Abstract
This chapter is about the model construction problem in some classes of models: models satisfying the conditions of reflexivity, seriality, symmetry, and combinations thereof. The corresponding logics are KT, KD, KB, KTB, and KDB. We also consider models whose accessibility relation is confluent (logic K.2) or is an equivalence relation (logic KT45, alias S5). While these conditions are only about a single relation, we also study properties involving two accessibility relations: inclusion and permutation.
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Notes
- 1.
So the hasNoSuccessor condition is only effective in subsequent applications of the Pos_deterministic1 rule: when some other ◊-formulas are added to the same node then hasNoSuccessor will make the Pos_deterministic1 rule fail. So it would for example succeed in closing the premodel for <> P & (<> ~P & Q). (Observe that parentheses are relevant: (<> P & <> ~P) & Q would not close.)
- 2.
A variant of that theorem was first stated in [GHS06]. However, the condition on node creation (the last condition of Theorem 1) was too weak and is strengthened here.
- 3.
We note in passing that the language of LoTREC contains an unmark action that erases marks and that is therefore not monotonic. However, we have banned it from this book, i.e., we suppose here that unmark is not part of the language.
References
C. Areces and B. ten Cate. Hybrid logics. In P. Blackburn, J. van Benthem, and F. Wolter, editors, Handbook of Modal Logic, volume 3. Elsevier Science, Amsterdam, 2006.
M. Fitting. Proof Methods for Modal and Intuitionistic Logics. D. Reidel, Dordrecht, 1983.
D.M. Gabbay. Labelled Deductive Systems, volume 1. OUP, London, 2002.
O. Gasquet, A. Herzig, and M. Sahade. Terminating modal tableaux with simple completeness proof. In G. Governatori, I. Hodkinson, and Y. Venema, editors, Advances in Modal Logic, volume 6, pages 167–186. King’s College Publications, London, 2006.
W. Lenzen. Recent Work in Epistemic Logic. North Holland, Amsterdam, 1978.
W. Lenzen. On the semantics and pragmatics of epistemic attitudes. In A. Laux and H. Wansing, editors, Knowledge and Belief in Philosophy and AI, pages 181–197. Akademie Verlag, Berlin, 1995.
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Gasquet, O., Herzig, A., Said, B., Schwarzentruber, F. (2014). Logics with Simple Constraints on Models. In: Kripke’s Worlds. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8504-0_4
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DOI: https://doi.org/10.1007/978-3-7643-8504-0_4
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