A little sleep, a little progress. The “fuzzy logic” approach that gives each statement a truth value between 0 and 1 can’t handle the obvious “this sentence is not true,” so it’s out. The other one-parameter approach I can think of is more clever. The thought was that each self-referential statement defines a transformation of it’s own “truth vector” (T, F), so consistency means that the statement should evaluate to eigenvectors of the transformation. Unfortunately, these transformations don’t always commute, so you can get inconsistent answers to “this sentence is not true and is not (1/sqrt(2),1/sqrt(2)).” Still working on that one.
Manfred’s log, stardate 11⁄30
A little sleep, a little progress. The “fuzzy logic” approach that gives each statement a truth value between 0 and 1 can’t handle the obvious “this sentence is not true,” so it’s out. The other one-parameter approach I can think of is more clever. The thought was that each self-referential statement defines a transformation of it’s own “truth vector” (T, F), so consistency means that the statement should evaluate to eigenvectors of the transformation. Unfortunately, these transformations don’t always commute, so you can get inconsistent answers to “this sentence is not true and is not (1/sqrt(2),1/sqrt(2)).” Still working on that one.