Why not ask your local friendly logician?

Logicians are perhaps of rather limited use to the world (as I’m occasionally reminded by Mrs Logic Matters). But they can be tolerably helpful when you are in danger of inadvertently confusing  use and mention, or if you want to avoid getting into muddles about variables, and so on.

Consider this:

A set is merely the result of collecting objects of interest, and it is usually identified by enclosing its elements with braces (curly brackets).

No: what gets surrounded by curly brackets in forming an expression identifying a set are expressions designating the elements, not the elements themselves. (And odd to say that sets are usually identified this way, using lists enclosed in curly brackets, when that only works for finite sets!)

Or how about this:

A property is a statement that asserts something about one or more variables. For example, the two statements “x is a real number” and “y \in \mathbb{R} and y \notin \mathbb{N}” are clearly properties that assert something, respectively, about x and y.

Ok: it makes sense to say e.g. that

\pi is a real number” asserts something about \pi,

because  \pi is a denoting term. But it doesn’t express any complete claim to say that

 ” x is a real number” asserts something about x

if “x” is left as a dangling free variable. And we can’t tidy up by imagining there is a governing quantifier, as you can’t quantify across quotes. Anyway, a property isn’t a statement of any kind (even if we allow open sentences with unbound variables to count as statements) — properties are what are expressed by open sentences, or are their semantic values, or some such.

How about this?

[An example of a compound sentence is]

P \land Q (means  “P and Q” and is called conjunction)

What convention on quotation marks is in play which would make it right to have the quotation marks this way round? And, being pernickety — but why not? —  it is certainly not the case that the wff is called “conjunction”! It is a conjunction.

Quite rightly, any logician would balk at write each of the above. Not so the mathematician Daniel Cunningham in his brand new book Set Theory: A First Course (CUP, 2016). Those quotations are all from the first six pages.

This seems a huge pity as the book later promises well as an introductory set theory text. I’ll report back in due course on the real content, once the book gets going. But it really is worth talking to your local friendly logician to avoid silly foul-ups like these.

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Solomon Feferman (13.xii.1928 — 26.vii.2016)

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I’m sad to hear that Solomon Feferman, an intellectual hero of mine, has died.

There is a very warm tribute here from the Stanford philosophy department.

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Those Brexit blues again

The admirable John Lanchester, writes on Brexit in the latest London Review of Books, which has made his piece freely available.

If you are not all Brexited out, then this strikes me as particularly insightful about some of the social currents at work, and more than usually worth a read.

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Reaching peak Apple: the Macbook 2016

I have not wanted to post about the really pretty miserable political news even though it has been much occupying my thoughts (for I have nothing new or insightful to add about Brexit,  about the incompetent narcissist supposedly leading the Labour Party, or about the new UK government, let alone about the disasters in the wider world). And I’m afraid that spending a lot of time tinkering with little presentational issues in the early chapters of my introductory text book as I slowly work towards a second edition doesn’t exactly provide inspiration for exciting logical posts here either.

So let me tell you about my new MacBook instead.

Headline news: if you are thinking of splurging out on new Apple kit, the revised 12″ Macbook is just terrific. Go for it.

Slow news: I’d been using as my portable (for libraries, cafés, sofa-surfing on my lap) a Macbook Air, now approaching five years old. Still a great bit of kit. But, and it’s quite a big ‘but’ for someone spending a lot of time writing, the on-screen text on the MBA now does seem decidedly blurry. Compared with modern retina screens on my iPhone, iPads, and a big and not-very-portable MacBook Pro used as my main home machine — yes, I really am reaching peak Apple here  — the elderly MBA screen is just not so great. Hence, initially pretty much for that reason alone, when a larger-than-expected royalty cheque arrived, I splashed out on the recently revised 12″ MacBook. Space gray, since you asked.

Six weeks in, I’m more than delighted. Some have commented adversely on the keyboard. But unless you are a very heavy-handed typist, you should find it excellent. Despite its different feel, going from this keyboard to the MacBook Pro and back really presented zero problem after just a day or two ‘s use. The screen is simply amazing. The lack of ports is also no problem at all for me, since all my work files live in DropBox, and everything else lives in various clouds too. And — now this was a very big surprise — the added portability (and added ease of use on a lap!) seems completely out of proportion to the actual change in size and weight compared to the MBA. Finally, the MacBook is even more gorgeous than the MBA, and I think aesthetics matter if you are spending so much time up close and personal with something!

Whether it would suit you as a sole machine, I can’t say: read the usual forums for advice on that. But for anyone looking for a very portable second machine to take out and about, this works a treat in every way. You’d have to be very hard to please not to be delighted!

Nerdy note: I initially got the entry-level model, but then returned it for the upgrade to the fastest m7 chip which does make a noticeable difference working with LaTeX: I’d recommend the small extra outlay for that reason. More on this, with some timings, here.

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The Arcanto Quartet with Maximilian Hornung play the Schubert Quintet

By chance, we just caught this BBC broadcast, available for 30 days, of a truly fine performance of the Schubert Quintet by the Arcanto Quartet with Maximilian Hornung as the additional cellist. An hour’s suitably reflective respite from the grimmest of news,  which others might appreciate too.

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IFL2: a first instalment

OK, I have been tinkering with the opening chapters of my Introduction to Formal Logic, trying to improve them for the planned second edition. Here then are the early chapters up to and including the first Interlude, in an initial re-draft [updated Aug 31]. Some quick notes:

  • I haven’t yet revised the end-of-chapter Exercises.
  • If you don’t know my book, then as in the first edition, Chapters 1 to 3 correspond roughly to e.g. the preamble chapter in Benson Mates’s book. Then Chapters 4 and 5 say something about showing invalid inference are invalid by the counterexample trick, and about showing valid inference are valid by coming up with multi-step proofs.
  • The old Chapter 6 has disappeared, however, with some material working into the end of Chapter. 5. I plan now to talk more about the validity of arguments with contradictory premisses later, and no longer think it good policy to muddy the waters by discussing this too soon.
  • This version, then, is 44 pages rather than 52 pages as before. I hope the result is overall crisper, clearer and better focussed, and certainly some repetition has disappeared. (To be honest, I cringe a  bit at some passages in the first edition!)
  • So … comments and corrections are most welcome! Regular readers here, please do, do chip in if you have anything useful to say. But also, if you have some students, beginners or recent beginners, who would be interested in giving feedback, please do point them this first excerpt from the book. In fact, encourage them by telling them that, when I asked for advice/comments on chapters from the second edition of my Gödel book, there was no correlation at all between seniority and the usefulness of suggestions.
  • Comments are probably better sent by email (rather than using the comments box — since this is much easier for your writing and a bit easier for my reading). If you have lots of comments, the ideal is perhaps to return a marked-up PDF. But whatever works for you! Use ps218 at cam dot ac dot uk
  • I’ll keep the current version fixed now for a few weeks, rather than revise piecemeal as comments arrive.

Enjoy, as they say!

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Most arguments are not arguments

Here’s a strange claim — or rather, something that ought to strike the uncorrupted mind as strange!

An argument consists of a set of declarative sentences (the premisses) and a declarative sentence (the conclusion) marked as the concluded sentence. (Halbach, The Logic Manual)

We are told more or less exactly the same by e.g. Bergmann, Moor and Nelson’s The Logic Book, Tennant’s Natural Logic, and Teller’s A Modern Formal Logic Primer. Benson Mates says the same in Elementary Logic, except he talks of systems rather than sets.

Now isn’t there something odd about this? And no, I’m not fussing about the unnecessary invocation of sets or systems, nor about the assumption that the constituents of arguments are declarative sentences. So let’s consider versions of the definition that drop explicit talk of sentences and  sets. What I want to highlight is what Halbach’s definition shares with, say, these modern definitions:

(L)et’s say that an argument  is any series of statements in which one (called the conclusion ) is meant to follow from, or be supported by, the others (called the premises). (Barker-Plummer, Barwise, Etchemendy, Language, Proof, and Logic)

In our usage, an argument is a sequence of propositions.We call the last proposition in the argument the conclusion: intuitively, we think of it as the claim that we are trying to establish as true through our process of reasoning. The other propositions are premises: intuitively, we think of them as the basis on which we try to establish the conclusion. (Nick Smith, Logic: The Laws of Truth)

And the shared ingredient is there too in e.g. Lemmon’s Beginning Logic, Copi’s Symbolic Logic, Hurley’s Concise Introduction to Logic, and many more.

Still nothing strike you as odd?

Well, note that on this sort of definition an argument can only have one inference step. There are premisses, a signalled final conclusion, and nothing else. Which seems to “overlook the fact that arguments are generally made up of a number of steps” (as Shoesmith and Smiley are very unusual in explicitly noting in their Multiple Conclusion Logic). Most real-world arguments have initial given premiss, a final conclusion, and stuff in-between.

In other words, most real-world arguments are not arguments in the textbook sense.

“Yeah, yeah, of course,” you might yawn in reply, “the textbook authors are in the business of tidying up ordinary chat — think how they lay down the law about ‘valid’ and ‘sound’, ‘imply’ and ‘infer’ and so on. So what’s the beef here? Sure they use ‘argument’ for one-step cases, and in due course probably use ‘proof’ for multi-step cases. So what? Where’s the problem?”

Well, there is of course no problem at all about stipulating usage for some term in a logic text when it is clearly signalled that we are recruiting a term which has a prior familiar usage and giving it a new (semi)-technical sense. That’s of course what people explicitly do with e.g. “valid”, which is typically introduced with overt warnings about no longer talking about propositions as valid, as we do, and so on. But oddly the logic texts never (almost never? — have I missed some?) seem to give a comparable explicit warning when arguments are being officially restricted to one-step affairs.

In The Argument Sketch, Monty Python know what an argument in the ordinary sense is: “An argument is a connected series of statements intended to establish a proposition.” Nothing about only initial premisses and final conclusions being allowed in that connected series!

So: I wonder how and why the logic texts’ restricted definition of argument which makes most ordinary arguments no longer count as such has continued to be propagated, with almost no comment? Any suggestions?

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Benson Mates wins!

Benson Mates starts the introductory chapter of his classic Elementary Logic as follows:

This chapter is designed to give an informal and intuitive account of the matters with which logic is primarily concerned. Some such introduction is surely required; otherwise, the beginner is likely to feel that he does not get the point of the formal developments later introduced.

Exactly! And Mates stresses that smoothing the way to a later grasp of technicalities doesn’t mean (at this stage) nailing everything down  in way that will in all respects survive later fine-grained philosophical scrutiny — again, we need a tolerably relaxed preamble  to help get the show on the road.

I’ll say something just a bit more detailed about three of the “pre-formal preambles” I picked out in the last couple of posts, from Mates, from Bergman, Moor and Nelson, and from my namesake Nick Smith.

Mates’s opening chapter, it seems to me, still works the best, covering some of the right things, at about the right level, with the right caveats (except, perhaps, that his §4 — where he pauses to, among other things, cast doubt on the notions of a proposition, statement, thought, and judgement — rather over-eggs the pudding). In §1 Mates defines an argument as valid [he says “sound”] if and only if it is not possible for its premisses to be true and conclusion false. He then elucidates the relevant notion of what is possible in terms of what is conceivable, and then explicates that to mean there are no lurking contradictions — which in turn is explicated in terms of no contradiction being derivable (so we have gone round in a circle, but Mates says why it is an illuminating one). In §2, the validity of an argument (with finitely many premisses) is related to the necessity of a related conditional. In §4 we meet the idea of a form of argument, and we get the idea of a logically necessary truth as being one that is necessary by virtue of its logical form, and a corresponding notion of logically valid argument. But Mates is clear that the question of which words should be considered logical, so what belongs to logical form, involves as he puts it, “a certain amount of arbitrariness”. §5 then notes just a few of the vagaries of ordinary language which mean that, once we want to start talking about patterns or forms of arguments, some degree of formalisation is more of less inevitable (“it is clear for the natural language that there are few, if any, matrices that literally have only necessary truths as substitution-instances). All this is done very, very clearly.

Turning to the later edition of The Logic Book,  Bergman, Moor and Nelson start by defining an argument as follows:

 An argument  is a set of two or more sentences, one of which is designated as the conclusion and the others as the premises.

They immediately tell us that sets are abstract objects (and introduce the brace notation).  But what work is this talk of sets really doing? It’s unnecessary — the classes here are virtual classes — and I’d say best avoided (like much pointless talk of sets).

Then we are told

 An argument is logically valid  if and only if it is not possible for all the premises to be true and the conclusion false.

So we don’t get the distinction we get in Mates, between truth-preserving arguments generally and those arguments which might be said to be purely logically valid. Indeed, we don’t get anything general about form at this early stage in The Logic Book (and a search seems to reveal that the authors oddly eschew all use of the phrase).

We next get something about logical necessity and logically consistent sets of sentences; and then a section which points out that the given definition of validity means an argument is valid if it has a necessary conclusion or has contradictory premisses. Now, I too had such a section towards the end of my longer preamble in IFL1: I think that was a  mistake. Of course, the point has to be made somewhere; but it now seems to me to be better discussed later (e.g. in the context of talking about tautological validity, when we consider whether all tautological valid are valid in the intuitive sense we were after in the preamble and tried to capture with the informal “necessary truth-preservation” definition). Certainly, this is not a point to be made very near the outset to students who still need to be won over!

Bergman, Moor and Nelson are lucid enough (though the prose can be a bit plonking). But having nothing to say here about any notion of logical form — if only to criticise it — means that I’m not going to be recommending their chapter as parallel reading for IFL2.

Turning to Logic: The Laws of Truth, Nick Smith writes with enviable accessibility. I do have a worry about his initial claim in §1.1 that logic is “the science of truth” (he quotes Frege to the effect that logic has the same relation to truth as physics has to weight or heat — which will puzzle those students who are in another course being sold a deflationary theory of truth!): But let that pass, as in context the message is that logic isn’t about the psychological process of reasoning but about relations of logical consequence between the propositions we reason with.

§1.2 is about the notion of a proposition as the bearer of truth. But there are six and a half pages on this, and I’m not sure that it’s best policy to pause on such matters which are in fact going to be side-stepped when we adopt logically perfect languages to play with. §1.3 defines arguments in the usual way (but still, as with Mates, only one inference step is allowed, which ought to strike students as a strange regimentation of the notion of an argument!). §1.4 like Mates gives us various formulations of the idea of being a necessarily-truth-preserving argument. But Smith reserves the word “valid” for those arguments which are necessarily truth-preserving and where the “form or structure” of the argument guarantees that it is necessarily truth-preserving. So Smith (like Mates but unlike the authors of The Logic Book) makes the distinction between the two notions,  but his terminology is minority usage, I think. However, Smith says surprisingly little about the notion of form here. Thus “John is Susan’s brother; hence Susan is John’s sister”  is supposed not to be valid — but isn’t it an instance of the form “X is Y’s brother; hence Y is X’s sister” which guarantees truth-preservation? Or if that sort of “form” doesn’t count, why not? Smith doesn’t really tell us: at least Mates indicates there is an issue here. §1.6 is about soundness. And then Smith’s  introductory chapter starts talking about propositional connectives.

As I said, Smith writes very well. But I still, in sum, prefer the way that Benson Mates covers the same ground.

 

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More pre-formal preambles

As I said in my previous logical post, I’ve been looking at what I called the “pre-formal preamble” that you get in (some) entry-level formal logic books like my IFL — i.e. the introductory chapter(s) which informally explain notions like deductive validity, consistency, the idea of arguments coming in families sharing the same inferential form, etc., before the book starts to introducing truth-functional connectives and so on. I noted that, among a selection of older books, Benson Mates has the most lucid and useful such preamble.

Now looking again along my shelves, and at one or two e-copies, how do some later books measure up? Taking the texts in alphabetical order, here’s a few quick headline thoughts for now (I’ll return to say more about Mates and the best of the following in another post).

Barker-Plummer, Barwise and Etchemendy’s, Language, Proof and Logic (1999/2011) has surprisingly little at the outset: three sections on ‘The special role of logic in rational inquiry’, ‘Why learn an articial language?’, ‘Consequence and proof’ take just four and a half pages. By my lights, beginners need more than that by way of scene-setting and motivation.

By contrast, Bergman, Moor and Nelson’s The Logic Book (1980, 1990, 1998) does have a preamble chapter of some 24 pages — or rather it did have such a chapter in the third edition. It oddly gets cut to 14 pages by the sixth edition (in part by dropping the section which contrasts deductive and inductive arguments). Since The Logic Book is widely praised, I’ll say more about this.

Copi’s Symbolic Logic (I’m looking at the 1973 version) gives rather short measure, just over six introductory pages, though they are reasonably crisp and clear. Copi and Cohen’s Introduction to Logic (the 1990 edition) goes to the opposite extreme, having 150 pages or so before turning to symbolic logic at all. Neither is a helpful model for me to follow in IFL! Though I should mention one thing; unlike many others, Copi and Cohen at least do mention early on that real-life passages of argument are usually multi-step affairs (while typically texts define arguments as one-step inferences having premisses, a conclusion, with nothing coming between them).

Goldfarb’s Deductive Logic (2003) has just two pages of preamble before starting in, albeit very gently and clearly, on propositional logic.

Hodges’s Logic (1977) starts with quite a few pages of discursive preamble before truth-functors eventually get introduced on p.86. But these are somewhat idiosyncratically done, and have e.g. some digressions into linguistics that seem less well-placed, forty years on.

Kahane’s Logic and Philosophy: A Modern Introduction (latterly co-authored with Hausman and Tidman; 12 editions from 1969 to 2013) is widely used. In the fourth edition, say, there is a preamble chapter of eight pages without much content, and what there is is pretty sloppy. In the eleventh edition, this has grown to a somewhat better 15 pages (though I do wish people wouldn’t write misleading things like this: “Logic is concerned primarily with argument forms, and only secondarily with arguments”).

Hurley’s Concise Introduction to Logic (also in its twelfth edition, 19??-2015) is again widely used. Whatever its virtues, it is hardly concise, getting to p. 650 before starting on the answers to exercises. There’s some 200 pages of informal preamble before turning to the traditional syllogism. Dipping in, this preamble looks clear enough with myriad examples, but I would have thought that this would badly test the patience of most students. Anyway, not a model to be emulated IFL2!

Lepore’s Meaning and Argument: An Introduction to Logic Through Language (2000) is a not-very-formal formal logic book, which just about belongs in this list. The first 30 pages are preamble before some elementary propositional logic gets under way: this is very accessible but perhaps a touch too elementary, perhaps?

Simpson’s Essentials of Symbolic Logic (1988, 3rd end. 2008) is a lucid book, recommended by a number of people: but it dives straight into propositional logic with almost no preamble at all.

(Nicholas. J.J.) Smith’s nice Logic: The Laws of Truth (2012) starts with some 23 pages (§§1.1-1.5) of very clear general remarks about the nature of propositions, of arguments in general, of necessarily-truth-preserving arguments, and arguments valid in the narrower sense of being necessarily truth-preserving in virtue of (logical) form.

Teller’s Formal Logic Primer (1989) is an earlier favourite of mine, but similarly to Simpson dives in to propositional logic with with only three pages of preamble.

OK, that’s just a selection of the available books out there: I was surprised how few do give a reasonably expansive preamble, scene-setting for students. Thinking ahead to the new webpages to accompany IFL2, I’m planning to add some recommendations for parallel reading for groups of chapters. At the moment, then, it would seem that the leading candidates for recommendations to accompany my preamble chapters might be the material in Mates and/or Nick Smith (or perhaps Bergman, Moor and Nelson in earlier editions) .  So let me take another look at these again, say more about these options, and see if they inspire — in a positive or negative way — any changes in rough draft preamble for IFL2.

To be continued

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Brexit blues

A miserable day.

Not so much for the result — who really knows what the effects will be, though I voted according to my best efforts at judging the likely upshots for the young and/or poor (for the well-off old like me will be fine either way).

But because we got here as we did … “The most depressing, divisive, duplicitous political event of my lifetime,” as Robert Harris put it.

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