I came across this question yesterday which is interesting. It asks whether a binary sole sufficient operator for the modal logic K exists. I tried to find an example of a sole sufficient operator for K for a little bit before taking a step back and realizing that I don't know any proof techniques for showing that sole sufficient operators (of a given arity) do not exist for a particular algebra.
What are some techniques for showing that a sole sufficient operator of a given arity does not exist?
So far, I have come up with a way to show that a unary sole sufficient operator does not exist for the $\square, \lnot$ fragment of S5. The result holds in a wider variety of modal logics than S5, but S5 is a good starting point.
Let the following equations axiomatize an S5-algebra. Let $\lozenge$ abbreviate $\lnot \square \lnot$.
- $ \lnot \lnot a \approx a $
- $ \square \square a \approx \square a $
- $ \lozenge \square a \approx \square a $
I claim that any algebra satisfying (1) and (2) that has a sole sufficient operator satisfies $\square a \approx a$ too; i.e. has modal collapse. Modal collapse is not in the deductive closure of S5, so asserting the presence of a sole sufficient operator has caused us to lose some models of our equation theory.
For proof, let $u$ be a sole-sufficient operator.
For some natural numbers $k$ and $l$, suppose $\lnot a = u^k(a)$ and $\square a = u^l(a)$.
Consider the formula $u^{2kl}(a)$.
- It is equivalent to $\lnot^{(2l)}(a)$ and thus $a$.
- It is also equivalent to $\square^{(2k)}(a)$ and thus $\square a$.
Thus it holds that $\square a \approx a$.