Incomplete normal modal logic systems

Apart from the classical example of KH, given by axiom $\Box(\Box p\leftrightarrow p)\to \Box p$, are there any other examples of incomplete propositional normal modal logic systems defined by axioms in 1 variable and of modal degree $\le 2$? If so, I would also appreciate a reference to where the proofs can be found. I am looking for the simplest possible examples. Van Benthem's is of modal degree 3, others are in 2 variables etc.

• You might be able to find what you're after in Priest's An Introduction to Non-Classical Logic: From If to Is. – Shaun Nov 10 '13 at 17:08

The logic $L$ that extends $\mathbf{K}$ with axioms (of degree at most 2 and with only 1 variable) $$\begin{array}{l} \Box p\to p\\ \Box\Diamond p\to\Diamond\Box p\\ \Box(p\to\Box p)\to(\Diamond p\to p) \end{array}$$ is even first-order definable, as it defines the class of frames satisfying the first-order sentence $\forall x\forall y(xRy\leftrightarrow x=y)$, but the modal formula ${\Box p\leftrightarrow p}$, which is valid on this class of frames, is not derivable in $L$. Therefore, the logic $L$ is not Kripke complete.
• Thank you for the answer. In the meantime I had found this system myself, and I suspected it was incomplete, only I was not able to prove(see this question) that ${\Box p\leftrightarrow p}$ is not a theorem, which seems to be the hard part. I understand the proof is in van Benthem's 1978 paper "Two simple incomplete modal logics", which I do not have. If anyone can sketch the proof here I would be interested. – JuneA Aug 21 '14 at 23:33