FLAP Journal 2016 Journal Article
More Modal Semantics without Possible Worlds.
- Daniel Skurt
- Hitoshi Omori
In one of his papers, John Kearns developed a semantics without possi- ble worlds for the normal modal logics T, S4, and S5 and even though his work challenged the orthodox view of Kripkean semantics, his work has never been widely recognized. This probably was due to the complexity of his non- deterministic hierarchic semantics. But contemporary developments in the area of non-deterministic semantics make his work most likely understandable for a bigger audience. In this paper we first try to clarify Kearns’ original approach within the context of non-deterministic semantics and give proofs of soundness and completeness with respect to Hilbert style calculi. Then, we simplify some of Kearns results and reveal its problematic aspects. Finally, we generalize the result to cover more modal logics such as K, KD and KTB. Hitoshi Omori was a Postdoctoral Fellow for Research Abroad of the Japan Society for the Promotion of Science (JSPS) at the time of submission, and now he is a Postdoctoral Research Fellow of JSPS. Daniel Skurt was supported by the Ruhr University Research School PLUS, funded by Germany’s Excellence Initiative [DFG GSC 98/3]. We would like to thank Thomas Ferguson, Rohan French, David Ripley, Christian Straßer and Heinrich Wansing for helpful discussions and/or comments on an earlier draft. Furthermore, we would like to thank Uwe Meinel and Michael De for proofreading our English. Finally, we would like to thank the anonymous referees for correcting our mistakes, and pointing out some important points which were missing in our draft.