consequently.org/news

About

I'm Greg Restall, and this is my website. I work in Philosophy at the University of Melbourne. [Email: greg at consequently.org; Skype: greg_restall; Post: Department of Philosophy, University of Melbourne, Parkville 3010, Australia.]

Writing

These are the three last modified entries on my writing page.

“Truthmakers, Entailment and Necessity 2008,” an addendum to “Truthmakers, Entailment and Necessity,” to appear in Truth and Truth-making, edited by E. J. Lowe and A. Rami, Acumen, 2008. →
[with Rebecca Kukla and Mark Lance] Appendix to Rebecca Kukla and Mark Lance ‘Yo!’ and ‘Lo!’: the pragmatic topography of the space of reasons, Harvard University Press, to appear. →
“Curry’s Revenge: the costs of non-classical solutions to the paradoxes of self-reference,” in The Revenge of the Liar, ed. JC Beall, Oxford University Press, pages 262–271, 2008. →
“Anti-Realist Classical Logic and Realist Mathematics,” under revision. →
“Proof Theory and Meaning: on second order logic,” to appear in the Logica 2007 Yearbook, Filosofia. →

Classes

In Semester 2, which starts on July 31, I’ll be teaching an honours seminar 161-438 Logic and Philosophy, in which we cover proof theory and its applications to semantics.

Events

AAL2007: the annual conference of the Australasian Association for Logic, University of Melbourne November 9 to 11, 2007.

Recent Past

University of Melbourne Philosophy Undergraduate Workshop, University of Melbourne September 21 to 23, 2007.

Logic Colloquium 2007, Wrocław, Poland, July 14-19, 2007.

1st GPMR Workshop on Logic & Semantics on Medieval Logic and Modern Applied Logic, Rheinische Friedrich-Wilhelms-Universität Bonn, Germany, on June 28-30, 2007.

Logica 2007, Hejnice Monastery, Czech Republic, 18-22 June 2007.

Heart of Philosophy Café talk and discussion on “What Marx, Freud and Nietzsche have taught me about belief in God”. Tuesday May 8, 7--9pm in the Merrick's General Store.

Masses of Formal Philosophy: Question 2

Here’s my (much delayed) answer to the second of Vincent Hendricks and John Symons’ five questions about Formal Philosophy.

What example from your work illustrates the role formal methods can play in philosophy?

I’ll focus on one example from some of my recent work.

In the last few years I have been working on topics in proof theory and connections between the way we can conceive of the structure of proofs and concerns in the theory of meaning. The idea that the meaning of a word or a concept might be usefully explicated by giving an account of its inferential role is a common one — the work of Ned Block, Bob Brandom and Michael Dummett are three very different examples of ways to take this idea seriously. It is a truism that meaning has some sort of connection with use, and use in reasoning and inference is a very important part of any account of use.

It has seemed to me that if we are going to take take inferential role as playing its part in a theory of meaning, then we had better use the best available tools for giving an account of proof. The theory of proofs should have something to teach philosophers who have interests in semantics. This is not a mainstream position — our vocabulary itself speeks against this, with the ready identification of model theory with ‘semantics’ and proof theory with ‘syntax’. The work of intuitionists such as Dummett, Prawitz, Martin-Löf and Tennant in conspicuous in its isolation at providing a contrary opinion to the mainstream. This has led to the opinion that semantically anti-realist positions — those that take proof or inference as the starting point of semantic theory, rather than truth-conditions or representation — are naturally revisionary and intuitionist. For intuitionistic logic has a clear motivation in terms of proof and verification, and it has seemed to many that orthodox classical logic does not.

I think that this is a mistake. It seems to me that natural proof-theoretic accounts of classical logic (starting with Gentzen’s sequent calculus, but also newer pieces of technology such as proof-nets) can have a central place in a theory of meaning that starts with inferential role and not with truth. We can think of the valid sequents (of the form X ⊢ Y, where X and Y are sets of statements) as helping us ‘keep score’ in dialectical positions. The validity of the sequent X ⊢ Y tells us that a position in dialogue in which each statement in X is asserted and each statement in Y is denied is out of bounds according to the rules of ‘the game.’ In fact, the structural rules in the sequent calculus can be motivated in this way. Identity sequents X,A ⊢ A,Y tell us that asserting and denying A (in the same context) is out of bounds. The rule of weakening tells us that if asserting X and denying Y is out of bounds then adding an extra assertion or extra denial would not aid the matter. The cut rule tells us that if a position in which X is asserted and Y is denied is not out of bounds, then given a statement A, either the addition as an assertion, or its addition as a denial will also not be out of bounds. If asserting A is out of bounds in a context, it is implicitly denied in that context. Explicitly denying is no worse than implicitly denying.

Thinking of Gentzen’s sequent calculus in this way gives an alternative understanding of classical logic. We think of the rules for connectives as ‘definitions’ governing assertions featuring the logical vocabulary. Proof-theoretical techniques such as the eliminability of the ‘cut’ rule tell us that these definitions are conservative. No matter what the rules of the game concerning our primitive vocabulary might be, we can add the classical logical connectives without disturbing the rules of assertion in that primitive vocabulary (the need for this point was made clearly in Nuel Belnap’s paper “Tonk, Plonk and Plink”). The logical vocabulary allows us to ‘make explicit’ what was merely ‘implicit’ before. The interpretation of the rules of the quantifiers is particularly enlightening. It allows us to sidestep the debate between ‘substitutional’ and ‘objectual’ accounts of quantification.

In my recent work I have tried to flesh out this picture, and to show how we can expand this story to take account of appropriate conditions for use for modal connectives such as possibility and necessity. The key idea is that in modal discourse we not only assert and deny, but we make assertions and denials in different dialectical contexts, and an assertion of a necessity claim in one context can impinge on claims in other dialectical contexts. This means that we can give a semantics of modal vocabulary that motivates a well-known modal logic (in the first instance, the simple modal logic S5, but the extension to other logics is not difficult) in which possible worlds are not the starting point of semantic explanation. Modal vocabulary needs not be conceived of as a way of describing possible worlds. It can be understood as a governing discourse in which we not only assert and deny to express our own commitments, but also to articulate the connections between our concepts. The structures of dialectical positions need not merely contain assertions and denials, but these may be partitioned into different ‘zones’ according to structure of the different suppositions and shifts of context in that discourse.

Posted 01:30 PM on September 16, 2006

Comments

That was an interesting post Professor Restall. I’m a grad student who is interested in phil. language and logic and I was wondering what would be a couple of good articles/books to look at for an introduction to proof theoretic semantics. What are the must-reads in that area? I’m taking proof theory right now and would love to connect it to my more central interests like that. Thanks

Shawn , September 17, 2006 01:52 AM

Hi Greg, I like this way of thinking about things! (I remember it from your talk in Nottingham last year). But I’ve a nagging worry. You give equal weight to assertion and denial: they’re just two sides of the same turnstile. But there seems to be much more to say than this; for instance, to first say what assertion is, and then to explain negation. Do you want to treat both assertion and denial as primatives, or to say more but leave what you do say independent of the rest of the story?

Mark Jago , September 17, 2006 03:38 AM

Hi Shawn:

Things to read? I’m writing a book on it at the moment, so you can have a look at the drafts available on the web.

For other stuff, I’d try looking at the work of Jaroslav Peregrin, who is connecting the inferentialist work of Brandom to other things in proof theory, pretty nicely. Neil Tennant’s work connects proof theory and meaning theory in a radically anti-realist direction.

Greg Restall , September 17, 2006 07:38 AM

Hi Mark!

I remember worrying about your question in Nottingham. The story I’d like to tell is that denial isn’t to be understood as the assertion of a negation, but is prior to that: in particular, assertion is pretty fundamentally a communicative action, and just as someone offers an assertion, it’s something that can be accepted (implicitly or explicitly reasserted by me) or rejected (implicitly or explicitly denied by me). And that this can happen without requiring the operator of negation, which is an essentially composible thing finding for each propositional content another content that is its negation. But I’m still figuring out the best way of thinking about it.

But, for the purposes of the book, I want to leave two options.

Leave the reader to take their favourite view of the connection between assertion and denial and use that.
Present my view (a hopefully more convincing version of what I’ve sketched here) as my preferred option.

The rest of the applications of the theory is pretty independent of the answer I give here.

Greg Restall , September 17, 2006 07:43 AM

I agree, denial isn’t simply asserting a negation; the whole point is to explain negation in terms of denial. I think my view is becoming something like: to explain assertion, you have to talk about attitudes such as belief and, ultimately, action. That’s what makes assertions more than just noises or marks on paper. Let’s say we’ve done that for assertion. Now, I wonder what to say about denial. Two options:

(1) Do the same for denial as assertion: explain it’s meaning in terms on analytic links with action. Or:

(2) Cash out denial in terms of assertyion + some expressive attitude, eg rejection. On this view, assertion is primary and denial (and so the Boolean connectives) become expressive attitudes to assertions of various types.

There’s probably more options; but out of these two, I’m guessing you go for the former?

Mark Jago , September 18, 2006 02:52 AM

Interesting post! Your work seems related to Lorenzen’s and Hintikka’s ideas of a game-theoetic semantics for logical languages (an idea of great interest recently to theoretical computer scientists developing theories of machine interaction). Have you explored the connections there?

On assertion and denial, I think any dialogic account would also need to deal with retraction (of prior assertions), something which I believe is neither the same as denial, nor as negation, of assertions.

Peter , October 8, 2006 04:17 AM

I’m busy and behind on everything, including replying to comments on my own site. Oh well, better late than never, I suppose…

Mark: I do want to say a little more on assertion and denial. I don’t want to think of assertion (or denial) purely in terms of links with action, but that should be a part of it. You’re right to think that I don’t want to explain denial in terms of assertion and negation. The direction of explanation for me goes the other way.

Peter: Yes, retraction is also a part of the story. To retract an assertion (or a denial) differs from the denial (or assertion) of the thing retracted.

In fact, retraction is useful in the connection with assertion and denial. Denying p means that any future assertion of p requires a retraction of that denial. Asserting p means that any future denial of p requires a retraction of that assertion. Or so it seems to me…

Greg Restall , October 23, 2006 09:59 PM