Choice inclusive general rough semantics

Transactions on Rough Sets XXI

2019

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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.

Section: Dialectical Negationmentioning

confidence: 99%

Dialectical Rough Sets, Parthood and Figures of Opposition-I

Transactions on Rough Sets XXI

2019

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“…Different formal definitions of granules have been used in the literature on rough sets and in granular computing. An improved version of the axiomatic theory of granules introduced in [18] is presented here. The axiomatic theory is capable of handling most contexts and is intended to permit relaxation of set-theoretic axioms at a later stage.…”

Section: Granules: An Axiomatic Approachmentioning

confidence: 99%

“…Exceptions to this trend include the Lesniewskimereology based approach [20]. Though Rough Y -systems have been introduced by the present author in ZF compatible settings [18], they can be generalised to semi-sets and variants in a natural way. This semi-set theoretical variant is work in progress.…”

Section: Introductionmentioning

confidence: 99%

“…The relevant concept of 'granules', the things that generate approximations, may also become opaque in these settings. To address these issues and for semantic compulsions, a new axiomatic theory of granules has been developed over a RYS in [18] and other recent papers by the present author. This theory has been used to formalise different principles, including the local clear discernibility principle in the same paper.…”

Section: Introductionmentioning

confidence: 99%

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Dialectics of Counting and the Mathematics of Vagueness

Transactions on Rough Sets XV

2012

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100

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.

“…We expect the reader to be aware of the different measures used in RST like those of degree of rough inclusion, rough membership (see [8] and references therein), α-cover and consistency degrees of knowledges. If these are not representable in terms of granules through term operations formed from the basic ones [7], then they are not truly functions/degrees of the rough domain. In [1], such measures are said to be noncompliant for the rough context in question and new granular measures have been proposed as replacement of the same.…”

Section: A Backgroundmentioning

confidence: 99%

Contamination-free measures and algebraic operations

2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE)

2013

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Abstract-An open concept of rough evolution and an axiomatic approach to granules was also developed in [1] by the present author. Subsequently the concepts were used in the formal framework of rough Y-systems (RYS) for developing on granular correspondences in [2]. These have since been used for a new approach towards comparison of rough algebraic semantics across different semantic domains by way of correspondences that preserve rough evolution and try to avoid contamination. In this research paper, we propose methods and semantics for handling possibly contaminated operations and structured bigness. These would also be of natural interest for relative consistency of one collection of knowledge relative other.