Abstract
New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (RST) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general RST recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical RST can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of rough naturals are also studied and variants are characterised in the penultimate section.
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References
Arnold, D., Fowler, K.: Nefarious numbers. Notices Amer. Math. Soc. 58(3) (2011), http://umn.edu/~arnold/papers/impact-factors.pdf
Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28, 211–221 (1996)
Banerjee, M., Chakraborty, M.K.: Foundations of vagueness: A categorytheoretic approach. Electronic Notes in Theoretical Computer Science 82(4), 10–19 (2003)
Banerjee, M., Chakraborty, M.K.: Algebras from rough sets – an overview. In: Pal, S.K., et al. (eds.) Rough-Neural Computing, pp. 157–184. Springer (2004)
Bunder, M.W., Banerjee, M., Chakraborty, M.K.: Some Rough Consequence Logics and their Interrelations. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets VIII. LNCS, vol. 5084, pp. 1–20. Springer, Heidelberg (2008)
Banerjee, M., Khan, M.A.: Propositional Logics from Rough Set Theory. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 1–25. Springer, Heidelberg (2007)
Banerjee, M., Yao, Y.: A Categorial Basis for Granular Computing. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 427–434. Springer, Heidelberg (2007)
Bazan, J., Skowron, A., Swiniarski, R.: Rough Sets and Vague Concept Approximation: From Sample Approximation to Adaptive Learning. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets V. LNCS, vol. 4100, pp. 39–62. Springer, Heidelberg (2006)
Bonikowski, Z., Wybraniec-Skardowska, U.: Vagueness and Roughness. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 1–13. Springer, Heidelberg (2008)
Carrara, M., Martino, E.: On the ontological commitment of mereology. Rev. Symb. Logic 2(1), 164–174 (2009)
Cattaneo, G.: Abstract approximation spaces for rough set theory. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, vol. 2, pp. 59–98. Physica, Heidelberg (1998)
Cattaneo, G., Ciucci, D.: Algebraic Structures for Rough Sets. In: Peters, J.F., Skowron, A., Dubois, D., Grzymala-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 208–252. Springer, Heidelberg (2004)
Cattaneo, G., Ciucci, D.: Lattices with Interior and Closure Operators and Abstract Approximation Spaces. In: Peters, J.F., Skowron, A., Wolski, M., Chakraborty, M.K., Wu, W.-Z. (eds.) Transactions on Rough Sets X. LNCS, vol. 5656, pp. 67–116. Springer, Heidelberg (2009)
Chajda, I., Niederle, J., Zelinka, B.: On existence conditions for compatible tolerances. Czech. Math. J. 26, 304–311 (1976)
Chakraborty, M.K., Samanta, P.: Consistency-Degree Between Knowledges. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 133–141. Springer, Heidelberg (2007)
Ciucci, D.: Approximation algebra and framework. Fundamenta Informaticae 94, 147–161 (2009)
Dubois, D., Prade, H.: Fuzzy Sets and Systems. Academic Press, New York (1980)
Dummet, M.: The Logical Basis of Metaphysics. Harvard University Press (1991)
Duntsch, I.: Rough sets and algebras of relations. In: Orlowska, E. (ed.) Incomplete Information and Rough Set Analysis, pp. 109–119. Physica, Heidelberg (1998)
Fine, K.: Vagueness, truth and logic. Synthese 30, 265–300 (1975)
Font, J.M., Jansana, R.: A General Algebraic Semantics for Sentential Logics. Association of Symbolic Logic, vol. 7 (2009)
Gomolińska, A.: On Certain Rough Inclusion Functions. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 35–55. Springer, Heidelberg (2008)
Grzegorczyk, A.: The systems of lesniewski in relation to contemporary logical research. Studia Logica 3, 77–95 (1955)
Hartonas, C.: Duality for lattice ordered algebras and for normal algebraizable logics. Studia Logica 58, 403–450 (1997)
Hodges, W.: Elementary predicate logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, pp. 1–130. Kluwer Academic (2004)
Inuiguchi, M.: Generalizations of Rough Sets and Rule Extraction. In: Peters, J.F., Skowron, A., Grzymala-Busse, J.W., Kostek, B., Swiniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 96–119. Springer, Heidelberg (2004)
Iwinski, T.B.: Rough orders and rough concepts. Bull. Pol. Acad. Sci. (Math.) (3-4), 187–192 (1988)
Jarvinen, J.: Lattice Theory for Rough Sets. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 400–498. Springer, Heidelberg (2007)
Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000)
Keefe, R., Smith, P.: Vagueness: A Reader. MIT Press, Cambridge (1999)
Keet, C.M.: A Formal Theory of Granules - Phd Thesis. Ph.D. thesis. Fac. of Comp. Sci., Free University of Bozen (2008)
Khan, M.A., Banerjee, M.: Formal reasoning with rough sets in multiple-source approximation spaces. Internat. J. Approximate Reasoning 49, 466–477 (2008)
Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets – a tutorial. In: Pal, S.K., Skowron, A. (eds.) Rough Fuzzy Hybridization, pp. 3–98. Springer (1999)
Konikowska, B.: A logic for reasoning with similarity. In: Orlowska, E. (ed.) Incomplete Information and Rough Set Analysis, pp. 461–490. Physica Verlag, Heidelberg (1998)
Lewis, D.K.: Parts of Classes. Basil Blackwell, Oxford (1991)
Li, T.-J.: Rough Approximation Operators in Covering Approximation Spaces. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Slowinski, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 174–182. Springer, Heidelberg (2006)
Lin, T.Y.: Neighbourhood systems- applications to qualitative fuzzy and rough sets. In: Wang, P.P., et al. (eds.) Advances in Machine Intelligence and Soft Computing, pp. 132–155. Duke University, Durham (1997)
Lin, T.Y.: Granular computing-1: The concept of granulation and its formal model. Int. J. Granular Computing, Rough Sets and Int. Systems 1(1), 21–42 (2009)
Mani, A.: V-perspectives, pseudo-natural number systems and partial orders. Glasnik Math. 37(57), 245–257 (2002)
Mani, A.: Super rough semantics. Fundamenta Informaticae 65(3), 249–261 (2005)
Mani, A.: Esoteric Rough Set Theory: Algebraic Semantics of a Generalized VPRS and VPFRS. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets VIII. LNCS, vol. 5084, pp. 175–223. Springer, Heidelberg (2008)
Mani, A.: Algebraic semantics of similarity-based bitten rough set theory. Fundamenta Informaticae 97(1-2), 177–197 (2009)
Mani, A.: Integrated dialectical logics for relativised general rough set theory. In: Internat. Conf. on Rough Sets, Fuzzy Sets and Soft Computing, Agartala, India, p. 6 (2009), http://arxiv.org/abs/0909.4876 (refereed)
Mani, A.: Meaning, choice and similarity based rough set theory. In: Internat. Conf. Logic and Appl., Chennai, pp. 1–12 (January 2009), http://arxiv.org/abs/0905.1352 (refereed)
Mani, A.: Towards an Algebraic Approach for Cover Based Rough Semantics and Combinations of Approximation Spaces. In: Sakai, H., Chakraborty, M.K., Hassanien, A.E., Slezak, D., Zhu, W. (eds.) RSFDGrC 2009. LNCS, vol. 5908, pp. 77–84. Springer, Heidelberg (2009)
Mani, A.: Choice inclusive general rough semantics. Information Sciences 181(6), 1097–1115 (2011), http://dx.doi.org/10.1016/j.ins.2010.11.016
Mani, A.: Dialectics of counting and measures of rough set theory. In: IEEE Proceedings of NCESCT 2011, Pune, Febuary 1-3, p. 17 (2011) Arxiv:1102.2558, http://arxiv.org/abs/1102.2558
Mani, A.: Towards logics of some rough perspectives of knowledge. In: Suraj, Z., Skowron, A. (eds.) Intelligent Systems Reference Library dedicated to the memory of Prof. Pawlak, p. 25 (2011) (in Press)
Mundici, D.: Generalization of abstract model theory. Fundamenta Math. 124, 1–25 (1984)
Novotny, M., Pawlak, Z.: Characterization of rough top and bottom equalities. Bull. Pol. Acad. Sci. (Math.) 33(1-2), 91–97 (1985)
Orlowska, E.: Kripke semantics for knowledge representation logics. Studia Logica XLIX, 255–272 (1990)
Orłowska, E., Rewitzky, I.: Discrete Duality and Its Applications to Reasoning with Incomplete Information. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 51–56. Springer, Heidelberg (2007)
Pagliani, P.: Pretopologies and dynamic spaces. Fundamenta Informaticae 59(2-3), 221–239 (2004)
Pagliani, P., Chakraborty, M.K.: Formal Topology and Information Systems. In: Peters, J.F., Skowron, A., Düntsch, I., Grzymała-Busse, J.W., Orłowska, E., Polkowski, L. (eds.) Transactions on Rough Sets VI. LNCS, vol. 4374, pp. 253–297. Springer, Heidelberg (2007)
Pagliani, P., Chakraborty, M.K.: A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer, Berlin (2008)
Pawlak, Z.: Rough sets. Internat. J. of Computing and Information Sciences 18, 341–356 (1982)
Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dodrecht (1991)
Pawlak, Z.: An inquiry into vagueness and uncertainty. PAS Institute of Computer Science 29/94 (1994)
Pawlak, Z.: Vagueness - A Rough Set View. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. LNCS, vol. 1261, pp. 106–117. Springer, Heidelberg (1997)
Pawlak, Z.: Some Issues on Rough Sets. In: Peters, J.F., Skowron, A., Grzymala-Busse, J.W., Kostek, B., Swiniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 1–58. Springer, Heidelberg (2004)
Perry, B., Dockett, S.: Young children’s access to powerful mathematical ideas. In: English, L., et al. (eds.) Handbook of International Research in Mathematics Education, pp. 81–112. LEA (2002)
Polkowski, L.: Rough Sets: Mathematical Foundations. Physica-Verlag, Heidelberg (2002)
Polkowski, L.: Rough Mereology as a Link between Rough and Fuzzy Set Theories. A Survey. In: Peters, J.F., Skowron, A., Dubois, D., Grzymala-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 253–277. Springer, Heidelberg (2004)
Polkowski, L.: Rough Neural Computation Model Based on Rough Mereology, pp. 85–108. Springer (2004)
Polkowski, L., Semeniuk-Polkowska, M.: Reasoning about Concepts by Rough Mereological Logics. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 205–212. Springer, Heidelberg (2008)
Polkowski, L., Skowron, A.: Rough mereology: A new paradigm for approximate reasoning. Internat. J. Appr. Reasoning 15(4), 333–365 (1996)
Pomykala, J.A.: Approximation, similarity and rough constructions. Tech. Rep. CT-93-07, ILLC, Univ. of Amsterdam (1993)
Qian, Y.H., Liang, J.Y.: Rough set method based on multi-granulations. In: Proc. 5th IEEE Conf. Cognitive Informatics, vol. 1, pp. 297–304 (2006)
Qian, Y.H., Liang, J.Y., Dang, C.Y.: Incomplete mutigranulation rough set. IEEE Transactions on Systems, Man and Cybernetics, Part A 20, 420–430 (2010)
Qian, Y., Liang, J.Y., Yao, Y.Y., Dang, C.Y.: Mgrs: A multi granulation rough set. Information Science 180, 949–970 (2010)
RamÃk, J., Vlach, M.: A Non-controversial Definition of Fuzzy Sets. In: Peters, J.F., Skowron, A., Dubois, D., GrzymaÅ‚a-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 201–207. Springer, Heidelberg (2004)
Mieszkowicz-Rolka, A., Rolka, L.: Variable Precision Fuzzy Rough Sets. In: Peters, J.F., Skowron, A., Grzymala-Busse, J.W., Kostek, B., Swiniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 144–160. Springer, Heidelberg (2004)
Russell, B.: Vagueness. The Australasian Journal of Psychology and Philosophy 1, 84–92 (1923)
Samanta, P., Chakraborty, M.K.: Covering Based Approaches to Rough Sets and Implication Lattices. In: Sakai, H., Chakraborty, M.K., Hassanien, A.E., Slezak, D., Zhu, W. (eds.) RSFDGrC 2009. LNCS, vol. 5908, pp. 127–134. Springer, Heidelberg (2009)
Satyanarayana, M.: Positively Ordered Semigroups. Lecture Notes in Pure and Applied Mathematics, vol. 42. Marcel Dekker Inc. (1979)
Sen, D., Pal, S.K.: Generalized rough sets, entropy and image ambiguity measures. IEEE Trans. -PartB: Cybernetics 39(1), 117–128 (2008)
Shapiro, S.: Vagueness in Context. Oxford University Press, Oxford (2006)
Simon, P.: Parts - A Study in Ontology. Oxford Univ. Press (1987)
Skowron, A.: Rough sets and vague concepts. Fund. Inform. 64(1-4), 417–431 (2005)
Skowron, A., Peters, J.: Rough-granular computing. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Granular Computing, pp. 225–328. John Wiley & Sons, Ltd., Chichester (2008)
Skowron, A., Stepaniuk, O.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)
Ślęzak, D., Wasilewski, P.: Granular Sets – Foundations and Case Study of Tolerance Spaces. In: An, A., Stefanowski, J., Ramanna, S., Butz, C.J., Pedrycz, W., Wang, G. (eds.) RSFDGrC 2007. LNCS (LNAI), vol. 4482, pp. 435–442. Springer, Heidelberg (2007)
Slowinski, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Trans. on Knowledge and Data Engg. 12, 331–336 (2000)
Takeuti, G.: Proof Theory, 2nd edn. North Holland, Amsterdam (1987)
Urbaniak, R.: Lesniewski’s Systems of Logic and Mereology; History and Re-evaluation. Ph.D. thesis, Department of Philosophy, Univ. of Calgary (2008)
Varzi, A.: Parts, wholes and part-whole relations: The prospects of mereotopology. Data and Knowledge Engineering 20, 259–286 (1996)
Vopenka, P.: Mathematics in the Alternative Set Theory. Teubner-Verlag (1979)
Vopenka, P., Hajek, P.: The Theory of Semisets. Academia, Prague (1972)
Wasilewski, P., Slezak, D.: Foundations of rough sets from vagueness perspective. In: Hassanien, A., et al. (eds.) Rough Computing: Theories, Technologies and Applications. Information Science Reference, pp. 1–37. IGI, Global (2008)
Weiner, J.: Science and semantics: The case of vagueness and supervaluation. Pacific Philosophical Quarterly 88(3), 355–374 (2007)
Yao, Y.: Relational interpretation of neighbourhood operators and rough set approximation operators. Information Sciences, 239–259 (1998)
Yao, Y.: A Partition Model of Granular Computing. In: Peters, J.F., Skowron, A., Grzymala-Busse, J.W., Kostek, B., Swiniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets I. LNCS, vol. 3100, pp. 232–253. Springer, Heidelberg (2004)
Yao, Y.: Semantics of Fuzzy Sets in Rough Set Theory. In: Peters, J.F., Skowron, A., Dubois, D., Grzymala-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 297–318. Springer, Heidelberg (2004)
Yao, Y.: The Art of Granular Computing. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 101–112. Springer, Heidelberg (2007)
Zakowski, W.: Approximation in the space (u, π). Demonstration Math. XVI, 761–769 (1983)
Ma, J.-M., Zhang, W.-X., Wu, W.-Z., Li, T.-J.: Granular Computing Based on a Generalized Approximation Space. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślęzak, D., et al. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 93–100. Springer, Heidelberg (2007)
Zhu, W.: Topological approaches to covering rough sets. Information Sciences 177, 1499–1508 (2007)
Zhu, W.: Relationship between general rough set based on binary relation and covering. Information Sciences 179, 210–225 (2009)
Ziarko, W.: Variable precision rough set model. J. of Computer and System Sciences 46, 39–59 (1993)
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Mani, A. (2012). Dialectics of Counting and the Mathematics of Vagueness. In: Peters, J.F., Skowron, A. (eds) Transactions on Rough Sets XV. Lecture Notes in Computer Science, vol 7255. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31903-7_4
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