In particular I'm interested in propositional logics with semantics given by truth tables which are not functionally complete, but which can define all their constant truth functions.
I think the restriction of Post's lattice to clones containing the constants shows that there are six two-valued cases.
One many-valued example is R. B. Angell's connexive logic CC1. (See [1].) Are there any other reasonably well-known or interesting examples here?
[1] McCall, S., Connexive implication, J. Symb. Log. 31, 415-433 (1966). ZBL0161.00405.
