Descriptive general frames are usually defined as general frames that are tight, compact, and differentiated. On p.91 of this paper by Skvortsov and Shehtman 1993, the authors omit the third condition, mentioning in a footnote that it is only required if the frame is non-reflexive. Is this a well-known fact? It's not mentioned in any of my textbooks.

For context, Skvortsov and Shehtman try to show that the quantified extension of every canonical propositional modal logic remains canonical under their "metaframe semantics", and they assume that a logic is canonical iff whenever it is sound wrt a descriptive general frame F,H then it is sound wrt F -- as in definition 6.11 of van Benthem, Modal Logic and Classical Logic (1983).

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    $\begingroup$ This is weird. The condition is redundant if the frame is a partial order (then it follows from tightness), but it definitely needs to be included if the frame may contain proper clusters. I suppose this is an error in the paper. $\endgroup$ Jun 9 at 15:21
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    $\begingroup$ For example, the logic $\mathbf{Triv=K}\oplus(\phi\leftrightarrow\Box\phi)$ (which extends S4) is canonical, but not under Skvortsov and Shehtman’s definition: for any set $W$, the frame $(W,W^2,\{\varnothing,W\})\models\mathbf{Triv}$ is tight and compact, but $(W,W^2)\nvDash\mathbf{Triv}$ if $|W|\ge2$. $\endgroup$ Jun 9 at 15:42


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