Mereology and uncertainty

Lech T. Polkowski

DOI: http://dx.doi.org/10.12775/LLP.2015.008

Abstract


Mereology as an art of composing complex concepts out of simpler parts is suited well to the task of reasoning under uncertainty: whereas it is most often difficult to ascertain whether a given thing is an element of a concept, it is possible to decide with belief degree close to certainty that the class of things is an ingredient of an other class, which is sufficient for carrying out the reasoning whose conclusions are taken as true under given conditions. We present in this work a scheme for reasoning based on mereology in which mereology in the classical sense is fuzzified in analogy to the concept fuzzification in the sense of L. A. Zadeh. In this process, mereology becomes rough mereology.

Keywords


knowledge; uncertainty; vaguenes; ambiguity; rough sets; fuzzy sets; mereology; rough mereology; granulation of knowledge; granular logics; spatial reasoning

Full Text:

PDF

References


Black, M., “Vagueness. An exercise in logical analysis”, Philosophy of Science, 4, 4 (1937): 427–455.

Bochenski, I. M., Die Zeitgenossischen Denkmethoden, A. Francke AG, Bern, 1954.

Casati, R., and A. C.Varzi, Parts and Places. The Structures of Spatial Representations, MIT Press, Cambridge MA, 1999.

Fleck, L., “O niektórych swoistych cechach myślenia lekarskiego” [“On some specific features of the medical way of thinking” in Polish], Archiwum Historji Medycyny i Historji Nauk Przyrodniczych, 6 (1927): 55–64. In English: pp. 39–46 in R. S. Cohen and Th. Schnelle (eds.), Cognition and fact: Materials on Ludwik Fleck, Reidel, Dordrecht, 1986. DOI: 10.1007/978-94-009-4498-5

Frege, G., Grundgesetze der Arithmetik, Band II, Jena, Verlag Hermann Pohle, 1903. In English: Ph. A. Ebert and M. Rossberg (eds.), Basic Laws of Arithmetic, Oxford University Press, 2013.

Gödel, K., “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik, 38, 1 (1931): 173–198. In English: pp. 144–195 in S. Feferman (ed.), “On formally undecidable propositions of Principia Mathematica and related systems I”, Kurt Gödel Collected works, vol. I., Oxford University Press, 1986.

Hájek, P., Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. DOI: 10.1007/978-94-011-5300-3

Heisenberg, W., “Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik”, Zeitschrift fuer Physik, 43 (1927): 172–198. In English: pp. 62–84 in J. A. Wheeler and W. H. Zurek (eds.), Quantum Theory and Measurement, Princeton University Press, Princeton NJ, 2014. DOI: 10.1515/9781400854554

Leśniewski, S., “Podstawy Ogólnej Teoryi Mnogości I” [“Foundations of General Set Theory I” in Polish], Prace Polskiego Koła Naukowego w Moskwie, Sekcya Matematyczno-przyrodnicza, No. 2, Moscow, 1916. In English: “On the foundations of mathematics”, Vito F. Sinis (transl.), Topoi, 2 (1983): 7–52.

Menger, K., “Statistical metrics”, Proceedings of the National Academy of Sciences USA, 28 (1942): 535–537. DOI: 10.1073/pnas.28.12.535

Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data, Kluwer, Dordrecht, 1991. DOI: 10.1007/978-94-011-3534-4

Polkowski, L., Approximate Reasoning by Parts. An Introduction to Rough Mereology, Springer Verlag, Berlin, 2011. DOI: 10.1007/978-3-642-22279-5

Polkowski, L., “A note on 3-valued rough logic accepting decision rules”, Fundamenta Informaticae, 61 (2004): 37–45.

Polkowski, L., and P. Osmiałowski, “Spatial reasoning with applications to mobile robotics”, pp. 433–453 in Jing Xing–Jian, (ed.), Mobile Robots Motion Planning. New Challenges, I-Tech Education and Publishing KG, Vienna, 2008. DOI: 10.5772/6002

Polkowski, L., and P. Ośmiałowski, “A framework for multi-agent mobile robotics: Spatial reasoning based on rough mereology in Player/Stage system”, pp. 142–149 in Chien-Chung Chan, J. W. Grzymala-Busse, and W.P. Ziarko (eds.), Rough Sets and Current Trends in Computing, Lecture Notes in Artificial Intelligence vol. 5306, Springer Verlag, Berlin, 2008.

Polkowski, L., and P. Ośmiałowski, “Navigation for mobile autonomous robots and their formations: An application of spatial reasoning induced from rough mereological geometry”, pp. 329–354 in A. Barrera (ed.), Mobile Robots Navigation, InTech, Zagreb, 2010. DOI: 10.5772/209

Polkowski, L., and M. Semeniuk-Polkowska, “On rough set logics based on similarity relations”, Fundamenta Informaticae, 64 (2005): 379–390.

Polkowski, L., and M. Semeniuk-Polkowska, “Granular mereotopology: A first sketch”, pp. 322–331 in Proceedings CS&P 2013 (Concurrency, Specification, Programming), 2013. Also as: “On the problem of boundaries from mereology and rough mereology points of view”, Fundamenta Informaticae, 133, 2–3 (2014): 241–255.

Polkowski, L., and M. Semeniuk-Polkowska, “Boundaries, borders, fences, hedges”, Fundamenta Informaticae, 129, 1–2 (2014): 149–159.

Polkowski, L., and A. Skowron, “Rough mereology: a new paradigm for approximate reasoning”, International Journal of Approximate Reasoning, 15, 4 (1996): 333–365. DOI: 10.1016/S0888-613X(96)00072-2

Simons, P., Parts. A Study in Ontology, 2nd ed., Clarendon Press, Oxford UK, 2003.

Smith, B., “Mereotopology: A theory of parts and boundaries”, Data and Knowledge Engineering, 20 (1996): 287–303.

Smith, B., “Boundaries: An essay in mereotopology”, pp. 534–561 in L. Hahn (ed.), The Philosophy of Roderick Chisholm, La Salle, Open Court, 1997.

Tarski, A., “Zur Grundlegung der Booleschen Algebra. I”, Fundamenta Mathematicae, 24 (1935): 177–198.

Tarski, A., “What is elementary geometry?”, pp. 16–29 in L. Henkin, P. Suppes, and A. Tarski (eds.), The Axiomatic Method with Special Reference to Geometry and Physics, Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam, 1959. DOI: 10.1016/S0049-237X(09)70017-5

van Benthem, J., The Logic of Time, Reidel, Dordrecht, 1983. DOI: 10.1007/978-94-010-9868-7

Zadeh, L. A., “Fuzzy sets”, Information and Control, 8 (1965): 338–353. DOI: 10.1016/S0019-9958(65)90241-X

Zadeh L. A., “Toward a unified theory of uncertainty”, pp. 3–4 in Proceedings of the IPMU the International Conference on Information Processing and Management of Uncertainty 2004, Perugia, Italy, vol. 1, Editrice Universitá La Sapienza, Rome, 2004.








Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism