Here’s a few items of news.

• I’ve been way too busy coordinating the Philosophy program at the University of Melbourne to be posting here. It’s been a busy ride for the year, but it’s been fantastic. It looks like I’ll be in the saddle again for 2011, so hopefully having learned how to hold the reins this year, I’ll be able to post here a little more next year. No promises though.

• I’ve been writing. Doing lots of writing. You’ll see a few changes on the writing page. I’m most happy with the new paper ”A Cut-Free Sequent System for Two-Dimensional Modal Logic: and why it matters.” There’s a little philosophy there, and a little proof theory, too. It’s not too long (just 24 pages), and I’d appreciate your comments. Head over to the paper page to download the paper and post your comments. (Make sure you admire the in-line diagrams on page 16. That’s LaTeX, with no postprocessing.)

• My next travels involve a trip to Guangzhou to take part in SELLC 2010. That’s going to be a blast, helping teach a Winter School to a bunch of enthusiastic students, and with a great slate of lecturers. It’ll be good to catch up with Samson, Robin, Mehrnoosh and Dag, and to meet the rest of the lecturers, as well as my colleagues at the ILC at Sun Yat-Sen University. On the way home from Guangzhou, I’ll take a quick side trip to Boston for the Eastern APA, to contribute to a session on the future of logic, with John Horty and Johan van Benthem. What fun! This will be my first ever APA, so if you have any advice for an APA neophyte, do let me know.

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!

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.

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?

First of all, in defence of Shoesmith and Smiley, the one example of proof by cases (in which the intermediate formula are all the same) is not the only consideration they provide in favour of taking their system to be a branch of logic. Showing that it is a codification of a notion of (let’s call it) quasi-proof which delivers logical consequence on Gentzen’s sequents, and which, when restricted to single conclusion deductions agrees with traditional natural deduction should count for something. If it’s not a branch of logic, what is it?

To be sure, a single case where the only multiple conclusions aren’t very multiple is not very satisfying. But there is a genuine sense, of course, in which even in this case the conclusions are multiple. Surely they have heard of the type–token distinction outside of Cambridge? There is a perfectly admissible sense in which the rule of conjunction introduction (in Gentzen’s natural deduction) requires two premises, even when the instance used is a derivation of P ∧ P from the premises P and P. (To rewrite this to be a proof from one instance of P to P ∧ P may radically change the structure of the proof, especially if both instances of P are proved in very different ways.)

But that is a minor point, not worth any more of our time. More important is the way that Rumfitt missed what is going on in proofs by cases. I don’t mind that he missed my example of a proof featuring multiple conclusions, discussed in ”Multiple Conclusions” on page 199 to address just the point Rumfitt raised here.

Suppose everyone is either happy or tired. Choose a person. It follows that this person is either happy or tired. There are two cases. Case (i) this person is happy. Case (ii) this person is tired, and as a result someone is tired. As a result, either this person is happy or someone (namely that person) is tired. But the person we chose was arbitrary, so either everyone is happy or someone is tired.

This seems to me to be perfectly valid reasoning: It’s a proof. It’s a renderng in (somewhat) natural language of a sequent derivation of “everything is an F or something is a G” from “everything is either F or G.” (For the sequent derivation I used, see the paper.)

In this proof, there are two conclusions active at a number of points. Stop the proof before the sentence starting “As a result”. Consider the state of play at this point in the derivation. We have a proof from “Everyone is either happy or tired” to the two conclusions: “Case (i) a is happy.” (where a is an eigenvariable standing for ‘that person’, which in turn points back to the choice made after the supposition.) “Case (ii) someone is tired.” In other words, we have a proof corresponding to the sequent (everything is either F or G ⇒ Fa, something is G.)

I cannot think of any way of understanding the deduction up to that point in such a way as to (a) see it as a single deduction, and (b) not introduce irrelevant connectives not explicitly used in the text. This example seems to me to be strong evidence that one can find in nature real-life uses of multiple conclusions in action, if you know where to look.

If you know where to look, you know that you don’t need tricky cases of intuitionistically invalid arguments to make the point. The point is there in Shoesmith and Smiley’s own examples of proof by cases. If you stop a proof by cases (in the middle of the ellipses eliding the interesting bits of the subproofs, in the bit of Shoesmith & Smiley quoted by Rumfitt) you will get more interesting cases where the conclusions in play differ. For example, take this proof of (p ∧ q) ∨ (p ∧ r) from p ∧ (q ∨ r). (Ignore the bracketed markers [a], [b] and [c] for the moment.)

Suppose p ∧ (q ∨ r). Then it follows that p. It follows that q ∨ r. So, we have two cases: (i) q, and (ii) r. [a] Consider case (i). Here, q, and we already have p, so p ∧ q. Consider case (ii). Here, r, and we already have p, so p ∧ r. [b]. Back in case (i), it follows that (p ∧ q) ∨ (p ∧ r). In (ii), it also follows that (p ∧ q) ∨ (p ∧ r) [c]. So, we conclude, (p ∧ q) ∨ (p ∧ r).

Again, this seems to me to be perfectly understandable reasoning. The point at which Rumfitt jokes about ‘multiple’ instances of the conclusion B corresponds to our point [c], where we have proved the conclusion (p ∧ q) ∨ (p ∧ r) in each case, but have not yet drawn them together to a single conclusion in the argument. True, at this point we have a proof with two instances of the one concluding formula. This is no better or worse than a proof with two instances of the one premise formula.

But [c] is only one place to stop the proof. Suppose we pause at point [b]. What do we have there? It’s a proof from one premise p ∧ (q ∨ r), leading to two cases, one in which we’ve proved p ∧ q, and the other in which we’ve proved p ∧ r. This corresponds to the sequent p ∧ (q ∨ r) ⇒ p ∧ q, p ∧ r, and the two concluding formulas are different.

Now, let me confess: I engineered this case by interleaving the two cases in the text, to get to an intermediate step in both of them. But this does not render the case any less salient. I could have stopped at point [a], and here, what we have got to is a sequent p ∧ (q ∨ r) ⇒ q, r. In other words, we have just got to the point at which the disjunction is broken up into two cases, and each case is ready for further processing using the rules.

That is the core idea of multiple conclusion natural deduction. The disjunction rule can be understood simply, as taking us from p ∨ q to the two conclusions p, q, which can then be operated on as usual, in the one proof. If you don’t like this, that’s fine. However, that is not an argument to the effect that multiple conclusion structures can’t be found in natural reasoning. They are there if you know where to look.

In a few days I’ll post a response to what I take to be the most interesting of Rumfitt’s arguments, his considerations against the Cut Rule. But that, I suspect, will take a little more time, and other duties are pressing for the next few days.

As ever, comments on these ideas are most welcome.

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.

Before I can tell you that story, I’ll have to tell you this story.^↓ What is multiple conclusion consequence? It is what is presented in Gentzen’s sequent calculus for classical logic. A multiple conclusion consequence links a number (maybe zero, maybe more) of premises X with a number (maybe zero, maybe more) of conclusions Y. We say that X entails Y if there’s a sequent derivation of Y from X. The soundness and completeness theorem linking Gentzen’s sequent calculus to models for classical logic tells us that X entails Y if and only if there is no model (and assignment of values to the variables, if we allow free variables in X and Y) in which each member of X is satisfied and no member of Y is satisfied. Formally speaking, multiple conclusion consequence is impeccable when it comes to classical logic, and its fragments such as distributive lattice logic.

In my favoured approach (discussed here, there and everywhere), the multiple conclusion consequence from X to Y makes sense as saying that a position in which every member of X is asserted and every member of Y is denied is self-defeating. (I’ll have more to say about what it is to be self-defeating in the salient sense, below.) For example, (p or q) entails p,q — and yes, asserting the inclusive disjunction of p and q while at the very same time denying p and denying q is to make a mistake—it’s for the very claims to undercut one another in a very special way.

Now, Rumfitt doesn’t like this approach. Let’s consider Rumfitt’s explicit criticism of the earliest exposition I have given of the ideas, in ”Multiple Conclusions”. He writes

Something like this case for multiple conclusions is presented in Restall 2005. But he overplays his hand in suggesting that ’Y is a multiple-conclusion consequence of X’ can be explained as meaning ‘The mental state of accepting all of X and rejecting all of Y would be self-defeating’. The mental state that consists of accepting that there will never be sufficient grounds for accepting or rejecting ‘There is a god’, while rejecting that very statement, is self-defeating. But ‘There is a god’ is in no sense a consequence of ‘There will never be sufficient grounds for accepting or rejecting “There is a god”’. (p. 80)

Here’s how I respond.

First: I just don’t agree with Rumfitt’s premise in the argument. I don’t agree that accepting (a) that there will never be sufficient grounds for accepting or rejecting ‘There is a god’ and (b) rejecting ‘There is a god’ are jointly self-defeating.

Of course, (a) and (b) jointly involve doing something for which there will never be sufficient grounds, but that’s not what is required for self-defeat in anything like any of the senses I was discussing.

Here’s why. You wouldn’t notice this from Rumfitt’s summary of the position in “Multiple Conclusions” but I was careful to not restrict the story of self-defeat (later traded in for the notion of incoherence) to occurrent beliefs or other mental states. In fact, though I talk about accepting and rejecting in the paper, I was careful to allow that this makes sense even in those cases where we accept claims hypothetically, or for the sake of the argument. “Suppse p”, I say. If you’re following along, then under the scope of this supposition, you accept p. You reaon from p. If you also accept q then, under this scope, it would be a mistake to reject q. This is clearly something we do, and we want a notion of logical consequence to apply under the scope of such suppositions or hypotheses. (It’s hard to see how we could make sense of the rule of conditional proof and what goes on when we reason like that without some such move.)

So, in this discussion, one should never limit accepting and rejecting to occurrent mental states like belief and disbelief. That will limit the range of applicability of the notion of logical consequence too much. This was one reason I used ‘accept’ rather than ‘believe’, and why I was careful to also talk of ‘assertion’ and ‘denial’, for one can similarly assert under the scope of a supposition, in a dialogue.

Now, consider what we might say about Rumfitt’s case and the position in which I accept that there is not and will never be any sufficient grounds for accepting or rejecting the statement ‘There is a god.’ OK. Let’s, for the sake of the argument, grant that. And let’s suppose that we believe it. Now, consider this: I ask you to consider what would be the case if, as a matter of fact, there were a god. You are quite well within your rights to say that in that case, there would be a god and that despite that fact, there would still be no grounds for accepting or rejecting the claim.

And I think you’d be right, and I think that in that discussion, under the scope of that supposition, we have a case where we (conditionally) accept ‘there is a god’ and accept ‘there is no grounds for accepting or rejecting the statement ‘there is a god’,’ and where we do so completely coherently, with no self-defeat at all. There is something epistemically defective, of course, in thinking that we are in such a circumstance as described under that supposition. But that’s not required. That’s the joy of assertion and denial, and of accepting and rejecting. We can make them fly beyond the limits of what we happen to believe (and disbelieve) at the present moment.

In other words, I don’t accept the premise of Rumfitt’s argument: I submit that it’s a too-narrow reading of ‘accepting’ and ‘rejecting’ (not allowing enough to count) and a too-wide reading of ‘self-defeat’ (allowing too much to count). If you understand ‘accept’ and ‘reject’ broadly enough—as I think we must—then the right notion of self-defeat is much easier to pin down.

What else do I say about the notion of self-defeat? In my later writing on this I have not used the term ‘self-defeat’ for it is too easily confused with epistemic notions of assertion without warrant. (Lloyd Humberstone pointed out a few years ago in conversation that you might worry about Moore paradoxical sentences on my approach, so I have been aware of this issue for some time.) Instead, I have used the more abstract terms of art such as involving a ‘clash,’ being ‘incoherent’ or being ‘out of bounds’ when talking about positions [X:Y] where X entails Y. What is involved in such a clash? The clash in the assertion of each member of X and the denial of each member of Y, whatever it is, is the kind of clash involved in asserting A and denying A. Its normative force, whatever it is, is not just any old force (like that of asserting something for which there is no evidence, or of denying something true). The connective rules for the sequent calculus show how other clashes can be reduced to this basic clash between assertion and denial of the one thing. Exactly how that reduction goes, and what one can say about the Cut Rule (which also connects clashes with other clashes), I’ll leave for another time. For now, I hope to have shown that Rumfitt’s case, and other Moore paradoxical cases, don’t cause any concern for the position in ”Multiple Conclusions.”

Later, I’ll consider other points from Rumfitt’s interesting article, but that seems like enough for now.

What do you think? I’d value feedback on this, as I’m trying to make sure that the position is clearly explained, convincing and correct. I’d settle for clearly explained though I’m aiming for all three.

I’ve got a prize to send to the first person who posts a comment explaining what highbrow cultural reference I just made: the offer is valid for one week, until June 8, 2009. I’ll post the answer if no-one has got it by then. ↑

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?

Bob will be sorely missed by many of us. His warmth and humour, his brilliance, and his willingness to talk logic (and much more) to anyone and everyone, will all be impossible to replace. If I manage to show a small fraction of both his logical insight, and his ability to communicate difficult concepts with good humour and wit, I’ll be a happy philosophical logician.

This weekend I’m off to Adelaide for one of our many Adelaide–Melbourne logic weekend funfests. This time it will be a bittersweet occasion, with many opportunities share our stories of the Maximum Leader of the Logicians’ Liberation League, and to toast his passing.

Elsewhere, David Chalmers remembers Bob fondly and links to some photos, and points to Bob’s notorious paper in which he proves that God’s existence is equivalent to the axiom of choice.

Thanks, Bob, for all you’ve done for me. My work and my life has been enriched in having you as a part of it.

It’s tempting to think of words, repeatable things that the are, as the types of token inscriptions, jottings, arrangements of pixels, utterances and all of the other kinds of ways we find to express words. Kaplan argues in “Words” that this isn’t right. I won’t rehearse the argument here, but I’ll tell you about one example he gives along the way.

Consider the special case of names. My name – ‘Greg’ – is, in one sense, the same name as Greg Currie’s name – we share the generic name ‘Greg’. In another sense, in the sentence

Greg wrote Arts and Minds but Greg wrote Logic.

the two uses of the string ‘Greg’ are different names, and different words. They don’t have the same referent, and hence, they don’t have the same meaning. Saying it like that (without the addition of surnames) is not necessarily the clearest way to say it, but if you get both of us together – last done in a pub in Nottingham in October 2005 – you could say exactly those words and make perfectly good, non-confusing sense.

Now, you might think that the two uses of the generic name ‘Greg’ are different words because they differ in referent, and that the only way two names of the same type could genuinely be two (and not one name after all) is that the case is like this – that they have different referents.

David Kaplan, in “Words” gives a delightful counterexample:

Let me tell you about the case of the mischievous Babylonian. One evening, the mischievous Babylonian looked up and saw Venus, and he thought to himself “This one is just as beautiful as Phosphorus, so let’s call it ‘Phosphorus’ too”.

The Babylonian names Venus (under the guise of the evening star) ‘Phosphorus’ in honour of ‘Phosphorus’, the morning star, little knowing that he has re-named the one and the same planet.

Just think: later, after learning some more celestial mechanics, the astronomer can say

My goodness! I didn’t know before, but now I see: Phosphorus is Phosphorus!

and this can make perfect sense as something that was discovered.

It turns out that individuating names is not as easy as we might hope. Genealogy, in all is contingent glory, plays a role.

I leave it as an exercise to the reader to apply this to the Paderewski case.

After arguing for years, unconvincingly, that semantic value (properly understood) is not affected by substitution, I hit upon a brilliant, new, and completely successful, strategy: argue, instead, that semantic value is affected by substitution.

Here’s a hint: the quote occurs after in the context of a discussion of proper names.

(Upon reflection, that’s not much of a hint, is it?)

Post your guess as to who I’m quoting, in the comments form on this post. (You can post an informed answer too, if you like, but I suspect a guess would be even more fun.) The most interesting answer will receive a prize in the honest-to-goodness snail mail.

Anyway, I’m not here to moan. I’m here to make you an offer. My friend, colleague, co-author (more like partner in crime) Jc Beall has gone and written another book Spandrels of Truth. (He’s one of these ridiculously prolific authors.) Anyway, it’s a cracker of a read, on the view that if we take seriously the idea that to say that <p> is true is no more and no less than to say that p, then the inconsistencies like the liar paradox are side-effects of the ‘design’ decisions, just like spandrels in architecture, and are neither to be worried about nor gotten rid of: instead, they’re to be lived with. Jc’s job is to convince us all of this by supplying us a paraconsistent logic and enough semantic machinery to show that living with them is not going to make the whole building fall down. I think he does a great job, and that this book is required reading for anyone thinking about theories of truth, paradoxes and stuff like that.

Thankfully, the required reading is not too expensive. Use this form for your ordering, and Oxford University Press will shave 20% off the price. Go buy it. Help Jc rocket up the best-seller lists. But be quick. The 20% discount lasts only until June 11, 2009.

The philosophy blogosphere is all a-flutter with the release of the 2009 Leiter Report. We have known for some time that the troubles here at Melbourne would impact our rating and the result is there for all to see. We’re out of the top ranking in Australia, sinking below Monash in the report’s league tables. Still, it’s nice to recognise that in the discipline rankings we’re notable in Applied Ethics, Feminist Philosophy, Mathematical Logic, Philosophy of Mathematics and Philosophical Logic. For a very small core of a department, we do rather well. We have a very nice group from which to rebuild a decent department – the trick will be managing the rebuilding phase. Please wish us luck (or if you can, lend us your support).

The trip was wonderful. I needed the break, and I enjoyed visting Pitt, CMU and UConn (thanks Shawn, Anil, Bob, Nuel, Kevin, Steve, Kohei, Horacio, Jc, Katrina, Marcus, Aaron, Colin, Reed, Michael, Scott, and everyone else, for your hospitality), and we had much fun at each stop on the way.

But now, I’m back in the saddle, confronted with a heavy class load – comprising first-year logic, a new upper-level logic subject I affectionately call “Kurt Gödel’s Greatest Hits”, and four classes in a fourth-year seminar on Epistemology and Metaphysics, where I’ll talk about recent work on truthmaking, from Lowe and Rami’s handy new collection – as well as some new postgraduate students (hi, Simon, hi Aaron!), significant administrative repsonsibilities, and writing and editing tasks I’m trying to keep on the boil.

So, work is busy. If you’re waiting on an email from me, hold on. I’m getting to it. I’m knocking the pile down each day, and at the moment, the pile is getting smaller, rather than bigger, so you can expect that I’ll get to yours at some time in the finite future.

After Zion National Park, we’re off to Las Vegas. There, we’re looking forward to meeting Greg Frost-Arnold in our couple of days there. (As well as the sensory overload of the Strip, of course.)

At Pittsburgh, I’ll finally get to meet Shawn whose blog I’ve read for ages, and Anil, whose work I’ve read and learned from for much longer. At UConn, I’ll catch up with Jc, which is always a blast, and Marcus, who’s moved there. I’ve also got the honour of giving the inaugural annual lecture of the UConn Logic Group.

I won’t be posting here unless I get web access, and I think that this isn’t a priority on the holiday. But after I’m back in February, I expect to be posting here more often.

• It’s exhausting, but lots of fun, to host visitors. Since just before Christmas, we’ve had visits from two good friends (separately) and four family members (one batch of three and one solo), as well as meals with other friends over the Christmas/Summer period. Much fun was had by all. It’s really easy to share good experiences in Melbourne with family and friends. Highlights have included Day 2 of the Boxing Day test, Shane Warne: the musical, and an evening watching (or re-watching) the Sound of Music and singing along.

• I’ve finally redesigned the site in a way I’m happy to see and to use. Again. There are two causes, one proximate, and one more long-term. The proximate cause was the superabundance of spam to moderate, on my own server. Logging in and cleaning out spam, arriving at the rate of at least one a minute, was depressing. So no more hosted weblog software for me. Instead, this site is now hand crafted on my computer (using Webby—it’s been fun to learn Ruby) and uploaded to the server as a static site. When I want comments on an item, I can implement them using intense debate or some other commenting service.

  The more long-term aim of the redesign was to make the information here more accessible. In particular, I’ve tried to make it easier to browse my papers, categorised by topic and these news items. Each news item like this one carry with themselves links to other news items from the same year. It was dead-simple to code up the scripts to do this, and I’ve found that it makes browsing around the past of this weblog a lot easier.

  The comments haven’t been transferred from the old site, yet, but they’re on my list of things to do. I hope you like the new site.

• It’s been a hard year at work, as I’ve hinted at before. There’s been a lot more going on than that going on. My leave over this summer break has been more than welcome. I’m away from the office for an extended period (for two weeks already, and I return only on February 23). For some of this leave, starting tomorrow, we’re going on a trip to the US, to get away from it all. The fact that we get to be there during Obama’s inauguration is a bonus! More on the trip, in the next post.

To share the Christmas spirit, I’ll share a favourite carol:

The tree of life my soul hath seen,
Laden with fruit and always green:
The trees of nature fruitless be
Compared with Christ the apple tree.

His beauty doth all things excel:
By faith I know, but ne’er can tell
The glory which I now can see
In Jesus Christ the apple tree.

For happiness I long have sought,
And pleasure dearly I have bought:
I missed of all; but now I see
’Tis found in Christ the apple tree.

I’m weary with my former toil,
Here I will sit and rest awhile:
Under the shadow I will be,
Of Jesus Christ the apple tree.

This fruit doth make my soul to thrive,
It keeps my dying faith alive;
Which makes my soul in haste to be
With Jesus Christ the apple tree.

The plan is to reboot the blog (and to fix the comment system, which is down for the count just now) for 2009. We’ll see how we go with that.

Are there any possible worlds? The idea of a point in logical space – at which every proposition is either True or False – seems at the one and the same time compelling and repellant. The notion plays a vital role in the semantics of logics of modality and conditionality, and so, is compelling. It is hard to take modal logic seriously without points in models that play the role of deciding every statement one way other the other. But to take possible worlds seriously as more than a useful fiction has seemed too great a price for many to pay. This squeamishness seems, to many, to have a distinctly ‘ontological’ flavour. Places at which there are blue swans or in which kangaroos have no tails seems to crowd the halls of being with blue swans and tailless kangaroos. We would do better without such things if we can. If we can explain possible worlds away – as propositions, or stories or abstracta or something else relatively tame – then we should. But ontological squeamishness is not my complaint about non-actual possible worlds. Instead of their non-actuality, my worry about possible worlds is their worldishness.

In particular, I worry about the way that these infinitely precise points are then put to use in semantics. Given that worlds are Deciders of Every Proposition, does it make sense to then turn around and think of a proposition as an arbitrary set of worlds: to think of any set of worlds as the kind of thing that can be a proposition? (David Lewis’ On the Plurality of Worlds (Chapter 1) is a very good defence of this position, but it is not just his. The view is everywhere.)

I’ll use this post to explain why this might not be a good idea.

I grant you: worldlike entities are rife in the study of logics. Tarski’s models for the first-order predicate logic decide every statement in the language as either true or false. A model is worldlike because it provides consistent and complete scenario, down to the finest detail the language can muster. It seems inherent in the very notion of classical logic, with its two valued semantics. Since every statement is either true or false, a world is exactly the kind of think we seem to be talking about when we use some arrangement of truth values consistent with the rules of the logic.

But maybe this is not how we need to think about things: consider the case of classical propositional logic, and think back to how you were taught truth tables. To show that ‘((p → q) → p) → p’ is a tautology, you showed that it received the value True in every row of a truth table. Now, how many rows did you use? Hopefully, four, one for each different possible assignment of true and false to the sentence letters ’p’ and ’q’. Any fewer than four rows would have left out some possibilities you should have checked, and any more than four rows would have been more than you needed to look at. The truth or falsity of ‘((p → q) → p) → p’ depends only on the truth or falsity of ’p’ and of ’q’, and not on the Values of ’r’, ’s’ or any other sentence letter. You check the value of your sentence not at a world, but at a row of a truth table which is a much blunter instrument: it cares not about most of the sentences in the language. It knows just enough to decide the truth of the sentences in its remit, in this case, sentences formed out of ’p’ and ’q’.

As for ‘((p → q) → p) → p’, so for every sentence. Assuming that the stock of sentence letters is boundless (say, p₁, p₂, p₃,…) no sentence will exhaust the vocabulary. Each sentence can be decided by a row of a truth table, and no (finite) truth table will decide every sentence. Each sentence in the vocabulary of propositional logic is finite, and so, no matter how long it goes on, there is always more vocabulary left to use.

In fact, we can formalise this. For any sentence A, let #A be the first sentence letter not included in A. So, #(p₁ ∧ ¬p₃) is p₂. A and #A are related in interesting ways. In particular, if you can show that

If #A ⊦ A then ⊦ A.

If any row of a truth table where #A is true is one where A is true too, then A is true everywhere. Why? Since #A is a sentence letter not occurring in A, the absence of a row in which #A is true and A false means that there’s no row where A is false. It’s a tautology.

In fact, we have three more facts like this:

If ⊦ A, #A then ⊦ A.

If A ⊦ #A then A ⊦ .

If A, #A ⊦ then A ⊦ .

Where, in general, X ⊦ Y holds when there is no row of a truth table where each member of X is true and each member of Y is false. In general, if there is no row of the table where A has some value (say false) and where #A has some value (say false, also) then there is no row where A is false, since the value of #A is completely independent from the value of A.

Is there an operator like ‘#’? It encodes the idea of indefinite extensibility: no matter what our sentence says, there is always something more to say. (Call these four conditions the rules of extensibility.)

If there is an operator like ‘#’, then if there are possible worlds, not every set of possible worlds counts as a proposition. Here is why. Take the set W of possible worlds, and take a particular world w from W. If {w} is a proposition, what about #{w}? Since {w} isn’t contradictory (it’s true somewhere, namely, at w), we don’t have {w} ⊦ #{w}, and similarly, we don’t have #{w},{w} ⊦. In other words, we need somewhere that {w} is true and #{w} isn’t, and we need somwhere that {w} and #{w} are both true. That’s impossible. There’s only one place {w} is true, namely w itself.

In other words, if singletons of worlds are propositions, then there is no operator like ‘#’. (There is a dual argument concerning complement propositions, which are true at every world except for one. They cause just as much havoc to ‘#’.)

If we have indefinite extensibility for propositions, then no statement can single out a world, for that would be a “final proposition” for which no extensibility is possible. If for every proposition A we can find some proposition #A independent of it, possible worlds are always idealisations, out of reach of a proposition. We can always approximate worlds relative to a given vocabulary, but if vocabularies an be extended, what we take to be a world is really a collection of worlds, to be further subdivided as new vocabulary is added. If ‘#’ is an operator, not all set of worlds (those infinitely precise “Deciders of Every Proposition”) count as a proposition.

Well, that would be all well and good, if there were an operator like ‘#’. Is there any operator like it?

Maybe not: the first problem is that ‘#’ as it stands is not really an operator or a connective like ‘¬’ or ‘∧’ or ‘∨’, since #A depends on how A is presented. For example, p₁ is equivalent to p₁ ∧ (p₁ ∨ p₂), but #p₁ is p₂, and #(p₁ ∧ (p₁ ∨ p₂)) should be p₃ instead.

We could smooth this wrinkle by defining ‘#A’ to be not the first sentence letter not occuring in ’A’, but rather, the first sentence letter not occuring in any sentence equivalent to ’A’. In that case, if A is equivalent to B, #A is equivalent to #B, so that is better. But now the question must remain: if ‘#’ is a genuine operator we can nest it: what is ##A? It is the first sentence letter not occurring in anything equivalent to #A. Consider ##p₁. The first sentence letter not occuring in anything equivalent to #p₁ is, of course, p₂, since #p₁ is equivalent to itself, and p₂ doesn’t occur in #p₁. But on the other hand, #p₁ is p₂, so ##p₁ should really be p₁ itself! Perhaps ‘#’ works as a way to transform sentences into other sentences, but it shouldn’t be thought of as an operator like the logical connectives. Maybe it cannot be coherently defined in such a way to satisfy the four rules of extensibility. Maybe there is no threat to propositions as arbitrary sets of infinitely precise worlds.

It might seem like that, but to conclude that there is no ‘#’ would be wrong. Although the definition of ‘#’ in terms of sentence letters is dubious, and does not give rise to a coherent operator when we attempt to nest it, it turns out that we can model a connective satisfying the rules of extensibility.

Here is a model:

Let’s think in terms of worlds, to start. Take the set W of worlds as the irrational numbers in the Real Line. The propositions at Level n are the unions of the any selection of irrational intervals of length 1/2ⁿ: (z/2ⁿ,(z+1)/2ⁿ) where z is an integer. These are closed under union (the union of any collections of intervals is a collection of intervals), intersection (there is no worry about endpoints of abutting intervals, as these don’t reach their endpoints, which are rational) and complement (the complement of some collection of intervals is the collection of the other intervals: since the endpoints are rational, they don’t occur in either a set or its complement). The propositions at each level are finer classifications of points than at any of the previous level. Propositions at Level 0 are collections of intervals such as (-2,-1), (0,1), (3,4), etc. Propositions at Level 1 are collections of finer intervals (-1.5,-1), (1.5,2), (3,3.5), etc., and so on….

Let’s interpret sentences in the language of propositional logic – enhanced with the operator ‘#’ – as propositions at some level or other. If A and B are interpreted as propositions at Level n, then ¬A, A∧B and A∨B are also interpreted as propositions at Level n, since the union, intersection or complement of propositions at Level n are also at Level n.

To interpret #A, where A is interpreted as a proposition at Level n, we will choose a proposition at Level n+1. In particular, we will choose an alternating proposition at Level n+1: the proposition consisting of all of the intervals (z/2ⁿ,(z+1)/2ⁿ) where z is even integer. The alternating proposition at Level 0 is

… (-4,-3) ∪ (-2,-1) ∪ (0,1) ∪ (2,3) ∪ (4,5) …

the alternating proposition at Level 1 is

… (-2,-1.5) ∪ (-1,-0.5) ∪ (0,0.5) ∪ (1,1.5) ∪ (2,2.5) …

and so on. It turns out that this choice for #A satisfies the four rules of extensibility.

Let A be interpreted as a proposition at level n. If A is not true everywhere, then it is false at some interval (z/2ⁿ,(z+1)/2ⁿ). Now consider #A. It is true at (2_z/2n+1,(2_z+1)/2ⁿ⁺¹), where A is not true. And #A is not true at ((2_z+1)/2n+1,(2_z+2)/2ⁿ⁺¹), where A is not true. Similarly, if A is not false everywhere, it is true at some interval (z/2ⁿ,(z+1)/2ⁿ). #A is true at (2_z/2n+1,(2_z+1)/2ⁿ⁺¹), where A is true. And #A is not true at ((2_z+1)/2n+1,(2_z+2)/2ⁿ⁺¹), where A is true. In other words, #A is truly independent of A. If A is true somewhere, at some such places, #A is true, and at others, #A is false. If A is false somewhere, at some such places, #A is true, and at others, #A is false.

This is completely general. We for any proposition A we have found another proposition #A. #A is more finely grained than A, and the four rules of extensibility are satisfied. In models like these, it makes sense to think of ‘#’ as an operator on propositions, and not merely a syntactic device for constructing sentences from other sentences.

The appeal to worlds in these models is not essential: we refrain from all talk of worlds and appeal instead to regions in a formal topological space. The definition of propositions in terms of sets of points – irrational numbers in our case – is not essential. The construction gives us an atomless boolean algebra, and these are well known algebraic structures. The value of the relatively concrete construction here is the manner in which extensibility corresponds to propositions being more and more finely grained, without that ever coming to an end. The model shows that the idea of indefinite extensibility of propositions is coherent.

I cannot tell if there are there any possible worlds, and if there are, if arbitrary sets of them count as propositions. What I can say, however, is that if there is indefinite extensibility, such as expressed by ‘#’ – and the logic of ‘#’ is coherent – then not all sets of worlds are propositions. For any proposition, there is always more.

[Thanks to Lloyd Humberstone, who got me first thinking about #, at a Melbourne Logic Seminar in 2006.]

Comments are, of course, most welcome. Is this crazy? Sensible? Anodyne? Let me know what you think.

In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic.

Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth.

This paper is a turnabout for me. While I’ve been making a slow turn for some time, from being a friend of naïve set theory in paraconsistent logics to a friend of truth-value gaps over gluts to a friendly critic of non-classical approaches to things like Curry’s paradox. In this paper, I present what I take to be much more problematic paradoxes around Frege’s Law (V), for which connectives like negation and the conditional do not feature at all, so questions of the design of the non-classical logic of negation or of the conditional are beside the point when it comes to the paradoxes.

I’d very much appreciate feedback on this paper. If you’re interested in this sort of thing, read the paper and post your comments on the paper’s page, over here.

Here’s the abstract:

In this paper I present an account of truth values for classical logic, intuitionistic logic, and the modal logic S5, in which truth values are not a fundamental category from which the logic is defined, but rather, feature as an idealisation of more fundamental logical features arising out of the proof theory for each system. The result is not a new set of semantic structures, but a new understanding of how the existing semantic structures may be understood in terms of a more fundamental notion of logical consequence.

I like these results, as they’re a mix of motivating philosophy, and formal proofs. I think I’m getting a better understanding of the relationship between the [Cut] rule in a sequent calculus and the maximality conditions involved in the behaviour of things like two-valued evaluations, possible worlds and points in model structures for other sorts of logics. It’s not a coincidence that sequents have two sides, and that there are two truth values. I don’t think I’ve plumbed the depth of the connections between proof theory and model theory, but at the very least in writing this paper up I’ve got a better idea of some of the interesting questions around this area.

It’s one of those ‘bonus’ results that I’ve got out of this research a uniform way of proving completeness for classical logic, Kripke and Beth models for intuitionistic logic, and universal models for S5, all with exactly the same sort of structure. That was a surprise to me when I saw those results just fall out.

Comments on the paper, of course, are welcome. Don’t comment on it here, but on the paper page.

Now I’ve got to figure out how to massage these results into the book. (It’s a pity my backlog of other writing-up tasks is so long, but that seems to be the natural state of the academic these days.)

Since then, I’ve been off to Canberra for a little workshop on hyperintensionality, organised by David Chalmers. Much fun was had there, too, but I must admit I’m even more confused than I was when I started. It must have been all that talking to Max Cresswell, Lloyd Humberstone and John Bigelow about how semantics works.