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
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.
I’m Greg Restall, and this is my personal website. I teach philosophy and logic as Professor of Philosophy at the University of Melbourne. ¶ From August 2021, I will be the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews. ¶ Start at the home page of this site—a compendium of recent additions around here—and go from there to learn more about who I am and what I do. ¶ This is my personal site on the web. Nothing here is in any way endorsed by the University of Melbourne.