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. 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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Intro to Formal Logic 2! — and writing a pre-formal preamble

News: I didn’t hesitate long — I’ve decided to take up the suggestion that I write a second edition of my Introduction to Formal Logic. The headline plan that CUP and I are both pretty happy with is to include chapters on natural deduction but to keep length under control by shifting some of the current content that goes beyond a usual first logic course to supplementary chapters available on the book’s webpages. So hopefully there will be a significantly improved new version, perhaps re-branded into the Cambridge Introductions to Philosophy series, out by the end of next year. Promises, promises … (So, talking of promises, enthusiasts for category theory probably won’t get the updates/additions to my ongoing Gentle Introduction that I was planning for this summer: those who think that logicians shouldn’t be fossicking about with category theory will be pleased I’m doing something possibly useful instead.)

In time, I’ll no doubt be making chunks of IFL2 available for comment and correction, suggestions and advice, as I did for the second edition of the Gödel book. Watch this space.

I’m making a start by reworking the first part of the book, call it the pre-formal preamble. At the moment, there are six (short) chapters talking in introductory terms about the business of logic, deduction vs induction, the general notion of a deductively valid argument, the way that valid arguments come in families sharing the same inference pattern, the informal counterexample method (roughly, how to show an argument is invalid by coming up with another argument sharing the same inferential form with true premisses and a false conclusion), and  informal proofs (roughly, how to show an argument is valid by breaking a big leap from premisses to conclusion into smaller, more obviously valid, steps). When I was lecturing the course on which the book was based, I’d start off with a few sessions on these very general themes before turning to the formal stuff. And I continue to think this is how you need to start, with some scene-setting and motivation for the formal turn. Still, I’m finding my current opening chapters a bit embarrassing in places (it is only when you return to something after a few years that you realize what you, of course, meant to say!). So I’m rewriting busily and I hope the result will be quite a bit better; it is certainly ought to be a bit shorter and tauter.

Having more or less done a rough initial rewrite over the last week or so, I’m pausing to take a look to see how other authors of formal logic texts have solved the problem of how to handle the pre-formal preamble. Let’s start by looking along my shelves, blowing the dust off the top of some intro books,  and revisiting first a few old texts from up to about 1970. I’ve actually been surprised by what I’ve found (which in some cases didn’t chime at all with my vague supposed memories).

Some authors solve the problem of the pre-formal preamble by the simple expedient of ignoring it, and diving straight into formal work. Richmond Thomason  does this in his  Symbolic Logic: An Introduction (1970) — this largely forgotten(?) book has some nice features, clearly presenting a Fitch-style system, but Ch. 1 is cheerfully titled ‘Uninterpreted Syntax of a Logical System’, which is diving straight in with a vengeance! There’s is a bit more, but in fact not a lot, at the beginning of Fitch’s Symbolic Logic: An Introduction (1952): but some of what there is seems skew to what beginners might need — and though the book tells us at some length about the nature of propositions,  we don’t get much about the point of deductive  logic before getting down to business.

Suppes’s Introduction to Logic (1957) goes straight into a very lucid introduction to sentential logic. There is very little preamble either in Lemmon’s once heavily used Beginning Logic (1965), which is already beginning to introduce a formal language by p. 7: and although the notion of validity is introduced at the bottom of Lemmon’s p.1, we don’t really get any of the usual informal explications. Jeffrey’s zestful Formal Logic: Its Scope and Limits (1967) is even faster off the mark than Lemmon. We do get an informal characterization of validity on his p.1, but then we are straight off to work with truth-functional logic.

My memory played me false with respect to Quine’s Methods of Logic (1952, 1966, 1974). Yes, there is an elegantly written introductory chapter, before we start working on propositional logic. But no, this is more an overview of aspects of Quine’s philosophy — his picture of e.g. the way ‘[o]ur statements about external reality face the tribunal of experience not individually but as a corporate body’ — than it is an informal account of notions of deductive validity etc.

So the first book from that era which I’ve picked out which does provide the kind of pre-formal preamble which (to my mind) it is so natural to want is Benson Mates’s Elementary Logic (1965, 1972). Here there is indeed a 19 page ‘Introduction’, with the first section invitingly titled ‘What logic is about’, followed by sections on ‘Soundness [i.e. validity] and truth’, ‘Soundness and necessary truth’, ‘Logical form’, and ‘Artificial languages’. (There is then a second chapter of ‘Further Preliminaries’). Mates’s first chapter here seems beautifully clear, and well geared to its audience, and very much in the spirit of my opening chapters. Or rather, I should say — since I must once upon a long time ago have read Mates carefully —  my opening chapters are in his spirit. So this goes on my list for a careful reread, to see if I’ve missed a trick here or there.

To be continued 

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Explaining Chaos, now freely available

John Earman wrote on the cover of my 1998 book Explaining Chaos “This book is a splendid achievement. With a minimum of technical apparatus, the author gives the reader a good feeling for the mathematics that underlies the collection of phenomena that collectively have come to be known as chaos. At the same time he provides a much needed debunking of the breathless claims about the revolutionary nature of chaos theory. There are also major positive contributions to our understanding of the nature of scientific methodology.”

I couldn’t have put it better myself. If by some unhappy chance you’ve missed out on reading the book (it’s quite short), then now you have no excuse not to rectify the omission. For by kind permission of Cambridge University Press, you can now freely download a copy, preferably from here (bandwidth considerations!) where you can preview the file. Alternatively, if you don’t have an academia.edu login, just click on the cover picture.

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“Multiversism and Concepts of Set” revisited

A month ago I posted here a link to an interesting paper here by Neil Barton. There’s now a discussion exchange, which it would be a pity to leave buried unread in comments on an old posting: so here it is.

From Rowsety Moid. It’s an interesting paper, but to me it seems there are many questionable steps in its arguments, and I would like to know what people who know more than I do about set-theoretic multiverses would say.

The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).

Or, page 11: “We do not make any claims as to what exists within the Multiverse, rather it is seen as an intuitive picture to facilitate algebraic reasoning concerning sets.” Even the sets don’t exist? “Given a structure” The structure doesn’t exist?

Also, it’s not clear whether he is addressing Hamkins’s actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.

Even when the aim doesn’t seem to be the maximalisation of radicalism, there are a number of questionable interpretations or restatements. On page 9: “One way to understand Hamkins’ suggestion is to hold that we refer to several universes at once via description”. By page 11, the “one way to understand” has dropped out. Hamkins saying “in this article I shall simply identify a set concept with the model of set theory to which it gives rise”, quoted on page 7, becomes “it was noted that the Multiversist thought that every model of set theory constituted a set concept” on page 13. The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work. Aren’t there more models than descriptions? (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)

Another important step in Barton’s argument is the idea that we (or at least “Hamkinsians”) can use only first-order descriptions and so “lack the conceptual resources to pin down a single universe precisely”. From that, via the “One way to understand Hamkins” mentioned above, we reach the idea that we end up referring to “clouds”. For some purposes, that restriction seems correct. But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.

In the background, there seems to be an ideological element to the argument. It’s difficult to pin it down, but I think it may become visible on page 15 when arguing that “the Hamkinsian can give no particular reason to focus on one stopping point rather than another” and then saying “the response that we simply stop somewhere (without being able to give any reason for a particular stopping point) seems, like Go ̈del, to ascribe unexplained powers to the human mind.”

The power to stop somewhere? Is that supposed to be mysterious? I supposed that, in a sense, it is unexplained; but only because pretty much everything about the mind is currently unexplained, if you push hard enough.

From Neil Barton First, let me say a big “Thank You!” to Peter for publicising my paper, and to Rowsety Moid for some excellent comments. Indeed, your remarks were very timely, as they highlighted a mistake that I corrected in the proofs (shameless self-promotion: the paper will come out in this volume.

You said, RM: “The “algebraic” interpretation strikes me as incoherent, or else slight of hand. When he explains it on page 10, he seems to be saying it is not involved with existence and reference, but he then talks of “a group G”, of “elements” of G, and of “constructing new groups from old”, which all involve existence and reference. His way out of this seem to be to say the algebraic view is not “concerned with” such things (which is largely a matter of attitude, focus and interest), and that we can understand operations on groups “not as making any claims about existence and reference” (as if that were the only way issues of existence or reference could come in).”

Sure: we can have the algebraic interpretation relate to issues of existence and reference. The point is just that one requires additional assumptions first. For example, if we assume that the relevant objects exist, then there is a class of structures which instantiate the relevant algebraic properties about which the algebraist talks. Indeed, then various claims she makes might be a good way of proving the existence of certain objects, and better than trying to construct these things `absolutely’ (I think non-standard models are possibility a vivid case here). But those additional assumptions are needed before her view gets off the ground.

To bring this out, imagine that nominalism about sets were true. The ontological interpretation would then be null and void, there simply is no multiverse. It seems that the algebraic interpretation might still live on, despite the fact that the relevant algebraic properties are uninstantiated. For, we could still say that IF we were given some objects satisfying the relevant properties, we would be able to do such and such operations (and similarly with other algebraic theories like group theory).

[N.B. It’s an interesting question how the algebraic interpretation relates to if-then-ism. It’s not clear that these are wholly the same because of the algebraists acceptance of indeterminacy in metalogical notions.]

RM: “Even the sets don’t exist? “Given a structure” The structure doesn’t exist?”

We say things like this all the time though. “If space-time is discrete, then such and such will hold, and it is theoretically possible to construct such and such kind of object.” That seems like a perfectly valid claim to make, even if space-time is not discrete. Similarly with sets. Give me a structure, and I will be able to do these sorts of operations. I make no claim on whether the structure exists.

[N.B. As someone who is generally of a realist persuasion, I tend to think that this sort of response depends on the existence of the structures anyway. But this is just a refusal to engage with the position, not a dialectically convincing response.]

RM: “Also, it’s not clear whether he is addressing Hamkins’ actual view or a maximally “radical” alternative. For instance, on page 8 “we are interested in Multiversism in its most radical form” is given as a reason for assuming that every level of metalanguage is indeterminate.”

You raise a good point here, that I think is a general feature of the debate: the exact positions on the table are unclear. The Hamkins paper, though both highly interesting and ingenious, is notoriously slippery when it comes to being fully precise about the commitments of his view. So: am I addressing Hamkins or a radical alternative? I don’t know: you’d have to ask Joel how well the view put forward coheres with his (who, it has to be mentioned, was very helpful in discussing the paper and I’ve found very approachable). However, the views I present in the paper are ones that can be extracted from some of the things that he says, and he’s often keen to embrace the radical consequences of his view. He’s very clear that he thinks that indeterminacy infects the metalanguage, and there is no definite concept of natural number.

RM: “there are a number of questionable interpretations or restatements.”

I think there are going to have to be with the literature as it stands. Nowhere is Hamkins explicit about the kinds of epistemology he envisages, or the full metaphysical character of his view. My paper is intended to be just as much filling out possible ways of taking Hamkins’ view as a criticism of some of the things he says. I would welcome it if there are alternative interpretations out there, or I have got something wrong in exegesis—that way we can be more precise about what views are available and tenable. But I need to see these additional interpretations before I can weigh them up against the ones I have put forward.

RM: “The idea that the concept gives rise to the model has been lost, and the idea that every model constitutes a concept has been added. Between pages 7 and 8, Hamkins’s “Often the clearest way to refer to a set concept is to describe the universe of sets in which it is instantiated” becomes “each model is correlated with a set concept, and we refer (to?) this concept through a description.” Such examples can be multiplied.

On page 14, it turns out to be important for Neil Barton’s argument that every model (or, perhaps, every “cloud” of models) corresponds to a concept and so to a description. I don’t see how that could work… (Hamkins seemed to have it that there was a model for every concept, not that there was a concept for every model.)”

This was sloppy on my part, and I made some alterations in the proofs as a result. You are right, we should say that we refer to the concept through the model. However, this is done through *description*: we refer to the concept by *describing* the model. However, since we can only use first-order descriptions, we can’t pin down a single model, so our reference must be indeterminate, but this requires fixing some other model (in which some concept is instantiated; I take it that every model instantiates a concept) and so on.

I think there’s a lot more to be said here (in fact it doesn’t seem impossible to me that we get a loop of concepts or models), but the challenge at least represents an invitation for the Hamkinsian Multiversist to be clear on their commitments, and explain why there’s no such problem. As it stands, the Ontological Interpretation is not developed in sufficient detail to explain how these problems are avoided.

RM: “Aren’t there more models than descriptions?”

That depends on where you live for the Hamkinsian. Since every universe’s multiverse is countable from the perspective of some other universe, the universes of a multiverse are bijective with the descriptions from a suitable perspective (though not through the natural mapping of a universe with a description of it, and not within any particular multiverse). In any case, to press the challenge I only require infinitely many concepts being used, so countable is enough. There’s also the question of whether or not Hamkins is allowing parameters, which would in turn make the issue a whole lot more complicated (given the emphasis on ultrapowers, I’m guessing he is allowing parameters for the ultrafilters), as then we could have proper-class-many descriptions. Again, this would be another area I would like clarification from the Hamkinsian.

RM: “But when we’re trying to ground reference and so avoid a vicious infinite regress? That’s not so clear. Natural language (English, for example) isn’t restricted to FOL, for a start.”

I’m in full agreement here! In fact I find the restriction to FOL excessive. Again though, this is a case where the dialectic with Hamkinsian is important. If he/she wants to admit non-FOL, s/he has to explain what is acceptable and what isn’t. Why is it okay for him/her to use non-FOL resources in giving an account of reference, yet an account of the semantic referents of the natural numbers or set theory in terms of properties/plurals/the ancestral relation/sets is not allowed? I want to know what the rules of the game are by which the Hamkinsian has a stronger position compared to the Universist.

RM:“but only because pretty much everything about the mind is currently unexplained, if you push hard enough.”

Sure! But here we’re seeing if the Hamkinsian has a decent response to Benacerraf’s challenge through description. So, given that this is the background for the paper, we can demand an explanation.

I think in general the paper shouldn’t be viewed as an all out attack on the Hamkinsian, but rather a request for him/her to clarify a number of points.

Thank you ever so much for the comments! I found them very helpful. Best Wishes, Neil

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Intro to Formal Logic 2?

I’m mulling over a proposal that I write a second edition of my Introduction to Formal Logic, first published thirteen years ago by CUP.

I’m tempted. I’m sure I could make a very much better job of it. Though, of course, I’m only too aware of how time consuming it would be to undertake, and I’d also like to crack on e.g. with the Gentle Introduction to category theory (which has been going slowly of late).

If I do write a second edition, I’d like to include chapters on natural deduction (Fitch-style, I think, for user-friendliness). But I wouldn’t want the book to get too unwieldy in length — and CUP wouldn’t let it! — so some stuff would also have to go. But I guess I could put online some of the current content that goes beyond a usual first logic course (e.g. the completeness proofs for trees for PL and QL!). Would that be a win-win solution? — a tighter book, with the stuff on natural deduction some people wanted, but with the “extras” still available for the small number of real enthusiasts?

I haven’t decided yet whether to take up the proposal, let alone how to reshape and add to the text if I do go with it. But any advice and suggestions (either here, or in emails, address at the end of the About page) — especially from those who have used the first edition or indeed, perhaps even better, from those who decided not to use it — will be very gratefully received!

Posted in Books, Logic | 3 Comments

The difference a bit of history makes


We keep car and bikes in the garage at the bottom of the garden. The lane there is very rough and ready. It is unclear who owns what and whose responsibility it all is. The potholes make cycling a hazard, and you have to inch a car along rather gingerly. A bit of a pain, we’ve always thought (though actually having a usable garage in central Cambridge is very unusual).

But now we find that the lane is medieval — a broad track used by people going along the river to cross over to Stourbridge Fair, once the largest fair in Europe (and in the sixteenth century, lasting a month).

Which puts the mud and rubble in a whole new light.

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