction texts that Ill be mentioning use.

4. Next and for philosophersthisis likely to be the key move beyondeven a substantial baby logic course, and for mathematicians thisis probably the point at which things actually get interesting you

need to know how to prove a soundness and a completeness theo-rem for your favourite deductive system for first-order logic. Thatis to say, you need to be able to show that theres a deduction inyour chosen system of a conclusion from given premisses only if thepremisses do indeed semantically entail the conclusion (the systemdoesnt give false positives), and whenever an inference is seman-tically valid theres a formal deduction of the conclusion from thepremisses (the system captures all the semantical entailments).

5. As an optional bonus at this stage, depending on what text you read,you might catch a first glimpse of e.g. the downward Lowenheim-Skolem theorem (roughly, assuming we are dealing with an ordinarysort of first-order language L, if there is any infinite model thatmakes the set ofL-sentences all true, then there is a model whichdoes this whose domain is the natural numbers), and the compact-ness theorem (if semantically entails then there is some finite

subset of which already semantically entails ). These are initialresults of model theory which flow quickly from the proof of thecompleteness theorem.

However, we will be returning to these results in5.1when we turnto model theory proper, so itwontbe assumed that your knowledgeof basic first-order logic covers them.

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6. Finally, you of course dont want to get confused by meeting variantversions of standard FOL too soon. But you should at some fairlyearly stage become aware of some of the alternative ways of doingthings, so you dont become fazed when you encounter different

approaches in different more advanced texts.For example:

(a) Syntaxes differ, for example, in which logical operators aretaken as basic, and which as defined shorthand. Thus some syn-taxes not only have a primitive absurdity sign K but take nega-tion to be defined rather than primitive (by putting def K). Again, some syntaxes distinguish variables from pa-

rameters, and then variables only ever appear bound, andtypographically different letters are used when we want, as itwere, names for arbitrarily selected objects.

(b) On the semantic side, there are variant ways of giving truthconditions for a quantified wff@xpxq. Some in effect say thatthis is true when pxq is true no matter how we interpret xwhentreated as a temporary demonstrative pointing to somethingin the domain. (Compare: All men are mortal is true justif he is mortal is true irrespective of which man we pointout.) Others in effect say that the quantified sentence is truewheneverpaq,pbq,pcq . . . are all true, where a, b, c, . . . arenew names added to the language, one for each element of thedomain (in a somewhat stretched, abstract, sense of name ifthe domain is uncountable).

You should also become aware of different proof-styles other thannatural deduction:

(c) First, and historically very importantly, there are old-schoolHilbert-style axiomatic linear proof-systems. A standard suchsystem for e.g. the propositional calculus has a single rule ofinference (modus ponens) and a bunch of axioms which can becalled on at any stage in a proof. A Hilbert-style proof is just

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a linear sequence of wffs, each one of which is a given premiss,or a logical axiom, or follows from earlier wffs in the sequenceby modus ponens. Such systems in their unadorned form arepretty horrible to work inside (proofs can be long and very

unobvious), even though their Bauhaus simplicity makes themeasy to theorize about from the outside. It does strike me aspotentially off-putting, even a bit masochistic, to concentrateprimarily on axiomatic systems when beginning serious logic Wilfrid Hodges rightly calls them barbarously unintuitive.But for various reasons they are often used in more advancedtexts, so you certainly need to get to know about them sooneror later.

(d) It is also fun and illuminating to meet a tree/tableau system ifyou havent already done so (such a system is particularly easyto teach not-very-good mathematicians whether philosophystudents or computers!). A tableau proof of the same illustra-tive inference as before would run as follows:

pP^ QqpP Qq

P

Q

P Q

What we are doing here is starting with the given premiss and

the negationof the target conclusion, and we then proceed tounpack the implications of these assumptions for their com-ponent wffs. In this case, for the second (negated-conditional)premiss to be true, it must have a true antecedent and falseconsequent: while for the first premiss to be true we have toconsider cases (a false conjunction must have at least one falseconjunct but we dont know in advance which to blame)

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which is why the tree splits. But as we pursue either branchwe immediately hit contradiction with what we already have(conventionally marked with a star in tableau systems), show-ing that we cant after all consistently have the premiss true

and thenegationof the conclusion true too, so the inference isindeed valid.

So, with that menu of topics to pursue, where to begin?

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Chapter 4

FOL: what to read

4.1 The main recommendationsLets start with a couple of stand-out books which strike me as particularlygood and, taken together, make an excellent introduction to the seriousstudy of FOL.

1. Ian Chiswell and Wilfrid Hodges,Mathematical Logic (OUP 2007).This very nicely written text is only a small notch up in actual dif-ficulty from baby logic texts like mine or Paul Tellers: but asits title might suggest it does have a notably more mathematicallook and feel (being indeed written by mathematicians). However,despite that, it remains particularly friendly and approachable andshould be entirely manageable even at this very early stage. It isalso pleasingly short. Indeed, Im rather tempted to say that if youcantcope with this lovely book then serious logic might not be foryou!

The headline news is that authors explore a (Gentzen-style) natu-ral deduction system. But by building things up in stages over threechapters so after propositional logic, they consider an interest-ing fragment of first-order logic before turning to the full-strengthversion they make proofs of e.g. the completeness theorem forfirst-order logic quite unusually comprehensible. For a more detaileddescription see mybook noteon C&H.

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Very warmly recommended, then. For the moment, you only needread up to and including 7.7; but having got that far, you mightas well read the final couple of sections and the Postlude too! (Thebook has brisk solutions to some exercises. A demerit mark to OUP

for not publishing C&H more cheaply.)

Next, complement C&H by reading the first half of

2. Christopher LearysA Friendly Introduction to Mathematical Logic(Prentice Hall 2000: currently out of print, but a slightly expandednew edition is being planned). There is a great deal to like aboutthis book. Chs. 13 do make a very friendly and helpful introductionto first-order logic, this time done in axiomatic style: and indeed atthis stage you could stop reading for the moment after 3.2, whichmeans you will be reading just over 100 pages. Unusually, Learydives straight into a full treatment of FOL without spending anintroductory chapter or two on propositional logic: but that happilymeans (in the present context) that you wont feel that you arelabouring through the very beginnings of logic one more time than isreally necessary so this book dovetails very nicely with C&H. The

bookisagain written by a mathematician for a mostly mathematicalaudience so some illustrations of ideas can presuppose a smatteringof elementary background mathematical knowledge; but you willmiss very little if you occasionally have to skip an example (andphilosophers can always resort to Wikipedia, which is quite reliablein this area, for explanations of mathematical terms). I like thetone very much indeed, and say more about this admirable book inanother book note.

Next, heres an alternative to the C&H/Leary pairing which is alsowonderfully approachable and can be warmly recommended:

3. Derek Goldreis Propositional and Predicate Calculus: A Model ofArgument* (Springer, 2005) is explicitly designed for self-study.Read Chs. 1 to 5 (you could skip 4.4 and 4.5, leaving them un-til you turn to elementary model theory). While C&H and Leary

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together cover overlapping material twice, Goldrei in much thesame total number of pages covers very similar ground once. Sothis is a somewhat more gently-paced book, with a lot of examplesand exercises to test comprehension along the way. A great deal of

thought has gone into making this text as helpful as possible. Soif you struggle slightly with the alternative reading, or just want acomfortingly manageable additional text, you should find this ex-ceptionally accessible and useful.

Like Leary, Goldrei uses an axiomatic system (which is one rea-son why, on balance, I recommend starting with C&H instead); butunlike Leary who goes on to deal with Godels incompleteness the-orem, Goldrei only strays beyond the basics of FOL to touch on

some model-theoretic ideas at the end of Ch. 4 and in the final Ch.6, allowing him to be more expansive about fundamentals. As withC&H and Leary, I like the tone and approach a great deal.

Fourthly in this section, even though it is giving a second bite to anauthor weve already met, I must mention

4. Wilfrid Hodgess Elementary Predicate Logic, in theHandbook of

Philosophical Logic, Vol. 1, ed. by D. Gabbay and F. Guenthner,(Kluwer 2nd edition 2001). This is a slightly expanded version ofthe essay in the first edition of the Handbook(read that earlier ver-sion if this one isnt available), and is written with Hodgess usualenviable clarity and verve. As befits an essay aimed at philosoph-ically minded logicians, it is full of conceptual insights, historicalasides, comparisons of different ways of doing things, etc., so it verynicely complements the more conventional textbook presentations

of C&H and Leary (or of Goldrei). Read at this stage the first twentysections (70 pp.): they are wonderfully illuminating.

Note (particularly to philosophers): If you have carefully read and mas-tered a book like Tellers introductory baby-logic-plus-a-little book, youwill actually already know about quite a few of the headline results cov-ered in the reading mentioned in this section. However, the big change is

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that you will now have begun to see the familiar old material being re-presented in the sort of mathematical style and with the sort of rigorousdetail that you will necessarily encounter more and more as you progressin logic, and that you very much need to start feeling entirely comfortable

with at an early stage.

4.2 Some other texts for first-order logic

The material covered in the last section is so very fundamental, and thealternative options so very many, that I really need to say at least some-thing about some other books.

So in this section I will list in rough order of difficulty/sophistication a small handful of recommended books that have particular virtues ofone kind or another. These are books that will either make for parallelreading or will usefully extend your knowledge of first-order logic or both.There follows a postscript where I explain why I havent recommendedcertain texts. Some of myBook Noteswill tell you a little about how someother Big Books on mathematical logic handle the basics.

First, a text by a philosopher, aimed at philosophers, though mathe-

maticians could still profit from a quicker browse:

1. David Bostocks Intermediate Logic (OUP 1997) goes wider butnot as deeply as Goldrei, for example, and in a more discursivestyle. From the preface: The book is confined to . . . what is calledfirst-order predicate logic, but it aims to treat this subject in verymuch more detail than a standard introductory text. In particular,whereas an introductory text will pursue just one style of seman-

tics, just one method of proof, and so on, this book aims to createa wider and a deeper understanding by showing how several al-ternative approaches are possible, and by introducing comparisonsbetween them. So Bostock does indeed usefully introduce you totableaux (trees) andan Hilbert-style axiomatic proof system andnatural deductionandeven a so-called sequent calculus as well (asnoted, it is important eventually to understand what is going on

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in these different kinds of proof-system). Anyone could profit fromat least a quick browse of his Part II to pick up the headline newsabout the various approaches.

Bostock eventually touches on issues of philosophical interest such

as free logic which are not often dealt with in other books at thislevel. Still, the discussions mostly remain at much the same levelof conceptual/mathematical difficulty as the later parts of Tellersbook and my own. In particular, he proves completeness for tableauxin particular, which I always think makes the needed constructionseem particularly natural.Intermediate Logicshould therefore be, asintended, particularly accessible to philosophers who havent donemuch formal logic before and should, if required, help ease the transi-

tion to work in the more mathematical style of the books mentionedin the last section.

Next let me mention a very nice, and freely available, alternative pre-sentation of logic via natural deduction:

2. Neil Tennant, Natural Logic** (Edinburgh UP 1978, 1990). Nowout of print, but downloadable as a scanned PDF (a rather large

file).All credit to the author for writing the first textbook aimed at

an introductory level which does Gentzen-style natural deduction.Tennant thinks that this approach to logic is philosophically highlysignificant, and in various ways this shows through in his textbook.Although not as conventionally mathematical in look-and-feel assome alternatives, it is in fact very careful about important details.It is not always an easy read, however, despite its being intended

as a first logic text for philosophers, which is why I didnt mentionit in the last section. However it is there to freely sample, somephilosophers could prefer Tennant to C&H, and anyone may wellfind it highly illuminating parallel reading on natural deduction.

Now, I recommendedA Friendly Introductionas a follow-up to C&H:but Learys book might not in every library. So as a possible alternative,

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I should mention an older and much used text which should certainly bevery widely available:

3. Herbert Endertons A Mathematical Introduction to Logic (Aca-

demic Press 1972, 2002) also focuses on a Hilbert-style axiomaticsystem, and is often regarded as a classic of exposition. However,it does strike me as a little more difficult than Leary, so Im notsurprised that some students report finding it a bit challenging ifused by itself as a first text. Still, it is an admirable and very reliablepiece of work which you should be able to cope with if you take itslowly, after you have tackled C&H. Read up to and including 2.5or2.6 at this stage. Later, you can finish the rest of that chapter to

take you a bit further into model theory. For more about this book,seethis book note.

I will also mention though with just a little hesitation another muchused text. This has gone through multiple editions and should also bein any library, making it a useful natural-deduction based alternative toC&H if the latter isnt available (though like Endertons book, this isperhaps half-a-notch up in terms of mathematical style/difficulty). Later

chapters of this book are also mentioned below in the Guide as possi-ble reading for more advanced work, so it could be worth making earlyacquaintance with . . .

4. Dirk van Dalen, Logic and Structure* (Springer, 1980; 5th edition2012). The chapters up to and including 3.2 provide an introduc-tion to FOL via natural-deduction, whichisoften approachable andwritten with a relatively light touch. However and this is why

I hesitate to include the book it has to be said that the bookisnt without its quirks and flaws and inconsistencies of presenta-tion (though maybe you have to be an alert and pernickety readerto notice them). Still the coverage and general approach is good, andmathematicians should be able to cope readily. I suspect the bookcould occasionally be tougher going for philosophers if taken froma standing start which is one reason why I have recommended

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beginning with C&H instead. Though if you have already workedthrough C&H, then reading van Dalen would certainly consolidateunderstanding. (See mymore extended reviewof the whole book.)

An aside: The treatments of FOL I have mentioned so far here and inthe last section include exercises, of course; but for many more exercisesand this time with extensive worked solutions you could also look at theends of Chs. 1, 3 and 4 of Rene Cori and Daniel Lascar, MathematicalLogic: A Course with Exercises(OUP, 2000). I cant, however, particularlyrecommend the main bodies of those chapters.

Next, here are three more books which elaborate on core FOL in dif-ferent ways. In order of increasing sophistication,

5. Jan von Platos Elements of Logical Reasoning* (CUP, 2014) isbased on the authors introductory lectures. Quite a lot of mate-rial is touched on in a relatively short compass as von Plato talksabout a range of different natural deduction and sequent calculi. SoI suspect that, without any classroom work to round things out, thismight not be easy if read as a first introduction. But suppose youhave already met one system of natural deduction (e.g., as in C&H),and now want to know more about proof-theoretic aspects of this

and related systems. Suppose, for example, that you want to knowabout variant ways of setting up ND systems, about proof-search,about the relation with so-called sequent calculi, etc. Then this isa very clear, approachable and interesting book. Experts will seethat there are some novel twists, with deductive systems tweakedto have some very nice features: beginners will be put on the roadtowards understanding some of the initial concerns and issues inproof theory.

6. Dont be put off by the title of Melvin Fittings First-Order Logicand Automated Theorem Proving (Springer, 1990, 2nd end. 1996).Yes, at various places in the book there are illustrations of how toimplement various algorithms in Prolog. But either you can easilypick up the very small amount of background knowledge about Pro-log thats needed to follow whats going on (and thats a fun thing

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to do anyway) or you can just skip.As anyone who has tried to work inside an axiomatic system

knows, proof-discovery for such systems is often hard. Which axiomschema should we instantiate with which wffs at any given stage of a

proof? Natural deduction systems are nicer: but since we can makeany new temporary assumption at any stage in a proof, again weneed to keep our wits about us, at least until we learn to recognizesome common patterns of proof. By contrast, tableau proofs (a.k.a.tree proofs, as in my book) very often write themselves, which iswhy I used to introduce formal proofs to students that way forwe thereby largely separate the business of getting across the ideaof formality from the task of teaching heuristics of proof-discovery.

And because tableau proofs very often write themselves, they arealso good for automated theorem proving.

Now (this paragraph is for cognoscenti!), if you have seen tableaux(trees) before, youll know that an open tableau for a single propo-sitional calculus wffA at the top of the tree in effect constructs adisjunctive normal form for A just take the conjunction of atomsor negated atoms on each open branch of a completed tree and dis-

join the results from different branches to get something equivalentto A. And a tableau proof that C is a tautology in effect works byseeking to find a disjunctive normal form for C and showing itto be empty. When this is pointed out, you might well think Aha!Then there ought to be a dual proof-method which works with con-

junctive normal forms, rather than disjunctive normal forms! Andyou of course must be right. This alternative is called the resolutionmethod, and indeed is the more common approach in the automated

proof community.Fitting always a most readable author explores both the

tableau and resolution methods in this exceptionally clearly writ-ten book. The emphasis is, then, notably different from most ofthe other recommended books: but the fresh light thrown on first-order logic makes the detour through this book vaut le voyage, asthe Michelin guides say. (By the way, if you dont want to take the

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full tour, then theres a nice introduction to proofs by resolution inShawn Hedman, A First Course in Logic(OUP 2004): 1.8, 3.43.5.)

7. Raymond Smullyan,First-Order Logic* (Springer 1968, Dover Pub-lications 1995) is a terse classic, absolutely packed with good things.In some ways, this is the most sophisticated book Im mentioningin this chapter. But enthusiasts can try reading Parts I and II, justa hundred pages, after C&H. Those with a taste for mathemati-cal neatness should be able to cope with these chapters and willappreciate their great elegance. This beautiful little book is thesource and inspiration of many modern treatments of logic based

on tree/tableau systems, such as my own though it packs in a lotmore than I do even at three times the length.

Not always easy, especially as the book progresses, but wonderfulfor the mathematically minded.

Lets finish by mentioning another, much more quirky, recent book bythe last author:

8. Raymond SmullyansLogical Labyrinths(A. K. Peters, 2009) wontbe to everyones taste. From the blurb: This book features a uniqueapproach to the teaching of mathematical logic by putting it inthe context of the puzzles and paradoxes of common language andrational thought. It serves as a bridge from the authors puzzle booksto his technical writing in the fascinating field of mathematical logic.Using the logic of lying and truth-telling, the author introduces thereaders to informal reasoning preparing them for the formal study of

symbolic logic, from propositional logic to first-order logic, . . . Thebook includes a journey through the amazing labyrinths of infinity,which have stirred the imagination of mankind as much, if not more,than any other subject.

Smullyan starts, then, with puzzles of the kind where you arevisiting an island where there are Knights (truth-tellers) and Knaves(persistent liars) and then in various scenarios you have to work out

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whats true from what the inhabitants say about each other and theworld. And, without too many big leaps, he ends with first-orderlogic (using tableaux), completeness, compactness and more. Not asubstitute for more conventional texts, of course, but for those

with a taste for being led up to the serious stuff via sequences ofpuzzles an entertaining and illuminating supplement.

Postcript Obviously, I have still only touched on a very small proportion ofbooks that cover first-order logic. The Appendix covers another handful. ButI end this chapter responding to some particular Frequently Asked Questionsprompted by earlier versions of the Guide.

What about Mendelson? Somewhat to my surprise, perhaps the most frequentquestion I have been asked in response to earlier versions of the Guide is Butwhat about Mendelson, Chs. 1 and 2? Well, Elliot Mendelsons Introduction toMathematical Logic(Chapman and Hall/CRC 5th edn 2009) was first publishedin 1964 when I was a student and the world was a great deal younger. The bookwas the first modern textbook at its level (so immense credit to Mendelson forthat), and I no doubt owe my career to it it got me through tripos! And itseems that some who learnt using the book are in their turn still using it to teachfrom. But lets not get sentimental! It has to be said that the book is often briskto the point of unfriendliness, and the basic look-and-feel of the book hasntchanged as it has run through successive editions. Mendelsons presentation ofaxiomatic systems of logic are quite tough going, and as the book progressesin later chapters through formal number theory and set theory, things if any-thing get somewhat less reader-friendly. Which doesnt mean the book wontrepay battling with. But unsurprisingly, nearly fifty years on, there are manyrather more accessible and more amiable alternatives for beginning serious logic.Mendelsons book is certainly a landmark worth visiting one day, but dont start

there. For a little more about it,see here.If you do really want an old-school introduction from roughly the same era,Id recommend instead Geoffrey Hunter,Metalogic* (Macmillan 1971, Universityof California Press 1992). This is not groundbreaking in the way e.g. SmullyansFirst-Order Logic is, nor is it as comprehensive as Mendelson: but it is stillan exceptionally good student textbook from a time when there were few tochoose from, and I still regard it with admiration. Read Parts One to Threeat this stage. And if you are enjoying it, then do eventually finish the book: it

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goes on to consider formal arithmetic and proves the undecidability of first-orderlogic, topics we revisit in 5.3.Unfortunately, the typography from pre-LATEXdays isnt at all pretty to look at: this can make the books pages appearrather unappealing. But in fact the treatment of an axiomatic system of logic isextremely clear and accessible. Worth blowing the dust off your librarys copy!

What about Bell, DeVidi and Solomon? I suggested in the previous chapterthat if you concentrated at the outset on a one-proof-style book, you would dowell to widen your focus at an early stage to look at other logical options. Andone good thing about Bostocks book is that it tells you about different stylesof proof-system. A potential alternative to Bostock at about the same level,and which initially looks promising, is John L. Bell, David DeVidi and GrahamSolomonsLogical Options: An Introduction to Classical and Alternative Logics(Broadview Press 2001). This book covers a lot pretty snappily for the moment,

just Chapters 1 and 2 are relevant and a few years ago I used it as a text forsecond-year seminar for undergraduates who had used my own tree-based bookfor their first year course. But many students found it quite hard going, as theexposition is terse, and I found myself having to write very extensive seminarnotes. For example, see my notes onTypes of Proof System,which gives a briskoverview of some different proof-styles (written for those who had first done logicusing by tableau-based introductory book). If you want some breadth, youd dobetter sticking with Bostock.

And what about Sider? Theodore Sider a very well-known philosopher haswritten a text called Logic for Philosophy* (OUP, 2010) aimed at philosophers,which Ive been asked to comment on. The book in fact falls into two halves. Thesecond half (about 130 pages) is on modal logic, and I will return to that in 6.1.The first half of the book (almost exactly the same length) is on propositionaland first-order logic, together with some variant logics, so is very much on thetopic of this chapter. But while the coverage of modal logic is quite good, I cantat all recommend the first half of this book.

Sider starts with a system for propositional logic of sort-of-sequent proofs

in what is pretty much the style of E. J. Lemmons 1965 book Beginning Logic.Which, as anyone who spent their youth teaching a Lemmon-based course knows,students do not find user-friendly. Why on earth do things this way? We thenget shown a Hilbertian axiomatic system with a bit of reasonably clear expla-nation about whats going on. But there are much better presentations for themarginally more mathematical.

Predicate logic gets only an axiomatic deductive system. Again, I cant think

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this is the best way to equip philosopherswho might have a perhaps shaky gripon formal ideas with a better understanding of how a deductive calculus for first-order logic might work, and how it relates to informal rigorous reasoning. But,as I say, if you are going to start with an axiomatic system, there are betteralternatives. The explanation of the semantics of a first-order language is quitegood, but not outstanding either.

Some of the decisions about what technical details then to cover in somedepth and what to skim over are pretty inexplicable. For example, there arepages tediously proving the mathematically unexciting deduction theorem foraxiomatic propositional logic (the result matters of course, but the proof is onesimple idea and all the rest is tedious checking): yet later just one paragraph isgiven to the deep compactness theorem for first-order logic, which a philosophystudent starting on the philosophy of mathematics might well need to know

about and understand some applications of. Why this imbalance? By my lights,then, the first-half of Siders book certainly isnt the go-to treatment for givingphilosophers a grounding in core first-order logic.

True, a potentially attractive additional feature of this part of Siders book isthat it does contain brief discussions about e.g. some non-classical propositionallogics, and about descriptions and free logic. But remember all this is being donein 130 pages, which means that things are whizzing by very fast. For example,the philosophically important issue of second-order logic is dealt with far tooquickly to be useful. And at the introductory treatment of intuitionistic logic is

also far too fast. So the breadth of Siders coverage here goes with far too muchsuperficiality. Again, if you want some breadth, Bostock is still to be preferred,plus perhaps some reading from 6.3below.

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Part III

Going beyond the basics

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Chapter 5

Elementary MathematicalLogic

We now press on from an initial look at first-order logic to consider othercore elements of mathematical logic. Recall, the three main topics we needto cover are

Some elements of the model theory for first-order theories.

Formal arithmetic, theory of computation, Godels incompleteness

theorems. Elements of set theory.

But at some point well also need to touch briefly on

Second-order logic and second-order theories.

A natural place to do this is after finding out just a little more about thecontrasting model theory of first-order theories.

As explained in 1.4, I do very warmly recommend reading a seriesof books on a topic which overlap in coverage and difficulty, rather thanleaping immediately from an entry level text to a really advanced one.You dont have to follow this excellent advice, of course. But I mention itagain here to remind you of one reason why the list of recommendationsin most sections is quite extensive and increments in coverage/difficultybetween successive recommendations are often quite small: this level of

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logic really isnt as daunting as the overall length of this chapter mightsuperficially suggest. Promise!

5.1 From first-order logic to elementary modeltheory

The completeness theorem is the first high point the first mathematicallyserious result in a course in first-order logic; and some elementary treat-ments more or less stop there. Many introductory texts, however, continue

just a little further with some first steps into model theory. It is clearenough what needs to come next: discussions of the so-called compact-

ness theorem (also called the finiteness theorem), of the downward andupward Lowenheim-Skolem theorems, and of their implications. Theresless consensus about what other introductory model theoretic topics youshould meet at an early stage.

As youll see, youll very quickly meet claims that involve infinite car-dinalities and also occasional references to the axiom of choice. In fact,even if you havent yet done an official set theory course, you may wellhave picked up all you need to know in order to begin model theory. If you

have met Cantors proof that infinite collections come in different sizes,and if you have been warned to take note when a proof involves makingan infinite series of choices, you will probably know enough. And Goldreischapter recommended in a moment in fact has a brisk section on the Settheory background needed at this stage. (If thats too brisk, then do askim read of e.g. Paul Halmoss very short Naive Set Theory*, or one ofthe other books mentioned at the beginning of5.4below.)

Lets start by mentioning a book that aims precisely to bridge the gapbetween an introductory study of FOL and more advanced work in modeltheory:

1. The very first volume in the prestigious and immensely useful Ox-ford Logic Guides series is Jane Bridges very compact BeginningModel Theory: The Completeness Theorem and Some Consequences(Clarendon Press, 1977) which neatly takes the story onwards just

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a few steps from the reading on FOL mentioned in our 4.1above.You can look at the opening chapter to remind yourself about thenotion of a relational structure, then start reading Ch. 3 for a veryclear account of what it takes to prove the completeness theorem

that a consistent set of sentences has a model for different sizes ofsets of sentences. Though now note the compactness theorem, and3.5 on Applications of the compactness theorem.

Then in the forty-page Ch. 4, Bridge proves the Lowenheim-Skolem theorems, which tell us that consistent first order theoriesthat have an infinite model at all will have models of all differentinfinite sizes. That shows that interesting first-order theories arenever categorical i.e. never pin down a unique model (unique up

to isomorphism, as they say). But that still leaves open the possi-bility that a theorys models of a particular sizeare all isomorphic,all look the same. So Bridge now explores some of the possibili-ties here, how they relate to a theorys being complete (i.e. settlingwhether or for any relevant sentence ), and some connectedissues.

The coverage is exemplary for such a very short introduction, and

the writing is pretty clear. But very sadly, the book was printed inthat brief period when publishers thought it a bright idea to savemoney by photographically printing work produced on electric type-writers. Accustomed as we now are to mathematical texts beingbeautifully LATEXed, the lookof Bridges book is really off-putting,and some might well find that the books virtues do not outweighthat sad handicap. Still, the discussion in 3.5 and the final chaptershould be quite accessible given familiarity with an elementary first-

order logic text like Chiswell and Hodges: just take things slowly.

Still, I am not sure that almost forty years on from publication this isstill the best place for beginners to start: but what are the alternatives?

2. I have already sung the praises of Derek GoldreisPropositional andPredicate Calculus: A Model of Argument* (Springer, 2005) for theaccessibility of its treatment of FOL in the first five chapters. Now

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you can read his 4.4 and 4.5 (which I previously said you couldskip) and then Ch. 6 on Some uses of compactness to get a veryclear introduction to some model theoretic ideas.

In a little more detail, 4.4 introduces some axiom systems de-

scribing various mat

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