1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 '13 at 17:08
1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ 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 '14 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. $\endgroup$ – Evgeny Zolin Aug 22 '14 at 21:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.