Thrice Denied: FLC Workshop

This weekend is the 2nd FLC Workshop, The Logic of Denial, at Arché, University of St Andrews. We've got a great line-up of logicians talking about a topic that is both philosophically deep and technically interesting. I've included the schedule at the end of this post. More information, including abstracts is here. I'm very happy to have lots of good friends and colleagues in town, too many, in fact, to list here. I hope to provide you with a report and some pictures later on. Even better, if the technology gods are on our side, we'll have recordings from the sessions. More on that later as well.

Saturday, 24th

11.00 - 12.30 Dave Ripley "Embedding Denial"
12.30 -13.30 Lunch
13.30 - 15.00 Heinrich Wansing: "Assertions, denials, proofs,disproofs, and their duals"
15.00 -15.30 Coffee break
15.15 - 16.45 Luca Tranchini: "Falsificationism: Dualising proof-theoretic semantics"
16.45 - 17.00 Coffee break
17.00 - 18.30 Ian Rumfitt: "Bilateralism superseded"

Sunday, 25th

09.30 - 11.00 Peter Schroeder-Heister: "Assertion and Denial in Proof-Theoretic Semantics"
11.00 - 11.15 Coffee Break
11.15 - 12.45 Michael De: "Assertion, Denial and two types of Consequence"
12.45 - 13.45 Lunch
13.45 - 15.15 Colin Caret: "Revenge Denied"
15.15 - 15.30 Coffee break
15.30 - 17.00 Greg Restall: "Derivation, Definition and Denial (together with some other speech acts)"

Logic of Denial

The 2nd FLC workshop (Foundations of Logical Consequence) is coming up this autumn. Theme: The Logic of Denial. What is the relationship between negation and denial in logic? Is denial best analysed as assertion of negation? If there are more than one sort of denial, how are their logical roles differentiated? Is denial helpful in an analysis of logical consequence? What is the relationship between multiple conclusion and denial? And more. I'll return later with details about dates and list of speakers.

Two links

A bit silent here lately. Let me just say that my new time management regime is improving my habits. Less procrastination, more work.

Regardless, I wanted to post a couple of links that I've seen today:

• Via the Experimental Philosophy blog I learned that Dave Ripley has a paper about ordinary people's reaction to some contradictory looking sentences like 'The circle is near the square, and it isn't near the square'. Apparently, it is good news for the dialetheists.
• Greg Restall has written a lengthy reply to Rumfitt's recent attack on multiple conclusion logic that I reported on some time back.
  

Time Management

Back in my undergraduate days I used to be obsessed with keeping track of how much time I spent reading and writing. I kept a timer in the library, and every minute of reading would be meticulously written down in a protocoll, weekly averages would be calculated, and procrastination would lead to death by statistics. It can be quite instructive seeing how much time you actually spend working (given that you're honest, and turn off the timer for coffee breaks and the like), but it comes with a price. Timing one's own life might easily result in all sorts of obsessive behaviour, leading to stress and poor work quality, ultimately culminating in social malfunction. That didn't happen to me -- or at least so I claim.

This was a time when my online life was still quite restricted. After disciplining myself with the timer for some years, I found that I did fine without it, and returned to a life unframed by graphs and averages. But, now, with interactive tools for time waste eroding away my good habits (twitter, blogs, Google Reader, email lists, online chess, wikipedia, Napoleon biographies, etc.), I've decided that it's time to return to a more time ascetic life style. However, re-introducing my old stop watch seems a tad anachronistic, so instead I've spent some valuable time looking for neat time management tools.

I've looked at a couple of them, and so far I've found two that I like. Both freeware, and both for OSX. The first is simply called Time Tracker, and provides an easy to use interface for timing different tasks. No heavy graphics here, but you can export a file with the data if so inclined. The other tool, Slife, is a bit more scary: It's a programme that keeps track of exactly which applications, documents, and websites you are paying attention to. There's really no cheating here, because you're not setting or controlling the timers yourself; the programme simply records what's surfacing on your screen. I've spent little time so far monitoring, but I'm already astonished by the abyss of time some of my apps constitute. On the plus side, my twittering doesn't seem to take up as much time as I expected; on the minus side, jeezes, when you add up all the minitasks - there's your whole day!

Hopefully, these new insights will allow me to regain some control of my day. Soarly needed control, now that I'm in the finishing stages of my thesis. Any advice or recommendations about programmes is accepted with thanks (Lifehacker is a pretty neat page for time management stuff in general).

Williamson on Williamson

This interview with Williamson has been finding its way around to different blogs lately. I thought it worth posting about a section I found particularly telling. Williamson writes short introductions to some of his main philosophical work, e.g., vagueness, epistemology, methodology. As a follow up to the question about vagueness, Williamson is asked about the principle of bivalence, a fixed point in many of his philosophical arguments.

3:AM: How far is your commitment to the principle of bivalence something that shapes your philosophical outlook and what are your thoughts about philosophical traditions that tend to dismiss it, such as Hegelianism?

TW: I regard classical logic, in a broad sense that includes the principle of bivalence, as the best guide we have in philosophy. That doesn’t mean that I think it crazy to challenge bivalence. Many able philosophers have argued against it in various interesting ways for various domains, including the past, the future, the infinite, and the quantum world, as well as vagueness. I don’t dismiss their arguments; I try to show in detail where they have gone wrong. I would put Hegelianism low on the order of challenges to bivalence, because Hegel was writing long before the development of modern logic, at a time when logic was in a terrible state, and so he had no idea of the resources of logic. There are profound things in Hegel, such as the master-slave dialectic in The Phenomenology of Spirit, but he was no logician. Although some contemporary advocates of non-classical logic refer to Hegel from time to time, I have never seen a powerful Hegelian critique of classical logic. [Emphasis added.]

It's the End of Logic As We Know It

In a recent issue of Grazer Philosophische Studien there is an interesting paper by Ian Rumfitt, 'Knowledge by Deduction', vol. 77 (2008), pp. 61-84. The paper deals with the vexed issue of how deductive reasoning can expand our knowledge, but what caught my attention was the last part of the paper. Rumfitt has hinted toward his dislike for multiple conclusion logics in earlier work, but this time he goes all out with a section long argument against it on epistemic grounds.

My intention is not run over the argument here, but just to call attention to a footnote. Rumfitt is considering the idea that multiple conclusion somehow captures the interplay between acceptance and rejection (or assertion and denial). In this context, we get the following footnote:

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 good', 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)

Puzzling. I'll have to go back and read Greg's paper again, but I suspect this is a strange way of reading what he's up to.

Online Talk by Crispin Wright

Crispin Wright's talk at the 1st FLC workshop is now available in the series of recordings mentioned here earlier. The title of the talk is 'Inferentialism and Harmony'. Go and listen.

Martin-Löf Conference on the Foundations of Mathematics: Day 3

One of the most engaging talks was Colin McLarty’s ‘What are the things of mathematics?’, a motivational blurb for structuralism-cum-category theory with some neat examples. McLarty wanted (at least in part) to make the case that mathematicians in practice talk about existence and identity in a manner closer to categorical set-theory à la Lawere rather than ZF speak. In principal, ‘elements’ of mathematical theories ought to be individuated not by ZF but by participation in structures. The foundations, then, is not membership as irreducible, but invariance up to isomorphism. McLarty’s preferred framework is CCAF, a classical first-order categorical set theory.

The major part of the talk was spent trying to deal with a prima facie problem for his foundational view. Mathematicians standardly talk about natural numbers as if they are identical to corresponding objects in the set of integers (similarly for integers with respect to the set of rationals, etc). However, from a structural point of view this equation is unfortunate. As far as these elements live inside different structures, equating them is a misunderstanding; the appropriate strategy would be to relate them by looking categorically at relations between the structures; and CCAF offers precisely such a perspective.

Since I spent a number of hours discussing with Stewart Shapiro on our way to Uppsala, his talk didn’t contain many surprises. It did, however, engender one of the more lively discussions during the conference. Stewart’s training in classical model-theory doesn’t prevent him from enjoying the occasional excursion into intuitionistic terrain. More precisely, Stewart subscribes to something along the lines of the Hilbertian slogan: “Consistency entail existence”. If it’s an interesting mathematical structure (‘interesting’ here largely a pragmatic issue), it matters little what the underlying logic is. His favourite example: smooth infinitesimal analysis which are classically inconsistent, and, furthermore, different from Bishop style constructive analysis (which is simply a sub-theory of classical real analysis) and intuitionistic analysis.

(And when Stewart had first opened the door to non-classicality, he didn’t shy away from opening for possibly interesting theories which require paraconsistent logics. Of course, as he pointed out, in that case ‘consistency entails existence’ ought to be ‘non-triviality entails existence’!)

In Stewart’s view, all of these intuitionistic structures are interesting in the sense that they not only underlie consistent mathematical theories, but in some cases they provide ‘better’ results than their classical counterparts when we consider pretheoretic intuitions about what’s being modelled. What does Stewart take to be the moral of this observation? Logical relativism: the issue of a One True Logic for which validity is a matter of truth-preservation in all structures is moot. Rather, Stewart is after a roughly naturalistic understanding of logic which allows for variations in what is correct reasoning depending on what counts as mathematically interesting structures. Clearly, this Hilbertian relativism has certain affinities with what is sometimes called pluralism, but Stewart prefers relativism so as not to confuse the position with the one recently advocated by JC Beall and Greg Restall.

Martin-Löf Conference on the Foundations of Mathematics: Day 2

The second conference day promised the greatest spectacle of them all. For years I’ve wanted to see Girard in action, in fact I’ve been to an event or two to see him only to find that he couldn’t make it. Day 2 finally delivered: the morning session was Girard with ‘Towards Non-Commutative Foundations’. I find it impossible to report reliably on the content of the talk – even if some of it was familiar from his writings (proof nets and the “geometry of interaction”), most of it was either mystical or mathematical. From what I could gather, Girard invokes a notion of ‘entanglement’, a label for the idea that proofs somehow affect or even distort the models (“the subject appears as a commutative window”), a thought explicitly inspired by quantum physics. The “object” -- in Girard’s view part of a non-commutative foundations -- is somehow elusive: “You have to become commutative to have a notion of mistake”... As a logician I find myself hopelessly lost, and as a philosopher I wonder if I’m in the presence of brilliancy or quakery, yet there is something seducing about his prophetic style. In any event, I recommend having a look at some of his writing, for example ‘Linear Logic, its syntax and semantics’. Incidentally, a signifcant part of his opus is availble online.

On a more sober note, Peter Pagin took us back to more mainstream philosophy later in the day. Pagin has been to St Andrews a number of times, and I’ve always thoroughly enjoyed his talks. ‘Assertion, truth, and judgement’ is an honest attempt at bringing together his work on assertion and assertoric content with Martin-Löf’s proposition/judgement distinction in a type-theoretic setting.

The starting point is a puzzle about propositions and disagreement: When a speaker asserts a proposition p, we might be tempted by the thought that the proposition is somehow about the world of utterance to include a world index in the structure of the proposition ‘In w, p’, where w is fixed by the context of utterance. Yet this threatens to trivialise propositional content since it is either true in every world or false in every world (compare the old puzzle about mathematical propositions in propositions-as-sets-of-possible-worlds). Thus, propositions are too coarse-grained to function as what is tracked in genuine disagreement.

Pagin is a proponent of the information account of assertion (an utterance is assertoric iff it is informative), and his general strategy for the above puzzle is to separate propositional content from assertoric content. This suggests, Pagin proposes, an analogy to Martin-Löf’s judgments, a : A where a is a proof object, or truth-maker, and A is the proposition (think w : A read ‘A is (actually) true’). The guiding idea is that type-theory offers, by analogy, a way to get more fine-grained distinctions by way of truth-makers. Since assertoric content can thus be used in a theory of disagreement, we avoid the problem of individuating propositional content further.

Martin-Löf Conference on the Foundations of Mathematics: Day 1

I’m in Uppsala, Sweden, for the Foundations of Mathematics Conference dedicated to Per Martin-Löf on the occasion of his retirement. I’ve travelled up here together with fellow Archean Stewart Shapiro who’s giving a talk on Wednesday (incidentally on logical pluralism). So far the event has been well-organised and well-attended, with an impressive venue at the Swedish Collegium for Advanced Study. I applaud the fact that there was no registration fee for the conference!

When you go to top heavy conferences – like this one – you often end up surprised by which talks catch your fancy. Day 1 had names in proof-theory like William Tait and Giovanni Sambin, both of which I much looked forward to hearing for the first time, but in the end others stole the show. Tait’s talk was a largely historical discussion entitled ‘The Myth of Intuition’, where the role of Kantian intuition in Hilbert’s finitism was analysed. Sambin’s talk was perhaps the biggest surprise of the conference: A proposal for a new foundations of mathematics (and most other things!) roughly based on principles of abstraction and application, a machinery which he employed for his Basic logic in the papers of his that I’m familiar with. I readily admit that I didn’t anticipate that these techniques would be extended to foundational issues. Apparently, there is a forthcoming book with OUP called The Basic Picture which will provide a framework for the fifty mintues we got at the conference. Meanwhile, I’ll have to return to his earlier work for the proof-theory.

That is not to say that we didn’t get more technical talks. Peter Aczel’s gave an impressive but impenetrable (for yours truly) talk about type setups. Michael Rathjen gave us a very interesting peek into the world of constructive set theory. In constructive ZF you gain the option of adding axioms which would render classical set theory inconsistent – weaker logic means more things are consistent. One example, used by Stewart Shapiro, is that of adding an axiom schema for intuitionistic Church’s Thesis.

However, adding full axiom of choice (AC) to constructive set theory yields excluded middle, so that option is ruled out. More intuitively, AC asserts the existence of a function without saying anything about its definability, a feature that is typically taken to be constructively objectionable. Instead, Rathjen discusses alternative weaker choice principles, and how the addition of these can be justified with the help of constructive type-theory.

Multimedia from Arché

Through the Arché website, it's just been announced that recordings of some of the talks from the 1st Workshop in the Foundations of Logical Consequence have been made available online. It's great that FLC is the first project out with media from workshops. For now only two talks are online: one by Graham Priest and one by yours truly. We're hoping to make more available soon, amongst others Crispin Wright's talk at the same workshop.

Hopefully, more talk will be put online from all the Arché projects. I'll keep you updated.

UPDATE: Some twittering philosophers have alerted me to a number of problems with the website and audio. Apparently it's not running well in IE.

Arché Summer School

If your idea of a good summer is spending time indoors, in rainy Scotland, while following lectures on epistemology, methodology, logic and philosophy of language, then your best bet is probably the Arché Summer School.

The summer school will be held from June 29th to July 3rd in idyllic St Andrews, Scotland. All of the Arché Projects will contribute with lectures by senior members of Arché and project post docs. For the logic interested, FLC has four lectures on nonclassical logics and related philosophical issues. All of our investigators, Stephen Read, Graham Priest, and Stewart Shapiro, will be lecturing.

Registration here. Course info here.