Algebraic Semantics of Similarity-Based Bitten Rough Set Theory

Transactions on Rough Sets XV

2012

Self Cite

103

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.

“…Generalised bitten upper approximation : A ubg = A ug \ A clg -this is a direct generalisation of the bitten approximation in [39,38]. Proof.…”

Section: Tolerance Spacesmentioning

confidence: 90%

Section: Semantic Domains Meta and Object Levelsmentioning

confidence: 99%

Section: Granules: An Axiomatic Approachmentioning

confidence: 99%

See 1 more Smart Citation

Dialectics of Counting and the Mathematics of Vagueness

Transactions on Rough Sets XV

2012

Self Cite

103

“…• The view that dialectical negation cannot be reduced to classical negation (see Ioan (1998)). Indeed, in rough sets many kinds of negations and partial negations have been used in the literature (see for example, Banerjee and Chakraborty (1996); Cattaneo and Ciucci (2004); Mani (2005Mani ( , 2018a; Pagliani (1998Pagliani ( , 2000, Cattaneo and Ciucci (2018); Cattaneo, Ciucci, and Dubois (2011); Mani (2008Mani ( , 2009aMani ( , 2011; Pagliani (2016); Ślȩzak and Wasilewski (2007) and these lead to many contradictions as in (1) contradictions Pagliani (1998) which are not false but that represent topological boundaries; (2) contradictions Pagliani (2000) which are not false but lie between an absolute and local false; (3) contradictions Pagliani (2016) which lead to at least a paraconsistent and a paracomplete logic. • The view that dialectical negation is glutty negation (example Brandom (2008);…”

Section: Dialectical Negationmentioning

confidence: 99%

Dialectical Rough Sets, Parthood and Figures of Opposition-I

Transactions on Rough Sets XXI

2019

Self Cite

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 antichain 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 formalised, 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. Her methods become more geometrical and encompass parthood as a primary relation (as opposed to roughly equivalent objects) for algebraic semantics.

“…This idea might work in some contexts -the developed/invented formalisms suggest some restrictions on possible contexts. Ideals and filters have been used by the present author in algebraic semantics of general rough sets in some of her earlier papers like [9,1,10,11]. Concepts of rough ideals have also been studied by different authors in specific algebras (see for example [12,13])-these studies involve the use of rough concepts within algebras.…”

Section: Introductionmentioning

confidence: 99%

Generalized Ideals and Co-granular Rough Sets

2017

Rough Sets

Self Cite

Abstract. Lattice-theoretic ideals have been used to define and generate non granular rough approximations over general approximation spaces over the last few years by few authors. The goal of these studies, in relation based rough sets, have been to obtain nice properties comparable to those of classical rough approximations. In this research paper, these ideas are generalized in a severe way by the present author and associated semantic features are investigated by her. Granules are used in the construction of approximations in implicit ways and so a concept of co-granularity is introduced. Knowledge interpretation associable with the approaches is also investigated. This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximations in spatial mereological contexts involving actual contact relations. The antichain based semantics invented in earlier papers by the present author also applies to the contexts considered.