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It seems well known in modal logic society that $\Box(\Box p\to p) \to \Box p$ in Kripke semantics of $GL$ implies well-foundedness of the relation i.e. no infinite ascending chains are allowed.

And validity of $GL$ additionally implies irreflexivity and transitivity of the relation.

But I am wondering what is the initial-like reference for the facts mentioned.

P.S. This question may not make clear anything interesting, this is just reference request to experts.

UPDATE It seems that Segerberg's thesis is tha earliest reference for the fist fact.

Concerning the second fact I reckon Dick de Jongh has proved this one, that said, unfortunately I could not find any reference.

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    $\begingroup$ Well-foundedness does not imply transitivity. Validity of GL implies transitivity. $\endgroup$
    – Emil Jeřábek
    May 31, 2022 at 19:00
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    $\begingroup$ @Emil Jeřábek Thank you! I agree, and I am sorry for mistype. $\endgroup$
    – Evgeny Kuznetsov
    May 31, 2022 at 19:33

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The paper Provability: the emergence of a mathematical modality by Boolos and Sambin says (bottom of page $9$) that the first fact was independently gotten by Kripke and Segerberg, and gives the latter's $1971$ thesis as a reference. However, I can't find a copy of Segerberg's thesis online; the only promising link I found, http://lpcs.math.msu.su/~zolin/ml/pdf/Segerberg_Essay_1971.pdf, seems broken.

The second fact you mention has trivial first conjunct and false second conjunct (non-transitive frames may still be well-founded), and so I doubt there is a citation for it.

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  • $\begingroup$ Thank you a lot! Yes, I suspected the same about the last fact and I agree with the remark of yours. $\endgroup$
    – Evgeny Kuznetsov
    May 31, 2022 at 19:06
  • $\begingroup$ I just realized that that time you addresed the direction well-foundedness -> transitivity. As Emil has noticed that was a mistype, it should be GL -> transitiviry. I recon Dick de Jongh has proved this one, that said unfortunately I could not find a reference. $\endgroup$
    – Evgeny Kuznetsov
    Jun 1, 2022 at 23:04

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