A note on the logic of eventual permanence for linear time (Q929630)

In [\textit{M. Byrd}, ``Eventual permanence'', Notre Dame J. Formal Logic 21, 591--601 (1980; Zbl 0426.03019)], Byrd claims that the logic of `eventual permanence' for linear time is the modal logic \textbf{KD5}. He argues that the logic of `eventual permanence' for linear time is a unique logic \(L\) such that for each formula \(\varphi\), \(\varphi\) is an \(L\)-theorem if and only if its \(\lozenge\square\)-translation is a theorem of the well-known modal logic \textbf{KD4.3}. In [\textit{L. Humberstone}, ``Weaker-to-stronger translational embeddings in modal logic'', in: G. Governatori et al. (eds.), Advances in modal logic. Vol. 6. Selected papers from the 6th conference (AiML 2006), Noosa, Australia, September 25--28, 2006. London: College Publications. 279--297 (2006; Zbl 1144.03013)], Humberstone noted that \textbf{KD5} does not satisfy the `if' condition of this statement. Using Segerberg's characterization of all normal extensions of the well-known modal logic \textbf{KD45} [\textit{K. Segerberg}, An essay in classical modal logic. Vol. 1, 2, 3. Filosofiska Studier. No.~13. Uppsala: University of Uppsala (1971; Zbl 0311.02028)] (see also \textit{N. Bezhanishvili}, ``Pseudomonadic algebras as algebraic models of doxastic modal logic'', Math. Log. Q. 48, No.~4, 624--636 (2002; Zbl 1016.03067)]), the author modifies Byrd's argument to show that in fact \textbf{KD45}, not \textbf{KD5}, is the logic with full and faithful \(\lozenge\square\)-tranlsation into \textbf{KD4.3}, and therefore the logic of `eventual permanence' for linear time is \textbf{KD45}.