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Time Flies

Tuesday, October 20, 2009

Time flies like an arrow.
Fruit flies like a banana.

If you’ve been following my twitter feed, you’d realise I’m still alive. You wouldn’t think that from the activity – or lack thereof – here. (Though a few papers have appeared – or changed their publication status – on my writing page.)

Here’s where we are: It’s been a busy, eventful semester, and the teaching period is almost done. I’ve had fun teaching proof theory to fourth-year students, tutoring intro philosophy to first years, and supervising graduate students (at last count, I have eight current research students in various stages of the degrees). One of the sadder things to befall us here at Melbourne is the departure of Allen Hazen, who as left our shores for the chillier climes of Edmonton. The Melbourne logic community’s loss is Canada’s gain here.

Tomorrow, I’m off on a short trip to Guangzhou, by way of St. Andrews and Bristol. It’s the long way around, but somebody has got to do it. I’m busy clearing the decks here of as much as I can before the trip. One of the decks to be cleared is this blog, so a post is in order.

Posting about not posting for a long time is so passé, so here’s a link to something you might like if you’re a logic person like me. Lately, I’ve enjoyed playing around with Wandering Mango’s program Deductions (Mac OS X only). It’s a very neat natural deduction educational tool: it helps you produce valid Fitch-style natural deduction proofs, using the format of the major texts used in intro teaching. Well, as far as I can tell, they’re the major texts ued in intro teaching in North America. In Australia, in Europe, in the UK, logic is taught in different ways: Smullyan-style tableaux, Gentzen tree-style natural deduction, Lemmon-style linear natural deduction (though see the update below) with labels, etc. There’s a lot you need to do if you’re going to cover the ground of all the ways of teaching introductory logic by way of ‘proofs’. I’ve been in touch with the developer, and he tells me this is only the beginning for Deductions. It’s built in a modular fashion, and it shouldn’t be too hard to start extending it to cover more systems.

So, if you teach logic, or if you’re learning logic and you’d like to learn it by having a proof assistant on side to keep your proofs on track take a look at Deductions.

Update on December 8, 2009: Jeff Pelletier reminds me in an email that Lemmon’s beginning logic was not the first to introduce what I called ‘Lemmon-style’ linear natural deduction. Patrick Suppes, in his Introduction to Logic. For more on ths history of natural deduction, a great place to start is Jeff’s own ”A Brief History of Natural Deduction.” Thanks for that, Jeff!

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Live from Hejnice

Saturday, June 20, 2009

Posting has been light, since I’ve been powering through work at the end of the semester, and getting ready for a quick trip west to Europe, for Non-Classical Mathematics 2009 and Logica 2009, preceded by a quick visit to Dresden to see Heinrich Wansing, and to break up the train trip from Frankfurt to Prague.

So, posting here has paused for a bit, but now that I’m settled in Hejnice and that there’s a wireless connection here, I can deal with some of my backlog of things I’ve promised to post. So, here’s a salad of links for you.

• My Faculty, the Arts Faculty, at the University of Melbourne, is holding a short Winter School on July 15 and 16, for students from Ausralia (but outside Victoria), to come and get a taste of the range of research done in the Faculty. If you’re from inside Australia but outside Victoria, you’re a ‘high achieving honours student’, and you’d like a trip to Melbourne to see what we do, please apply. Applications close June 22, so you’ve got to be quick!

• I’m helping organise ICLC2009, the Intenational Conference on Logic and Cognition at Sun Yat-Sen University in Guangzhou, held from November 2 to 4, 2009. The deadline for getting your paper in for this conference is a leisurely October 1, 2009. Guangzhou is great (I was there last year and had a wonderful time at the Institute for Logic and Cognition), and if you can come along, please do submit a paper.
  

• Nick Griffin and Bernard Linsky are hosting PM@100, a conference on the 100th Anniversary of the publication of Principia Mathematica. That conference is from 21–24 May, 2010 at McMaster University in Ontario, and paper submissions are due only on January 1, 2010, so for this you have more time to get things together.

• There is no deadline at all for joining Friends of the SEP Society. Are you a Friend of the Stanford Encyclopedia of Philosophy? So many of us – students, academics, interested readers – use it for our research, and it’s a great resource for everyone. If you’re a regular user of the SEP (and if you’re interested in philosophy, who wouldn’t be?), consider joining the Society to help support the work of the Encyclopedia. For a small fee, you support the encyclopedia, you help it keep up its mission of free, high quality introductions to philosophical themes – and you get access to great quality PDF versions of the entries in the SEP, which are just ideal for printing out and reading (and annotating) offline. You also (if you like) get email notifications whenever the articles you’ve downloaded get updated. It’s a good deal, and it’s much cheaper (at US$25 a year for a full subscription, down to US$5 a year for a student subscription) than a journal subscription.

I’ll get back to posting on more substantial things later. Now I’ve got a conference to attend. For little quips along the way, you can follow the twitter feed.

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Rumfitt on Multiple Conclusions, Part 2

Tuesday, June 2, 2009

This is Part 2 of a series of comments on Ian Rumfitt’s paper “Knowledge by Deduction” (Grazer Philosophische Studien, vol. 77 (2008) pp. 61–84). In Part 1, I focussed on Rumfitt’s direct criticism of my approach in ”Multiple Conclusions,” and I tried to show that his criticism missed the mark, and that it missed the mark in an important way. The norms of logical consequence and logical coherence apply not only to occurrent beliefs but to all manner of states of accepting and rejecting (or acts of assertion and denial), whether they express our deep standing beliefs or hypotheses we simply entertain lightly.

In this part, I want to consider the comments on the possibility of genuine proofs with multiple conclusions. Rumfitt writes (and I’ll quote him at some length here), on page 79:

The rarity, to the point of extinction, of naturally occurring multiple-conclusion arguments has always been the reason why mainstream logicians have dismissed multiple-conclusion logic as little more than a curiosity. (See e.g. Tennant 1997, 320.) And attempts by enthusiasts to alleviate the embarrassment here have often ended up compounding it. In the introduction to their textbook on the subject, Shoesmith and Smiley concede that multiple-conclusion proofs can scarcely be said to form part of the everyday repertoire of mathematics. ‘Perhaps the nearest one comes to them’, they go on, “is in proof by cases, where one argues “suppose A₁ then B; … suppose A_m then B; but A₁ ∨ … ∨ A_m, so B”. A diagrammatic representation of this argument exhibits the downward branching which we shall see is typical of formalised multiple-conclusion proofs … But the ordinary proof by cases is at best a degenerate form of multiple-conclusion argument, for the different conclusions are all the same (in our example they are all instances of the same formula B)” (Shoesmith and Smiley 1978, 4–5). “At best degenerate”, though, hardly says it. I do not know how the word “multiple” is used in Cambridge, but in the rest of the English-speaking world it is understood to mean “more than one”. So an example of an argument in which all the conclusions (sic) are identical provides little justification for taking multiple-conclusion logic seriously. But since this is all that Shoesmith and Smiley provide by way of a positive case for deeming their system to be a branch of logic, readers of their book may be forgiven for closing it with a sigh on reaching p. 5 of the introduction.

What can I say about that?

Read on …

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Rumfitt on Multiple Conclusions, Part 1

Monday, June 1, 2009

Thanks to Ole Hjortland, I’ve been alerted to Ian Rumfitt’s paper “Knowledge by Deduction” (Grazer Philosophische Studien, vol. 77 (2008) pp. 61–84.). In it, he makes a number of critical comments on multiple conclusion accounts of logical consequence, and in particular, he makes some critical remarks on my paper ”Multiple Conclusions.” Now, the criticism of mutiple conclusion consequence isn’t the main point of the paper—the main topic is how one can acquire knowlege by deduction, as the title indicates. On that topic, it’s a really interesting paper, and I hope to comment on those parts at some time.

However, since the paper ends with the sentence

But we have found reason to leave multiple-conclusion logics to the boy racers, and focus on the single-conclusion rules, by following which we can splice together the deliverances of various sources of knowledge to come to know things that we could not know otherwise. (page 83)

I’ve got to respond. It’s clear that the criticism of multiple conclusion consequence plays a significant role in the paper, and in how Rumfitt thinks of the topic of acquiring knowledge by deduction. The ‘boy racer’ image—which I think is not intended to be flattering to people like me who have advocated multiple conclusion logics—arises out of an argument to the effect that multiple conclusion logics are finely tuned machines, which are fiddly to maintain, like a sports car. I’ll leave the metaphor for readers to judge.

Given the criticism I should reply in some kind of public forum, to get a response out there. It doesn’t seem appropriate to write an extensive essay just in response to a few points made in one paper, though I may make the remarks in some other paper I’m writing if it is appropriate to the topic at hand. But I have a weblog, it seems like the appropraite avenue for responding.

I’ve got three comments to make. They are, in turn.

1. On Rumfitt’s explicit criticism of my ‘overplaying my hand,’ as cited by Ole in the post that drew my attention to Ian’s paper. This is taken up here.

2. On Rumfitt’s point that multiple conclusion deductions aren’t found in nature. (This point is, of course, not limited to Rumfitt. It’s found throughout the literature, predominantly in response to Shoesmith and Smiley’s book.) This is taken up in Part 2.

3. On Rumfitt’s interesting argument concerning the multiple conclusion Cut Rule being properly stronger than mere transitivity. (This is the point at which the metaphor of the sports car appears.)

Making all three comments in the one post seems excessive. So I’ll make the first comment here, and leave the other two for later posts in the next little while.

Read on …

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Problems for Naïve Property Theories

Thursday, May 21, 2009

I’ve been thinking about generalisations of Russell’s paradox, cleaning things up so you can’t get around the problem by changing the logic of connectives. I don’t think that mucking around with negation or implication gets to the heart of the issue. (This view is shared by some very insightful people. I haven’t come to it alone.)

Getting around negation and conditionals is surprisingly easy, once you get the proof theory sorted out. I’ve been noodling about with this issue for a year or so now. I presented on this in a talk at the World Congress of Paraconsistency last year, and a bit of it has appeared in my draft discussion of some themes from Hartry Field’s Saving Truth From Paradox.

There, the paradoxical derivations are done in sequent calculi, and they’re not the most perspicuous presentation. I managed to sharpen it up a bit tonight, and the resulting proof is here. It’s not explained in the text of that note: that gives just the definitions and the proof. I hope to get to that soon. But let me use this site to get the ideas out in a rough and ready form.

The gist of the idea is this. Folks like Graham Priest, Hartry Field and Jc Beall think that for every description φ(x) there’s a property <x:φ(x)> of being an x such that φ(x). An object a instantiates the property <x:φ(x)> if and only if φ(a). The traditional problem is this: consider the property <x:x doesn’t instantiate x>. Does this instantiate itself or not? If it does, it doesn’t. If it doesn’t, it does.

The solutions favoured by Priest, Field and Beall (and my former self), though they differ in details, all agree that we should muck around with the logic of negation. (And also the logic of the conditional, as the property <x: if x instantiates x then I’m a monkey’s uncle> is just as problematic: see Curry’s paradox.)

Now, it’s a pain to worry about each different tweak to the logic of negation and the logic of the conditional, and worry about whether this patch or that fix really does solve the problem. (It’s a fun pain, if you like that kind of thing, but a pain nonetheless.)

I’ve been looking at formulations of the problem that avoid all talk of negation, conditionals and other stuff my friends and colleagues can argy bargy about. Instead, I’m trying to make do with the logic of instantiation (that’s implicit in the so-called naïve theory of properties, for which each description φ(x) has a corresponding property <x:φ(x)> of being an x which is φ. An object a instantiates the property <x:φ(x)> if and only if φ(a).) So, we adopt two inference rules:

[εI] From φ(a) infer a ε <x:φ(x)>

[εE] From a ε <x:φ(x)> infer φ(a)

for each open sentence φ( ). (The ‘ε’ is our shorthand for ‘instantiates.’)

Then, we need two more things. First, a sentence that is pretty bad. One from which we can infer everything will do the trick. (If you have a universal quantifier around, ‘everything instantiates everything’ will do nicely. But it isn’t mandatory.) In other words, we have a ‘⊥’ for which

[⊥E] From ⊥ infer any sentence you like.

Finally, we need the logic of identity for properties. You need to have some account of when <x:φ(x)> = <x:ψ(x)> for different sentences φ and ψ. It’d be odd to say that the property of being red and square was a different property from the property of being square and red, wouldn’t it? (The extant naïve theories of properties say little about this. The extant consistency or non-triviality proofs for naïve theories of properties, alas, make different descriptions denote different properties, which is not what you should want.)

So, what can we say that would rule out out distinctions where there is no difference at all? What identity condition works for this sort of property? Extensionality is the identity condition for sets. If the things in set A are the same as the things in set B, then A and B are the same set. That’s clearly too strong for properties. (Think renates and cordates, or featherless bipeds and humans.) But if I can deduce that a ε S from a ε T, and vice versa (where a is aribtrary), using deduction alone and no contingent side conditions, then what difference could there be between property S and property T? None that I can see, that’s for sure. This motivates the following condition.

[=I] If I can deduce a ε S from a ε T, and a ε T from a ε S, with no other side conditions, discharge those assumptions and infer S = T.

(Parenthetical remark: that doesn’t mean that being H₂O is the same property as being water, unless you think you can infer that a is H₂O from a is water, and vice versa, using logic alone. You can think that they are necessarily coextensive without thinking that. We’re not identifying properties coarsely.)

The rule [=I] tells us when two properties are identical. We need to know what we can infer from the claim that two properties are identical. That seems straightforward. You only get out what you put in:

[=I] From t ε S and S = T, infer t ε T.

That’s five simple inference principles.

Those five inference principles are enough for you to deduce ⊥.

This is bad, since from ⊥ one can validly deduce everything.

How can we deduce ⊥? We use identity and ⊥ to do what we wanted negation to do before our friends and colleagues said negation didn’t do that. That is, consider this property:

<x:<y:x ε x> = <y:⊥>>

That is, consider the property of being an x such that the property that anything has when x instantiates itself as a property is the same thing as the property that nothing has. (In other words, consider the property of not being self instantiating, but we won’t say that, since we have nice arguments about the logic of negation.)

Using [⊥E], [εI], [εE], [=I] and [=E] alone, we can deduce ⊥. Here’s the proof. It has fifteen steps, each one of which is one of those five rules.

I think that this is a serious problem for anyone who likes naïve theories of properties. You’ve got to say which of those rules break down: and by ‘break down’ I mean something very precise. For which of the rules [⊥E], [εI], [εE], [=I] and [=E] are you prepared to accept the premise and reject the conclusion? If you can’t do that, then a forced march down the proof suffices to commit you to ⊥.

So, what will it be?

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