A long time ago, when the world was young and UK philosophy departments nearly all taught an amount of formal logic to their first year students, Peter Millican (I think it was) wrote round — yes, this was before email — to ask what text books we used in our courses. The majority answer was “Lemmon but …”, as in “I use Lemmon, but the students find it tough/I have to supplement it with handouts/I don’t really like it because …” There wasn’t a huge amount of choice back then, however, and Lemmon’s 1965 *Beginning Logic* was indeed notably better than most of the alternatives.

Nowadays, by contrast, there’s a ridiculous number of introductory logic books to choose from. If you ask around, no book stands out (partly no doubt because there’s no longer any kind of consensus about what should go into an entry-level logic course). However, the *form* of the answer still tends to be the same — “I use XYZ, but …”. Teachers of first year logic do seem to be a *very* picky bunch, rarely satisfied by someone else’s text. The unwise and perhaps over-confident among us of course think we could do better. The *really* unwise actually try do so. We spend an inordinate amount of time writing our introductory masterpiece, not believing the friends who kindly warn us that it will all take three times longer than we’d ever planned. And then an ungrateful world quite mysteriously fails to rush to welcome the result of all our efforts as the One True Logic Text.

So it goes. I speak from hard-won experience. It is with warm fellow-feeling for the author, then, that I note the arrival on the scene of Yet Another Introductory Logic Book (I rather wanted to use that as the title for mine, but thought better of it). My namesake Nicholas J. J. Smith’s *Logic: The Laws of Truth* is now out with Princeton UP: the publisher’s web page for the book is here. If you are thinking of adopting a new logic text, you can get an e-inspection copy. Full disclosure: Nick has very kindly indeed sent me a hardback copy.

I’ve browsed through quite a bit, dipping in and out. In general terms, I do much like the tone and the level and the coverage and the writing. And there’s the right amount of more ‘philosophical’ discussion alongside the formal work. Though I’m now going to be picky. Of course. Read “below the fold” for some comments. Still, the headline news is that if you are a teacher in the market for new logic text, or a student looking for very helpful reading, this could indeed be the book for you.

(1) *First appearances* It isn’t the most important thing, of course, but it is not trivial either: it should be said straight away that the book has been very attractively typeset. It immediately *looks *more appealing than many logic books — not an easy thing to pull off with symbol-laded texts (though see (3) below, where I in effect complain that there are not enough symbols!). Credit to the publishers for this.

(2)* Coverage * The first 350 pages or so are logic-by-trees, for propositional logic (Part I) and predicate logic (Part II). The last hundred and a bit pages are on ‘Foundations and Variations’. The first chapter in Part III gives soundness and completeness proofs for the tree system (and comments on decidability, etc.). The second chapter discusses in fifty pages ‘Other Methods of Proof’ (axiomatic, ND, sequent calculi). The third and last of the additional chapters offers thirty pages on set theory.

I have to say that I’m quite green with envy that Nick got the space for his Part III! In my case, CUP balked at the idea of going much over 350 pages: so though I too start with trees and I sneak in a completeness proof for them, the chapters I’d have liked to have included on natural deduction in particular had to be axed.

Obviously, then, I do *very* much approve of the overall pattern of this book: trees first (students do find them easier to manage), then expand the story somewhat to other proof systems. It is one good way of arranging things. If, on the other hand, you are wedded to the belief that natural deduction should come more or less first, then you won’t like this book any more than you like other tree-first texts.

Parts I and II and a bit of Part III of Nick’s book cover much the same material, then, as my book, in about the same number of pages: so, at least initially, we pretty much agree on the major issue of pace too. But Part III does ratchet up the pace quite a bit: so, more generally, how well has Nick used his extra page budget? I’d have said that sequent calculi are definitely a more ‘advanced’ topic, and don’t belong in a first logic book. And as for axiomatic presentations of logics, I’d not say much about them (or just leave them till later too, until axiomatic theories are being discussed more generally). Instead, I would have used quite a bit of the extra space for a more generously paced discussion of natural deduction, first Fitch-style, then Gentzen-style. Nick’s squeezed fourteen pages are (I believe) just too quick to make a comfortable introduction for beginners, or to give them a real feel for the *naturalness* of certain systems: as I say, the pace has noticeably speeded up. (And would a student pick enough to be able to see what is going on in Dummettian discussions of harmony etc. that she might later encounter?) So I wouldn’t have used the fifty pages of his penultimate chapter as Nick does. But you might prefer the breadth against the depth.

As to the chapter on set theory, my feelings about this are pretty mixed. On the one hand, I agree that beginners need to pick up a reading-knowledge of set notation (because they are going to repeatedly meet it later). On the other hand, at this level we really shouldn’t be corrupting the youth by getting sets into the story unnecessarily soon, or e.g. by talking of relations or functions as sets of tuples (Great-uncle Frege will be *very* cross).

(3) *Talk’n’chalk *In logic lectures, you introduce the Big Ideas, and then do some worked examples on the board to illustrate them. In a book, you can give more detailed and expansive explanations of the Big Ideas (with the *i*‘s dotted and the *t*‘s crossed, and twiddly bits added); and you’ve got room too for more worked examples with running commentaries which the reader can review in her own time.

Nick is, I think, very good at the talk, at the discursive explanations: patient and very lucid. But in the book he isn’t particularly generous with the chalk, with the worked-examples-with-running-commentary. In fact, you might say he seems to be downright stingy. At a rough estimate, my book has — for one example — well over twice as many examples of propositional trees, and they tend to have more running commentary too. For another example, take that stumbling block for beginners, the translation into predicate logic of sentences involving multiple quantifiers. There are almost no worked examples here as compared with my book (or Paul Teller’s or many others). For a third example, there are remarkably few worked examples of predicate trees. (Ah: perhaps this explains in part the initially friendly look of the book — it just doesn’t have as many displayed arrays of scary symbols as you’d expect!)

True, there *are* lots of exercises in *Logic: The Laws of Truth*, and there will soon be a long answer-book freely available on the web. And that might well go some way to soothing worries here. But — from the draft document I’ve seen at the moment — the solutions are given in the answer-book, but again without much commentary (I mean without the class-room remarks like “At this point in the tree, we could instantiate the quantifier at line 4 with either *a* or *b*: the second is the better bet because …”, “The translation …. is tempting, but that’s wrong because ….”, and so on). Overall, I still think that the off-line student sitting with the book should have rather more by way of commented worked examples of all kinds immediately to hand to refer to as paradigms, before she embarks on exercises. But I agree that’s a debatable question of teaching style.

(4) *Getting pernickety *Comments (2) and (3) are about overall features of the book — its general shape, and the number of detailed worked examples as you go through. When we turn to fine details, then the fun can start!

I don’t think you should define “argument” as on p. 11 so that arguments can only have one inference step. On p. 41, the symbols of PL (the language of propositional logic) lack the ‘therefore’ sign; by p. 64, PL has acquired the sign unannounced, for “we translate [a certain] argument into PL as follows”, In my experience, you have to say quite a bit more than Nick does on p. 78 to make students swallow *ex falso quodlibet*. On p. 82, Nick oddly continues to talk about *the* logical form of a proposition when he has just argued there is typically no such thing (so he finds himself setting the exercise “Give three correct answers to the question ‘what is the form of this proposition?’ …”!). And so it goes.

Then there are little sins(?) of omission: for a start, you might query the lack of any discussion of the proper use of quotation marks or of use/mention (unless I have missed it!).

A somewhat more serious niggle concerns defining a valid argument at the outset to be one which is necessarily truth preserving “by virtue of its form or structure” — for I just didn’t pick up a clear account of what in general makes for a structural or formal feature of an inference as presented in the vernacular. We can tell stipulative stories about regimented arguments in certain formal languages, of course: *this*, we say, is deemed to be logical vocabulary, *that* isn’t. But the definition of validity is offered before we are supposed to know about such things, as a general story. Nick doesn’t say anywhere near enough to make it fly.

But you can nag away at any logic book like this, and I’m not going to do more of that here and now (though I may return to some themes)! Whether the book will work as a first formal logic text for you — as a teacher or a student — will surely depend on the bigger, structural, things. Do you want to start with trees? If you do, do you also want a book that tells you something about ND too (but not a lot)? How many commented worked examples do you want, or is it most important to you to get good discursive explanations of the Big Ideas? Nick’s whole book will take you to almost the same place as e.g. my book plus Bostock’s, with a bit of set theory thrown in, in considerably few pages than our two books together. Do you want to travel a bit more speedily with him? Alternatively, if what you want in a first course stops after Part II, then Nick’s very nicely written book and mine would work well together with one as the supplementary back-up reading for the other. The choice is yours!

Nick SmithThanks Peter. Just a quick follow-up to say that the promised answer book is now finished and available here.

Nick SmithThanks so much for your post, Peter: I’m really grateful to you for taking time to look at the book and to comment on it. I’ve been following your blog for a long time and have got a lot out of your book notes — they’re a real service to the community.

Without getting too pernickety, I’ll just make a few responses to some of your bigger-picture comments.

Worked examples: My approach was to get the reader into the exercises as soon as possible — to get them trying to do the logic for themselves. If some readers have little idea on how to proceed on a given kind of problem, they can first look at a few answers (i.e. use them as examples); those who are getting an idea of what is going on can then start to try to answer questions for themselves, checking against the given answers as they go; and then those getting more confident can do their own answers and then check the given answers afterwards. Maybe you’re right that some of the answers could benefit from some additional commentary! The nice thing about having them online is that this sort of thing can be added down the track, if there is a demand for it. (As for offline readers, the online answers are in PDF format, so they can be printed out.)

Set theory: It’s a lingua franca; students of the formal sciences won’t get far without some basic knowledge of it — so I do think this chapter needs to be there. Personally my positive feelings towards set theory go further than this: for example, I think that seeing how all sorts of different things (numbers, functions, formal languages, and so on) can all be seen as living in the world of sets is conceptually enlightening (whether or not one ultimately accepts any given such reduction as the truth of the matter). But I think I basically agree with you about not introducing sets

toosoon: this chapter is the final one, and is billed as being more of an appendix than a regular chapter.Validity as necessary truth preservation

in virtue of form: I know it’s pretty common nowadays for logic books and teachers to introduce the intuitive conception of validity as the idea of necessary truth preservation (the conclusionmustbe true if the premisses are true). But historically, I don’t think thisisthe guiding conception of validity — and furthermore it runs into the problem (cf. Etchemendy) that the model-theoretic definition of logical consequence cannot rightly be seen as a precise spelling-out ofthisnotion (whereas itcan, I think, be seen as a precise spelling-out of the notion of necessary truth preservationin virtue of form). So I do think it’s important to mention the idea of form in the initial intuitive motivating story.Peter SmithThanks Nick. I’m not unsympathetic to your remarks about validity and form. But I’d be more cautious, more hedged, signalling more explicitly to the beginner that the issue of how to apply the idea of truth-preservation in virtue of form to ordinary arguments is messy to say the least. Still, that doesn’t matter, fortunately, ‘cos our concern is going to be regimented arguments in formal languages where we

cangive a sharp account. And I’d want to bring out that contrast more.As to sets, there’s the kind of set talk which is virtual (in Quine’s sense) and can be traded in for plural talk: and there’s more committal set talk. The first is harmless. But you get pretty committing, talking about the iterative conception, no less. That’s overkill, I’d still say, in an introductory book!

Aldo AntonelliMy first reaction upon reading about N. Smith’s new book was that there is no way to justify writing 576 pages to teach students about truth trees. I think I have said in comments on this site that one of the main virtues of texts I like, for instance Teller’s

Primeris its brevity. I am with Callimachus here:mega biblion, mega kakon.But on reflection I think we might have a disagreement as to what logic at this level is like. I tend to think of it more as a specialized skill, rather than a body of knowledge. We teach students

howto translate arguments,howto show them valid by truth trees or natural deduction etc., rather than, e.g.,whatlogical form is.Sure, recognizing validity requires some grasp of logical form. But students can achieve that without a full-fledged comprehensive theory of logical form. In fact, professional logicians think they know what logical form is, in spite of there not being available a complete, definitive, theory of logical form.

I grant that this way of conceiving of logic is really more suited for the classroom than for self-study. There you can get away with minimal textbook support, and fill in the details at the chalkboard. I have in fact in the past taught intro logic with

notextbook; I don’t do that anymore, not because it doesn’t work, but because it makes students feel better if they have a textbook as a security blanket; nonetheless I always tell them that they shouldn’t think they can learn logic by reading the book, any more than they can learn how to swim by reading on the theory of buoyancy (you gotta jump in the water).Peter SmithWell, to be accurate, it is 528 pages in all, but under 370 of them are on logic-by-trees. And those include a fair amount of ‘philosophical’ discussion en route (e.g. about the relation between ordinary language connectives and the formal counterparts, about definite descriptions, and so on — the usual kinds of thing).

But yes, I agree, N. Smith’s book, like P. Smith’s, may well strike the more mathematically minded as unnecessarily laborious. Still, even in a place like Cambridge, there’s a pretty substantial proportion of students who aren’t so mathematical, and who do seem to need to take logic pretty slowly — who need take-home texts with patient explanations and worked examples before they can “jump in the water”. So I think there’s certainly a place for our kind of book.

Nick SmithYes, I do think we have a pretty fundamental disagreement here regarding

desiderata. I say at the beginning of the Preface of my book that the aim

is to discuss the

whyof logic (what it all means, what is reallygoing on) as well as the

how(the formal tools and techniques). Sothe idea was, explicitly, to write a philosophically rich introduction to

logic — as opposed to a sparse introduction to the mechanics. (Of course I

am not the only person to attempt such a thing. For example, Peter has a

lot of interesting discussion along these lines in his intro book.)