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Topological perspective on the hybrid proof rules. (English) Zbl 1278.03051
Blackburn, Patrick (ed.) et al., Proceedings of the international workshop on hybrid logic (HyLo 2006), Seattle, WA, USA, August 11, 2006. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 174, No. 6, 79-94 (2007).
Summary: We consider the non-orthodox proof rules of hybrid logic from the viewpoint of topological semantics. Topological semantics is more general than Kripke semantics. We show that the hybrid proof rule BG is topologically not sound. Indeed, among all topological spaces the BG rule characterizes those that can be represented as a Kripke frame (i.e., the Alexandroff spaces). We also demonstrate that, when the BG rule is dropped and only the Name rule is kept, one can prove a general topological completeness result for hybrid logics axiomatized by pure formulas. Finally, we indicate some limitations of the topological expressive power of pure formulas. All results generalize to neighborhood frames.
For the entire collection see [Zbl 1273.68016].

MSC:
03B45 Modal logic (including the logic of norms)
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[1] Aiello, Marco; van Benthem, Johan; Bezhanishvili, Guram, Reasoning about space: the modal way, Journal of logic and computation, 13, 889-920, (2003) · Zbl 1054.03015
[2] Guram Bezhanishvili, Ray Mines, and Patrick J. Morandi. Topo-canonical completions of closure algebras and Heyting algebras. Under submission · Zbl 1135.06009
[3] Patrick Blackburn and Balder ten Cate. Pure extensions, proof rules and hybrid axiomatics. Studia Logica, 84(2):281-326, 2006. To appear · Zbl 1115.03009
[4] Blackburn, Patrick; de Rijke, Maarten; Venema, Yde, Modal logic, (2001), Cambridge University Press Cambridge, UK · Zbl 0988.03006
[5] Balder ten Cate, David Gabelaia, and Dmitry Sustretov. Modal languages for topology: expressivity and definability. Under submission · Zbl 1172.03013
[6] ten Cate, Balder; Marx, Maarten; Viana, Petrucio, Hybrid logics with sahlqvist axioms, Logic journal of the IGPL, 13, 3, 293-300, (2005) · Zbl 1080.03006
[7] Chellas, Brian F., Modal logic, (1980), Cambridge University Press Cambridge Massachusetts · Zbl 0431.03009
[8] Došen, Kosta, Duality between modal algebras and neighbourhood frames, Studia logica, 48, 219-234, (1989) · Zbl 0685.03013
[9] Gargov, George; Passy, Solomon; Tinchev, Tinko, Modal environment for Boolean speculations, (), 253-263 · Zbl 0701.03008
[10] Andrey Kudinov. Difference modality in topological spaces. A talk at Algebraic and Topological Methods in Non-Classical Logics II, June 2005, Barcelona. Abstract available at http://atlas-conferences.com/c/a/p/u/76.htm
[11] Kudinov, Andrey, Topological modal logics with difference modality, (), 319-332 · Zbl 1148.03019
[12] Litak, Tadeusz, Isomorphism via translation, (), 333-351 · Zbl 1144.03015
[13] Makowsky, Janos A.; Ziegler, Martin, Topological model theory with an interior operator: consistency properties and back-and-forth arguments, Arch. math. logic, 21, 1, 37-54, (1981) · Zbl 0472.03027
[14] McKinsey, J.C.C.; Tarski, Alfred, The algebra of topology, Annals of mathematics, 45, 141-191, (1944) · Zbl 0060.06206
[15] Katsuhiko Sano. Topological completeness for hybrid logics without the global modality. In preparation · Zbl 1290.03006
[16] Thomason, Steve K., Categories of frames for modal logic, Journal of symbolic logic, 40, 439-442, (1975) · Zbl 0317.02012
[17] Venema, Yde, Derivation rules as anti-axioms in modal logic, Journal of symbolic logic, 58, 1003-1034, (1993) · Zbl 0793.03017
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