# On Sider’s Logic for Philosophers — 1

It hasn’t been mentioned yet in the Teach Yourself Logic Guide, so I’ve predictably been asked a fair number of times: what do I think about Ted Sider’s Logic for Philosophy (OUP 2010)? Isn’t it a rather obvious candidate for being recommended in the Guide?

Well, I did see some online draft chapters of the book a while back and wasn’t enthused. But, yes, I am more than overdue to take a look at the published version. So here goes …

The book divides almost exactly into two halves. The first half (some 132 pages), after an initial chapter ‘What is logic?’, reviews classical propositional and predicate logic and some variants. The second half (just a couple of pages longer) is all about modal logics. I’ll look at the first half of the book for this post, and leave the second half (which looks a lot more promising) to be dealt with a follow-up.

OK. I have to say that the first half of Sider’s book really seems to me to be ill-judged (showing neither the serious philosophical engagement you might hope for, or much mathematical appreciation).

Here is one preliminary point. The intended audience for this book is advanced philosophy students, so presumably students who have read or will read their Frege and their Tractatus. So what, for example, will they make of being baldly told in §1.8, without defence or explanation, that relations are in fact objects (sets of ordered pairs), and that functions are objects too (more sets of ordered pairs)? There’s nothing here about intension and extension, and about why we should identify functions with their graphs. We are equally baldly told to think of binary functions  as one-place functions on ordered pairs (and the function that maps two things to their ordered pair …?). Puzzled philosophers might well want to square what they have learnt from Frege and the Wittgenstein with modern logical practice as they first encountered  it in their introductory logic courses: so you’d expect a second level book designed for such students to proceed more cautiously and address the obvious worries. But that doesn’t happen here.

And in fact we get a pretty skewed description of modern logic anyway, even from the very beginning, starting with the Ps and Qs. Sider seems stuck with thinking of the Ps and Qs as Mendelson does (the one book which he says in the introduction that he is drawing on for the treatment of propositional and predicate logic). But Mendelson’s Quinean approach is actually quite unusual among logicians, and certainly doesn’t represent the shared common view of ‘modern logic’. I won’t rehearse the case again now, as I’ve explained it at length here. But students need to know there isn’t a uniform single line to be taken here.

When Sider turns to looking at formal systems for propositional logic we get sequent proofs in what is pretty much the style of Lemmon’s book. Which as anyone who spent their youth teaching a Lemmon-based course knows, students do not find user friendly. Why do things this way? And how are we to construe such a system? One natural way of understanding what is going on is that the system is a formalized meta-theory about what follows from what in a formal object-language. But no: according to Sider sequent proofs aren’t metalogic proofs because they are proofs in a formal system. Really? (Has Sider not noticed that in Mendelson too, the formal proofs are all metalogical?)

OK, so the philosophical student is introduced to an unfriendly version of a sequent calculus for propositional logic, and then to an even more unfriendly Hilbertian axiomatic system. Good things to know about, but probably not when done like this. And not — in a book addressed to puzzled philosophers — without a lot more discussion of how this all hangs together with what the student is likely to already know about, natural deduction and/or a tableau system. And not without a better discussion, too, of the way the conception of logic changed between e.g. Principia and Gentzen, from being seen as regimenting a body of special truths to being seen as regimenting inferential practice. Further, the decisions about what to cover and what not to cover are pretty inexplicable. For example, why pages actually proving the deduction theorem for axiomatic propositional logic, and later just one paragraph on the compactness theorem for FOL, which students might really need to know about and understand some applications of?

Predicate logic is then dealt with by an axiomatic system (apparently because this approach will come in handy in the second half of the book — I’m beginning to suspect that the real raison d’être of the book is indeed the discussion of modal logic). I can’t think this is the best way to equip philosophers who have a perhaps shaky grip on formal ideas with a better understanding of first-order logic. The explanation of the semantics of a first-order language isn’t bad, but not especially good either. This certainly isn’t the go-to treatment for giving philosophers what they need.

True, a nice feature of this part of Sider’s book is that it does have a discussion of some non-classical propositional logics, and has a little about descriptions and free logic.  But actually the philosophically serious issues of intuitionistic logic and second-order logic are dealt with far too quickly to be useful, so the breadth of Sider’s coverage goes with superficiality.

I could go on. But the headline summary about the first part of Sider’s book is that I found it (whether wearing my mathematician’s or philosopher’s hat) irritating and unsatisfactory. Sorry to be carping!

Comments from those who have used/taught/learnt from the book?

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### 8 Responses to On Sider’s Logic for Philosophers — 1

1. Daniel Nagase says:

Prof. Smith,

I didn’t read Sider’s published version, only the draft version. Still, considering it was my first real contact with “serious” logic (after reading what I think is the wonderful Suppes book), I found it pretty helpful. It’s true that some of the things are rather hurried or non-philosophical. Nevertheless, it taught me a number of good techniques (including induction) that have proved useful now that I’m pursuing more advanced studies. For example, the proof of the deduction theorem itself, while a little bit tedious, was pedagogically useful, as I could see more clearly in this simple setting how to proceed with the appropriate induction. The chapter on FOL also helped me to understand more clearly what’s at stake in the transition from propositional to predicate logic, specially with its focus on the semantics.

So, although the first half of the book is certainly not a thorough treatment of the technicalities involved, it did serve me as a good transition from “baby logic” to more advanced stuff.

• Peter Smith says:

Thanks for this! Though I guess this is not really inconsistent with the summary judgement that there are better options here.

2. Peter Smith says:

Gary Williams tweets: “Yes, thank you! I had the great displeasure of taking a course based on Sider’s book. Still have nightmares about Hilbert proofs.”

3. Geoffrey Ferrari says:

Hi Peter,

I learned at least two important things from Sider’s book. First, it was in Sider’s book that I first learned that one can interpret the semantic theory of logical validity as providing a more precise understanding of what it is for an argument to be truth-preserving “no matter how things are in the world and no matter how one interprets the meanings of the non-logical vocabulary” (my words, not Sider’s). I’m not so familiar with other logic books to be able to make a comparison, but Sider’s explanation of this point seemed to me to be admirably clear and instructive to the beginner insofar as it allows one to connect formal logical theory to one’s pretheoretical understanding.

Second, it was in Sider’s book that I first saw an explanation of why the truth value of sentences in predicate logic does not depend on one’s initial choice of variable assignment.

As I say, I can’t make any comparison with other books that may treat these topics. But these points seem to me very important for the beginner and I found Sider’s treatment nice and clear and helpful.

4. There’s nothing here about why we should treat functions that have the same graph as the same

OK, but does he need to argue why we should treat 4⁄8 the same as 1⁄2? I guess I feel the same about your comment in your linked PDF

the sentence ‘Either grass is green or grass is not green’ – at least
once we pre-process it as ‘Grass is green ∨ ¬ grass is green’ – is an instance of both .

which is beautifully written, by the way.

But the schema P ∨ ¬P and the schema Q ∨ ¬Q are isomorphic in the appropriate way. Can we fix the problems you’re saying by just inserting the phrase “up to isomorphism” in the appropriate places?

identify functions with their graphs

(or at least identify &Ropf;→&Ropf; etc functions with their graphs … we don’t have a standard “Graph” for functions from &Copf;→&Ropf;&sup3;)

• Peter Smith says:

The complaint about Sider on functions was that, in a book directed to philosophers you’d at least expect a discussion of functions-as-rules (i.e. treated intensionally) vs functions construed extensionally.

The complaint about schemas in the note about Goldfarb et al can indeed be approached by talking about isomorphism classes and so on (I wasn’t saying it was an unsolvable problem, just that more needed to be said.

5. Rowsety Moid says:

I’m finding it tricky to work out what the problem’s meant to be with the way Sider handles Ps and Qs. The blog post says:

Sider seems stuck with thinking of the Ps and Qs as Mendelson does (the one book which he says in the introduction that he is drawing on for the treatment of propositional and predicate logic). But Mendelson’s Quinean approach is actually quite unusual among logicians, and certainly doesn’t represent the shared common view of ‘modern logic’.

TextbooksOnLogic.pdf says:

It isn’t difficult to guess a major influence on Goldfarb. His one-time colleague W.V.O. Quine’s Methods of Logic was first published in 1950, and in that book – much used at least by philosophers – logical notions like consistency, validity and implication are indeed defined in the first instance for schemata.

But on p 37 of Sider’s book, he writes: “… the notion of validity does not apply to schemas” and explains that it’s defined instead in terms of truth in interpretations, and that that’s defined only for wffs. He notes that schemas contain metalinguistic variables and that that they’re aren’t part of the language of PL. So it looks like he isn’t taking the Quinean approach.

• Peter Smith says:

First, I should say that I’d originally intended to write a much long piece on Sider, but got cheesed off: so, yes, what I posted ended up disproportionately quibbling about $P$s and $Q$s — which isn’t at all my main worry about his presentation of first-order logic.

Looking again, though, I’m now not at all sure that Sider has a clear view about the $P$s and $Q$s. He says that they “represent declarative natural language sentences”. But what does “represent” mean here? It doesn’t mean “stand in for” or “abbreviate” particular natural language sentences (in a fixed context), else they would have meaning, and Sider thinks they don’t have meaning (though can be given meaning). Are they schematic? Hmmmmmm.

But anyway, that’s not my main worry (though his unclarity about this in a book by a philosophy for philosophers is not very encouraging). My main beef about the presentation of FOL, as will come out in the next update of the Guide, is his choice of systems to discuss, and his mode of discussing them. There are better options.