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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.

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  • $\begingroup$ You might be able to find what you're after in Priest's An Introduction to Non-Classical Logic: From If to Is. $\endgroup$
    – Shaun
    Nov 10, 2013 at 17:08

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The answer can be found in the Volume 3 of the Handbook of Philosophical Logic, 2nd Edition, 2001. See the "Correspondence Theory" chapter by Johan van Benthem, pp. 325-408. On page 333, Fact 5, he gives an example of an incomplete modal logic:

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.

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    $\begingroup$ 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. $\endgroup$
    – JuneA
    Aug 21, 2014 at 23:33
  • $\begingroup$ Here is van Benthem's 1978 paper: mirror1 | mirror2 Also you will find the following paper relevant to your question: "A Simple Incomplete Extension of T which is the Union of Two Complete Modal Logics with f.m.p." by Roy A. Benton (2002): link. $\endgroup$ Aug 22, 2014 at 21:19

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