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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.

Degrees of Truth, Degrees of Falsity

Toby Ord (a former student of mine, now taking Oxford by storm), has written up a nice short essay on degrees of truth and degrees of falsity. It shows how you can get a very nice little algebra if you extend the usual non-classical idea of a 4-valued logic in which truth and falsity are somewhat independent with the “fuzzy” idea of degrees of truth between zero and one. Both ideas have a heritage. The idea of considering the interval [0,1] as a lattice of truth values goes back to Łukasiewicz, and the four-valued algebra, now known as BN4, traces back at least to some early work by Mike Dunn.

Toby considers nice properties of this little algebra. It seems to me that a good exercise for someone who likes fiddling with concrete algebras would be this: define a conditional → on the algebra such that

when restricted to the fuzzy interval [0,1] it agrees with Łukasiewicz’s conditional.
when restricted to the values t, b, n and f agrees with the usual BN4 conditional.
has as many natural properties as possible. In particular, defining ‘A fuse B’ as ~(A → ~B) gives an associative and commutative operator, and fusion is connected with the conditional by means of the usual residuation postulates.

If you do this, you’ll have a nice concrete lattice which is a model for multiplicative and additive linear logic (and a little bit more — it’s distributive), and I’ll have a nice example to talk about in Non-Classical Logic.

So, please go and read Toby’s Essay, complete my exercise, and let me know what you come up with.

Posted 08:57 AM on February 19, 2006

Comments

Greg,

I do have a sort of inquietude with regard to the truth table of the negation of the values neither and both. Intuitively I can think of a negation where n and b remain unchanged when negated. But I intuitively can also think of a system where they swap places, i.e., the negation of neither is both and vice versa. And, thirdly, I can also think of a system where not-both means simply that a proposition cannot be both true and false, but it does mean that it is neither or either, and so on. I expect that Toby talked a little bit about these other possibilities.

Tony Marmo , February 23, 2006 01:00 AM

Toby doesn’t consider these things explicitly — he takes the BN4 choice for negation as a starting point, and then looks at how to extend it to the continuum. The BN4 choice is your first one: n and b are both fixed points for negation. Think of n as the (under-defined) empty set of values, and b as the (over-defined) set {0,1} of falsity and truth. Then evaluate negation like this: the evaluation of ~p contains 1 iff the evaluation of p contains 0, and the evaluation of ~p contains 0 iff the evaluation of p contains 1. It’s a natural thought, though perhaps, as you indicate, not the only possible thought.

Greg Restall , February 23, 2006 05:24 PM

Greg,

Here is my attempt at a solution. Given truth-values of the form [x, y] where x is the degree of truth and y is the degree of falsity, define a conditional operation as follows:

[x, y] -> [z, w] = [min(df(x,z), df(w,y), max(0, x -(1 - w))]

where df(x,y)= min(1,1 - (x - y))

This definition is based on two considerations:

(1) A conditional should be false just to the extent that the antecedent is true and the consequent false. That gives the falsity component of the truth-value for the above conditional; max(0,x - (1 - w)) is supposed to be an appropriate measure of how false a conditional is, based on how true the antecedent is (x) and how false the consequent is (w).

(2) A conditional should be true to the extent that it preserves truth from antecedent to consequent AND preserves falsity from consequent to antecedent. The consequent should not be less true than the antecedent and the antecedent should not be less false than the consequent. The function df(x, z) is supposed to be a measure of truth preservation from antecedant (x) to consequent (z), while df(w, y) measures the degree of falsity preservation from consequent (w) to antecedant (y).

I have been able to establish that this definition corresponds to the BN4 conditional when truth and falsity values are restricted to 0 and 1. Also, you can show that [x,0]->[y, 0] = [min(1, 1 - (x - y)), 0], which corresponds to the truth condition for the Lukasiewicz conditional.

I haven’t had much luck yet in establishing other properties of this conditional. Proving that various implicational principles are theorems seems quite tricky (A -> A is an easy one). I have established that fusion - defined as ~(x->~y) - is commutative, but associativity and residuation are still open problems. I don’t even know if this conditional satisifies modus ponens! Perhaps the definition could be simplified, which would make proofs easier. Or perhaps there’s a better solution …

Sam Butchart , February 26, 2006 05:58 PM

boring techie type question, but: how do you and Toby generate these beautiful diagrams, i.e. lattices, when formatting papers and tex? any recommendations? will you make the tex src to any of the papers available for discussion? could make for a nice blog post for people trying to format logic papers nicely.

Postmodernist , February 27, 2006 10:56 AM

I’m pretty sure Toby’s not using TeX, but is using something else. But I use pgf to generate pictures with TeX. You can find details of how I do this over on the wiki.

Greg Restall , February 27, 2006 11:02 AM

to Postmodernist:

I generated them with a program called omnigraffle. As far as I can tell it is only available for Macs, and can be found at:

http://www.omnigroup.com/applications/omnigraffle/

Of course I could have got by in the discussion without any diagrams at all, but when you can get them to look so nice and clear, I think they really help. Especially for people like me who think quite spatially.

to Sam Butchart:

That seems pretty good, although I have to say that my intuition is really at sea on this one, even with a plausible conditional right in front of me. Still I guess there are only a few ways to sensibly meet Greg’s proposed constraints and maybe it wouldn’t be that hard to decide between the few. Or we might even want to treat them as equally reasonable conditionals.

Toby Ord , March 5, 2006 11:30 AM