# zbMATH — the first resource for mathematics

Multimodal logics of products of topologies. (English) Zbl 1113.03018
If modal logics $$L_1,L_2$$ with modalities $$\square_1,\square_2$$ are determined by classes $$\mathbb{F}_1,\mathbb{F}_{2}$$ of Kripke frames, then $$L_1 \times L_2$$ is determined by the class of products $$\mathbb{F}_1 \times \mathbb{F}_{2}= \langle W_1 \times W_2,R_1,R_2\rangle$$ and is axiomatized (by D. Gabbay and V. Shekhtman, under suitable conditions) by Fusion $$L_1+L_2$$ plus $$com=\square_1\square_2p\to\square_2\square_1p$$ and $$chr=\diamondsuit_1\square_2p\to\square_2\diamondsuit_1p$$. In the topological semantics, when $$\square$$ is interpreted as the interior of a set, products $$X\times Y$$ not always validate $$com$$ and $$chr$$. The authors prove that both $$X$$ and $$Y$$ being Alexandrov spaces (intersection of an arbitrary family of open sets is open) is sufficient. However, for rationals $$\mathbb{Q}$$, the logic of $$\mathbb{Q}\times \mathbb{Q}$$ is complete for the fusion $$\text{S}4+\text{S}4$$, hence much weaker than $$\text{S}4 \times \text{S}4$$. A new completeness proof of S4 for $$\mathbb{Q}$$ is presented. The authors introduce and investigate new kinds of topologies, horizontal and vertical, which they call coordinate topologies. In the typical case of $$\mathbb{R}\times \mathbb{R}$$ a set $$A$$ is horizontally open if it contains with each point $$(x,y)$$ a horizontal interval $$(a,b)\times \{y\}$$ for $$a<x<b$$ and similarly for the vertical topology.

##### MSC:
 03B45 Modal logic (including the logic of norms) 54B10 Product spaces in general topology
Full Text:
##### References:
 [1] Aiello, M., and J. van Benthem, ’A modal walk through space’, Journal of Applied Non-Classical Logics, 12:319–363, 2002. · Zbl 1185.03060 [2] Aiello, M., J. van Benthem, and G. Bezhanishvili, ’Reasoning about space: the modal way’, Journal of Logic and Computation, 13:889–920,2003. · Zbl 1054.03015 [3] Barwise, J., ’Three views of common knowledge’, Proc. of TARK, 1988, pp. 365–379. · Zbl 0704.03008 [4] van Benthem, J., and D. Sarenac, ’The geometry of knowledge’, in J-Y. Béziau, A. Costa-Leite and A. Facchini (eds.), Trends in Universal Logic, 2005. · Zbl 1087.03011 [5] van Benthem, J., ’Information as correlation and information as range’, ILLC manuscript, 2003. [6] van Benthem, J., Exploring logical dynamics, CSLI publications, 1997. [7] Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge University Press, 2001. [8] Gabbay, D. M., and V. B. Shehtman, ’Products of modal logics I’, Log. J. IGPL, 6:73–146, 1988. · Zbl 0902.03008 [9] Gabbay, D. M., A. Kurucz, F. Wolter, and M. Zakharyaschev, Manydimensional modal logics: theory and applications. Studies in Logic and the Foundations of Mathematics, Volume 148. Elsevier, 2003. · Zbl 1051.03001 [10] Gabelaia, D., A. Kurucz, F. Wolter, and M. Zakharyaschev, ’Products of ’transitive’ modal logics’, J. Symbolic Logic, 70:993-1021, 2005. · Zbl 1103.03020 [11] Goldblatt, R., ’Diodorean modality in Minkowski spacetime’, Studia Logica, 39:219–237, 1980. · Zbl 0457.03019 [12] Löwe, B., and D. Sarenac, ’Cardinal spaces and topological representations of bimodal logics’, J. of IGPL, 13:301–306, 2005. · Zbl 1077.03012 [13] Kuratowski, K., and A. Mostowski, Set theory, Studies in Logic and the Foundations of Mathematics, Vol. 86. North-Holland Publishing Co., Amsterdam-New York-Oxford; PWN–,Polish Scientific Publishers, Warsaw, 1976. [14] McKinsey, J. C. C., and A. Tarski, ’The algebra of topology’, Ann. of Math. 45:141–191, 1944. · Zbl 0060.06206 [15] Mints, G., ’A completeness proof for propositional S4 in Cantor space’, Ch. 6, in Logic at Work, Kluwer Publishing, 1998. · Zbl 0923.03026 [16] Sarenac, D., Modal logic and topological products, PhD Thesis, Stanford University, 2005. · Zbl 1077.03012 [17] Segerberg, K., ’Two-dimensional modal logic’, J. Philos. Logic, 2:77–96, 1973. · Zbl 0259.02013 [18] Shehtman, V.B., ’Two-dimensional modal logics’, Mathematical notices of the USSR Academy of Sciences, 23:417–424, 1978. (Translated from Russian.) · Zbl 0403.03015 [19] Spaan, E., Complexity of modal logics, PhD thesis, University of Amsterdam, Institute for Logic, Language and Computation, 1993. · Zbl 0831.03005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.