We’re coming semi-live from Tartu, as long as my computer’s battery lasts, anyway.
For live off-the-cuff comments, try my experimental twitter stream, which is profoundly silly, but fun to fiddle with on my phone when my fingers are itchy.
The lineup today.
Me, giving “Pluralism and Proofs.” I think I went well. Look at Ole’s sneak-preview of what I said, which is pretty-well what I said here. The fact that Dag Prawitz and Per Martin-Löf didn’t eat me alive, but were actually interested, was profoundly reassuring. I’ve only done the slides, not a written version of the paper.
Graham Priest on “Logical Pluralism as another application of Chunk-and-Permeate.” This is an interesting project, conceiving of logic’s application to a domain of reasoning as variegated – a domain of discourse can be ‘chunked’ into different bits, where different logics are operative, and then ‘permeation’ relations tell us how facts in one chunk transfer over to facts in another. Graham was using this general setup (which he introduced in earlier work with Bryson Brown, on the infinitesimal calculus – one chunk is where the infinitesimal is zero, another chunk is where it’s nonzero, and you only permeate facts about non-infinitesimals, or something like that).
Graham’s take on pluralism here is that you can think of the different chunks as different kinds of things we’re reasoning about: quantum objects (with their quantum logic) and middle-sized dry goods and their non-quantum (say, classical logic).
I think that this application is really interesting, and it’s not what we (JC and I) talking about in our pluralism. We’re pluralists about the logic used for evaluating “reasoning about” the same collections of things.
The top question (or was it three?) came from Johan van Benthem, who pointed out the parallel with channel theory (I should have warned Graham about that because I should have spotted that!), and interesting applications in database combination, and legal reasons why information maybe shouldn’t flow from one to the other.
This was all about general elimination harmony: the notion that we can take the introduction rule for a connective as defining it, and a general elimination rule (defined in a systematic way, at least for simple cases of introduction rules) can be reconstructed out of the introduction rule if that connective is to be defined.
Read’s favourite “connective” (in shudder quotes, as it doesn’t connect anything) is the bullet (or blob) – • – with the introduction rule allowing us to infer • from a proof from • to absurdity. He showed that one could define an appropriate general elimination rule in harmony with that introduction rule. So it’s OK in some sense – it’s harmonious — but not a conservative extension of the underlying pre-• consequence relation.
He then went on to talk of the application of these notions to modality (which I’d heard from Stephen before in the paper “Harmony and Autonomy in Classical Logic”), but then he went on to talk of conditionality. Here, the discussion took an interesting turn, for Stephen pointed out that the necessary discussion of structural rules (which came up in the discussion of conjunction, disjunction, etc). Ringing the changes on this – or at least attending to questions about contraction and weakening – give us a way to understand the logical behaviour of conditionality, for the structural rules feature prominently in the behaviour of the conditional rules.
If that was fun, even more fun happened in discussion, with zingers of questions from Per Martin-Löf and Dag Prawitz. Per pushed on the definition of the blob – this definition literally makes no sense for Per, because for Per, you can’t prove that • is a proposition. This is right, and it helped bring out the difference between Per’s constructive type theory and Stephen’s logic. Then Dag Prawitz pointed out that the choice of structural rules cannot be arbitrary, but must be constrained in some way: what is the ground for choosing one set of structural rules rather than another. Stephen indicated that this is a really interesting question worth sorting out.
Well, it wasn’t ominous. It was a lovely example of how very different perspectives on what logic is (or does) result in making pluralism look very implausible. Manuel’s starting point was that logic has something to do with cognitive architecture–in particular, that inferences arising out of generative grammar rules–are genuinely valid, and that logic deviating from that just doesn’t get a grip on what we’re thinking.
That’s right, I think, given the starting point, but I don’t agree with Manuel’s starting point. This means, of course, that I should say something about the connection is between cognitive architecture and logical validity. Yes, I should. I haven’t said much, except to say that the connection between deductive consequence relations and the inferences one does or should make are generally complicated matters, for the usual Harman-esque reasons.
Discussion, I must admit, got quite pointed at times, as you’d expect when people disagree about fundamental issues, and want to sort out where you disagree. It was all in good fun, and Manuel sweetly came up to me afterwards to say “I hope you weren’t offended.” Of course I wasn’t, and as Johan van Bentham mentioned in discussion time, Manuel lavished us with attention to bring out why so much of JC’s and my take on logical pluralism is wrong-headed from his view, and so much critical attention is – indeed – a form of praise. Too right it is! Thanks, Manuel!
Now I’m back. Hartry’s talk featured lots of interesting concerns I can’t recount here (neither the time nor the space), but he major point for me was his connection between the notion of logical validity and constraint of degrees of belief. Hartry argues (as he did in the Locke lectures) that if one takes the inference from A to B to be valid, then the degree of belief of A should be no higher than the degree of belief of B (and this generalises for arguments with more premises), and that you should not take validity to be necessary truth preservation, for this ensnares you in paradox. Then since validity is a normative notion, and since Hartry thinks we should be anti-realists about norms, there is room for a kind of plurality here.
But there is not much room for plurality where it’s exciting, as far as Hartry can see, since that kind of wiggle room isn’t there when it comes to truth value gaps and truth value gluts. The disagreement between a gap view and a glut view and classical view isn’t up for pluralism on this account.
That’s completely right, by my lights. My disagreement with Graham Priest over whether or not a liar sentence is both true and false is not up for a pluralist reconciliation. We just disagree.
Where I think there’s scope for more work is the nature of the norms, and whether or not the connection between degree-of-belief management and deductive validity is two way, or only one way. Is an argument from A to B valid (or we know it’s valid, or whaever) if and only if we ought regulate our degree of belief to have D(A) never over D(B)? I don’t think that’s right, but making the case for this pro or con requires close attention to what an attribution of degrees of belief involves, I reckon.