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.

In this paper, I construct a logic, extending classical logic with a single unary operator, which has no complete Boolean algebras as models. If the family of propositions we are talking about in (1) and (2) has the kind of structure described in that logic, then (1) and (2) cannot jointly hold. I then explain what this might mean for theories of propositions and possible worlds.

This paper is dedicated to Professor Robert K. Meyer.

In this note, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose a correction.

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!

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.

In this essay we will look at the topic of what can be known a priori.

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.