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.

I’m *Greg Restall*, and this is my *personal website*. ¶ I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology ¶ I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.

To receive updates from this site, subscribe to the RSS feed in your feed reader. Alternatively, follow me at @consequently@hcommons.social, where most updates are posted.

- Social media: @consequently@scholar.social.
- Personal email: greg@consequently.org.
- St Andrews profile.
- My office is in Edgecliffe, The Scores.