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.
