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Dialectical Rough Sets, Parthood and Figures of Opposition-I

Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10810)

Abstract

In one perspective, the main theme of this research revolves around the inverse problem in the context of general rough sets that concerns the existence of rough basis for given approximations in a context. Granular operator spaces and variants were recently introduced by the present author as an optimal framework for anti-chain based algebraic semantics of general rough sets and the inverse problem. In the framework, various sub-types of crisp and non-crisp objects are identifiable that may be missed in more restrictive formalism. This is also because in the latter cases concepts of complementation and negation are taken for granted - while in reality they have a complicated dialectical basis. This motivates a general approach to dialectical rough sets building on previous work of the present author and figures of opposition. In this paper dialectical rough logics are invented from a semantic perspective, a concept of dialectical predicates is formalized, connection with dialetheias and glutty negation are established, parthood analyzed and studied from the viewpoint of classical and dialectical figures of opposition by the present author. The proposed method become more geometrical and encompass parthood as a primary relation (as opposed to roughly equivalent objects) for algebraic semantics.

Keywords

Rough objects Dialectical rough semantics Granular operator spaces Rough mereology Polytopes of dialectics Antichains Dialectical rough counting Axiomatic approach to granules Constructive algebraic semantics Figures of opposition Unified semantics 

Notes

Acknowledgement

The present author would like to thank the referees for detailed remarks that led to improvement (especially of the readability) of the research paper.

References

  1. 1.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning About Data. Kluwer Academic Publishers, Dodrecht (1991)zbMATHGoogle Scholar
  2. 2.
    Mani, A.: Algebraic methods for granular rough sets. In: Mani, A., Düntsch, I., Cattaneo, G. (eds.) Algebraic Methods in General Rough Sets. Trends in Mathematics, pp. 157–336. Birkhäuser, Basel (2018).  https://doi.org/10.1007/978-3-030-01162-8_3Google Scholar
  3. 3.
    Mani, A.: Antichain based semantics for rough sets. In: Ciucci, D., Wang, G., Mitra, S., Wu, W.-Z. (eds.) RSKT 2015. LNCS (LNAI), vol. 9436, pp. 335–346. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-25754-9_30Google Scholar
  4. 4.
    Mani, A.: Knowledge and consequence in AC semantics for general rough sets. In: Wang, G., Skowron, A., Yao, Y.Y., Ślȩzak, D., Polkowski, L. (eds.) Thriving Rough Sets. SCI, vol. 708, pp. 237–268. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-54966-8_12Google Scholar
  5. 5.
    Mani, A.: Pure rough mereology and counting. In: WIECON 2016, pp. 1–8. IEEXPlore (2016)Google Scholar
  6. 6.
    Mani, A.: On deductive systems of AC semantics for rough sets. ArXiv. Math (arXiv:1610.02634v1), pp. 1–12, October 2016
  7. 7.
    Mani, A.: Dialectics of counting and the mathematics of vagueness. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XV. LNCS, vol. 7255, pp. 122–180. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31903-7_4Google Scholar
  8. 8.
    Polkowski, L.: Approximate Reasoning by Parts. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22279-5Google Scholar
  9. 9.
    Mani, A.: Super rough semantics. Fundamenta Informaticae 65(3), 249–261 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mani, A.: Algebraic representation, duality and beyond. In: Mani, A., Düntsch, I., Cattaneo, G. (eds.) Algebraic Methods in General Rough Sets. Trends in Mathematics, pp. 459–552. Birkhäuser, Basel (2018).  https://doi.org/10.1007/978-3-030-01162-8_6Google Scholar
  11. 11.
    Banerjee, M., Chakraborty, M.K.: Rough sets through algebraic logic. Fundamenta Informaticae 28, 211–221 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ioan, P.: Logic and Dialectics. Al. I. Cuza Universities Press (1998)Google Scholar
  13. 13.
    Schang, F.: Opposites and oppositions around and beyond the square of opposition. In: Beziau, J.Y., Jacquette, D., et al. (eds.) Around and Beyond the Square of Opposition. Studies in Universal Logic, vol. I, pp. 147–174. Birkhauser, Basel (2012)zbMATHGoogle Scholar
  14. 14.
    Ciucci, D., Dubois, D., Prade, H.: Oppositions in rough set theory. In: Li, T., et al. (eds.) RSKT 2012. LNCS (LNAI), vol. 7414, pp. 504–513. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31900-6_62Google Scholar
  15. 15.
    Mani, A.: Integrated dialectical logics for relativised general rough set theory. In: International Conference on Rough Sets, Fuzzy Sets and Soft Computing, Agartala, India, 6 p. (Refereed) (2009). http://arxiv.org/abs/0909.4876
  16. 16.
    Mani, A.: Ontology, rough Y-systems and dependence. Int. J. Comput. Sci. Appl. 11(2), 114–136 (2014). Special Issue of IJCSA on Computational IntelligenceMathSciNetGoogle Scholar
  17. 17.
    Mani, A.: Algebraic semantics of proto-transitive rough sets. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XX. LNCS, vol. 10020, pp. 51–108. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53611-7_3Google Scholar
  18. 18.
    Chakraborty, M.K.: On some issues in the foundation of rough sets: the problem of definition. Fundamenta Informaticae 148, 123–132 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Grätzer, G.: General Lattice Theory. Birkhauser, New York (1998)zbMATHGoogle Scholar
  20. 20.
    Koh, K.: On the lattice of maximum-sized antichains of a finite poset. Algebra Universalis 17, 73–86 (1983)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mani, A.: Algebraic Semantics of Proto-Transitive Rough Sets, 1st edn. arXiv:1410.0572, July 2014
  22. 22.
    Mani, A.: Approximation dialectics of proto-transitive rough sets. In: Chakraborty, M.K., Skowron, A., Kar, S. (eds.) Facets of Uncertainties and Applications. Springer Proceedings in Math and Statistics, vol. 125, pp. 99–109. Springer, New Delhi (2015).  https://doi.org/10.1007/978-81-322-2301-6_8Google Scholar
  23. 23.
    Ciucci, D.: Approximation algebra and framework. Fundamenta Informaticae 94, 147–161 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    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).  https://doi.org/10.1007/978-3-642-03281-3_3Google Scholar
  25. 25.
    Yao, Y.Y.: Relational interpretation of neighbourhood operators and rough set approximation operators. Inf. Sci. 111, 239–259 (1998)zbMATHGoogle Scholar
  26. 26.
    Iwinski, T.B.: Rough orders and rough concepts. Bull. Pol. Acad. Sci. (Math.) 3–4, 187–192 (1988)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Cattaneo, G.: Algebraic methods for rough approximation spaces by lattice interior-closure operations. In: Mani, A., Düntsch, I., Cattaneo, G. (eds.) Algebraic Methods in General Rough Sets. Trends in Mathematics, pp. 13–152. Birkhäuser, Basel (2018).  https://doi.org/10.1007/978-3-030-01162-8_2Google Scholar
  28. 28.
    Pagliani, P., Chakraborty, M.: A Geometry of Approximation: Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns. Springer, Berlin (2008).  https://doi.org/10.1007/978-1-4020-8622-9zbMATHGoogle Scholar
  29. 29.
    Polkowski, L., Skowron, A.: Rough mereology: a new paradigm for approximate reasoning. Int. J. Approx. Reason. 15(4), 333–365 (1996)MathSciNetzbMATHGoogle Scholar
  30. 30.
    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).  https://doi.org/10.1007/978-3-540-85064-9_9Google Scholar
  31. 31.
    Mani, A.: Choice inclusive general rough semantics. Inf. Sci. 181(6), 1097–1115 (2011)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mani, A.: Axiomatic granular approach to knowledge correspondences. In: Li, T., et al. (eds.) RSKT 2012. LNCS (LNAI), vol. 7414, pp. 482–487. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31900-6_59Google Scholar
  33. 33.
    Mani, A.: Granular Foundations of the Mathematics of Vagueness, Algebraic Semantics and Knowledge Interpretation. University of Calcutta (2016)Google Scholar
  34. 34.
    Moore, E.F., Shannon, C.E.: Reliable circuits using less reliable relays-I. II. Bell Syst. Tech. J. 191–208, 281–297 (1956)zbMATHGoogle Scholar
  35. 35.
    Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(379–423), 623–656 (1948)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lin, T.Y.: Granular computing-1: the concept of granulation and its formal model. Int. J. Granular Comput. Rough Sets Int. Syst. 1(1), 21–42 (2009)Google Scholar
  37. 37.
    Mani, A.: Approximations from anywhere and general rough sets. In: Polkowski, L., et al. (eds.) IJCRS 2017. LNCS (LNAI), vol. 10314, pp. 3–22. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-60840-2_1Google Scholar
  38. 38.
    Ciucci, D.: Back to the beginnings: Pawlak’s definitions of the terms information. In: Wang, G., Skowron, A., Yao, Y., Ślȩzak, D., Polkowski, L. (eds.) Thriving Rough Sets. Studies in Computational Intelligence, vol. 708. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-54966-8_11Google Scholar
  39. 39.
    Cattaneo, G., Ciucci, D.: Algebraic methods for orthopairs and induced rough approximation spaces. In: Mani, A., Düntsch, I., Cattaneo, G. (eds.) Algebraic Methods in General Rough Sets. Trends in Mathematics, pp. 553–640. Birkhäuser, Basel (2018).  https://doi.org/10.1007/978-3-030-01162-8_7Google Scholar
  40. 40.
    Yao, Y.Y., Yao, B.: Covering based rough set approximations. Inf. Sci. 200, 91–107 (2012)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Düntsch, I., Gediga, G.: Rough Set Data Analysis: A Road to Non-invasive Knowledge Discovery. Methodos Publishers, Bangor (2000)Google Scholar
  42. 42.
    Pagliani, P.: Covering rough sets and formal topology – a uniform approach through intensional and extensional constructors. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets XX. LNCS, vol. 10020, pp. 109–145. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53611-7_4zbMATHGoogle Scholar
  43. 43.
    Sambin, G.: Intuitionistic formal spaces - a first communication. In: Skordev, D. (ed.) Mathematical Logic and Its Applications, pp. 187–204. Plenum Press, New York (1987)Google Scholar
  44. 44.
    Sambin, G., Gebellato, S.: A preview of the basic picture: a new perspective on formal topology. In: Altenkirch, T., Reus, B., Naraschewski, W. (eds.) TYPES 1998. LNCS, vol. 1657, pp. 194–208. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48167-2_14Google Scholar
  45. 45.
    da Costa, N., Wolf, R.G.: Studies in paraconsistent logic-1: the dialectical principle of the unity of opposites. Philosophia - Philos. Q. Israel 15, 497–510 (1974)Google Scholar
  46. 46.
    Pagliani, P.: Rough set theory and logic-algebraic structures. In: Orłowska, E. (ed.) Incomplete Information: Rough Set Analysis, pp. 109–190. Physica Verlag, Heidelberg (1998).  https://doi.org/10.1007/978-3-7908-1888-8_6Google Scholar
  47. 47.
    Pagliani, P.: Local classical behaviours in three-valued logics and connected systems. Part 1. J. Multiple Valued Log. 5, 327–347 (2000)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Cattaneo, G., Ciucci, D.: Algebras for rough sets and fuzzy logics. In: Peters, J.F., Andrzej, S., Grzymała-Busse, J.W., Kostek, B., Świniarski, R.W., Szczuka, M.S. (eds.) Transactions on Rough Sets. LNCS, vol. 3100, pp. 208–252. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  49. 49.
    Mani, A.: Algebraic semantics of similarity-based bitten rough set theory. Fundamenta Informaticae 97(1–2), 177–197 (2009)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Ślȩzak, 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).  https://doi.org/10.1007/978-3-540-72530-5_52Google Scholar
  51. 51.
    Cattaneo, G., Ciucci, D., Dubois, D.: Algebraic models of deviant modal operators based on De Morgan and Kleene lattices. Inf. Sci. 181, 4075–4100 (2011)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Ficara, E.: Hegel’s Glutty negation. Hist. Philos. Log. 36, 1–10 (2014)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Brandom, R.: Between Saying and Doing. Oxford University Press, Oxford (2008)Google Scholar
  54. 54.
    McGill, V.P., Parry, W.T.: The unity of opposites - a dialectical principle. Sci. Soc. 12, 418–444 (1948)Google Scholar
  55. 55.
    Mani, A.: Towards formal dialectical logics. Technical report (1999)Google Scholar
  56. 56.
    Priest, G.: Doubt Truth to be a Liar. Oxford University Press, Oxford (2008)zbMATHGoogle Scholar
  57. 57.
    Gabbay, D.: Oxford Logic Guides, vol. 1, 1st edn, p. 33. Clarendon Press, Oxford (1996)Google Scholar
  58. 58.
    Batens, D.: Narrowing Down Suspicion in Inconsistent Premise Set (2006, preprint)Google Scholar
  59. 59.
    Batens, D.: Against global paraconsistency. Stud. Soviet Thought 39, 209–229 (1990)Google Scholar
  60. 60.
    Apostol, L.: Logique et Dialectique. Gent (1979)Google Scholar
  61. 61.
    Hoffmann, W.C.: A formal model for dialectical psychology. Int. Log. Rev. 40–67 (1986)Google Scholar
  62. 62.
    Gorren, J.: Theorie Analytique de la Dialectique. South West Philos. Stud. 6, 41–47 (1981)Google Scholar
  63. 63.
    Marx, K., Engels, F.: Marx and Engels: Collected Works, vol. 24. Progress Publishers, Delhi (1989)Google Scholar
  64. 64.
    Batens, D.: Dynamic dialectical logics. In: Paraconsistent Logic - Essays. Philosophia Verlag, Munich (1989)Google Scholar
  65. 65.
    Tzouvaras, A.: Periodicity of negation. Notre Dame J. Form. Log. 42(2), 88–99 (2001)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Priest, G.: What not? A defence of a dialetheic theory of negation. In: Gabbay, D. (ed.) What is Negation?, pp. 101–120. Kluwer, Dordrecht (1999)zbMATHGoogle Scholar
  67. 67.
    Priest, G.: To be and not to be: dialectical tense logic. Studia Logica 41(2/3), 249–268 (1984)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Zeleny, J.: Paraconsistency and dialectical consistency. Log. Point View 1, 35–51 (1994)zbMATHGoogle Scholar
  69. 69.
    Priest, G.: Logicians setting together contradictories - a perspective on relevance, paraconsistency and dialetheism. In: Jacquette, D. (ed.) Blackwell Handbook to Philosophical Logic. Blackwell, Hoboken (2007)Google Scholar
  70. 70.
    Grim, P.: What is contradiction? In: Priest, G. (ed.) The Law of Non-contradiction. Oxford Universities Press, Oxford (2007)zbMATHGoogle Scholar
  71. 71.
    Woods, J.: Dialectical Considerations on the Logic of Contradiction: Part I, II (2004, preprint)Google Scholar
  72. 72.
    Priest, G.: Dialectic and dialetheic. Sci. Soc. 53(4), 388–415 (1990)Google Scholar
  73. 73.
    Priest, G.: Contradictory concepts. In: Weber, E., Wouters, D., Meheus, J. (eds.) Logic, Reasoning and Rationality. Interdisciplinary Perspectives from The Humanities and Social Sciences, vol. 5, pp. 197–216. Springer, Dordrecht (2014).  https://doi.org/10.1007/978-94-017-9011-6_10Google Scholar
  74. 74.
    Swaminathan, M., Rawal, V.: A Study of Agrarian Relations. Tulika Books, New Delhi (2015)Google Scholar
  75. 75.
    Zimmerman, B. (ed.): Lesbian Histories and Cultures: An Encyclopedia. Garland Reference Library of the Social Sciences, vol. 1008. Garland Publishers, New York (2000)Google Scholar
  76. 76.
    Saha, A., Sen, J., Chakraborty, M.K.: Algebraic structures in the vicinity of pre-rough algebra and their logics II. Inf. Sci. 333, 44–60 (2015)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Mani, A.: Contamination-free measures and algebraic operations. In: 2013 IEEE International Conference on Fuzzy Systems (FUZZ), pp. 1–8. IEEE (2013)Google Scholar
  78. 78.
    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).  https://doi.org/10.1007/978-3-540-85064-9_1Google Scholar
  79. 79.
    Pawlak, Z.: Rough logic. Bull. Pol. Acad. Sci. (Tech.) 35, 253–258 (1987)MathSciNetzbMATHGoogle Scholar
  80. 80.
    Hyde, D., Colyvan, M.: Paraconsistent Vagueness-why not? Aust. J. Log. 6, 207–225 (2008)MathSciNetzbMATHGoogle Scholar
  81. 81.
    Avron, A., Konikowska, B.: Rough sets and 3-valued logics. Studia Logica 90, 69–92 (2008)MathSciNetzbMATHGoogle Scholar
  82. 82.
    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 Professor Pawlak ISRL, vol. 43, pp. 419–444. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-30341-8_22Google Scholar
  83. 83.
    Rasiowa, H.: An Algebraic Approach to Nonclassical Logics. Studies in Logic, vol. 78. North Holland, Warsaw (1974)zbMATHGoogle Scholar
  84. 84.
    Ciucci, D.: Orthopairs and granular computing. Granular Comput. 1(3), 159–170 (2016)Google Scholar
  85. 85.
    Belnap, N.D.: A useful four-valued logic. In: Dunn, J.M., Epstein, G. (eds.) Modern Uses of Multiple-valued Logic. Episteme, vol. 2, pp. 5–37. Springer, Dordrecht (1977).  https://doi.org/10.1007/978-94-010-1161-7_2Google Scholar
  86. 86.
    Moretti, A.: Why the logical hexagon? Logica Univers 6, 69–107 (2012)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of Calcutta, International Rough Set SocietyKolkata (Calcutta)India

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