Pagliani / Chakraborty A Geometry of Approximation

Rough Set Theory: Logic, Algebra and Topology of Conceptual Patterns
1. Auflage 2008
ISBN: 978-1-4020-8622-9
Verlag: Springer Netherland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 704 Seiten, Web PDF

Reihe: Mathematics and Statistics

'A Geometry of Approximation' addresses Rough Set Theory, a field of interdisciplinary research first proposed by Zdzislaw Pawlak in 1982, and focuses mainly on its logic-algebraic interpretation. The theory is embedded in a broader perspective that includes logical and mathematical methodologies pertaining to the theory, as well as related epistemological issues. Any mathematical technique that is introduced in the book is preceded by logical and epistemological explanations. Intuitive justifications are also provided, insofar as possible, so that the general perspective is not lost.

Such an approach endows the present treatise with a unique character. Due to this uniqueness in the treatment of the subject, the book will be useful to researchers, graduate and pre-graduate students from various disciplines, such as computer science, mathematics and philosophy. It features an impressive number of examples supported by about 40 tables and 230 figures. The comprehensive index of concepts turns the book into a sort of encyclopaedia for researchers from a number of fields.

'A Geometry of Approximation' links many areas of academic pursuit without losing track of its focal point, Rough Sets.

Zielgruppe

Research

Autoren/Hrsg.

Pagliani, Piero

Chakraborty, Mihir

Inhaltsverzeichnis

Perception and Concepts: A Phenomenological.- Observations, Noumena and Phenomena.- Concrete and Formal Information Constructions.- Pre-Topological and Topological Approximation Operators.- Frames (Part I).- The Logico-Algebraic Theory of Rough Sets.- Logic and Rough Sets: An Overview.- Basic Logico-Algebraic Structures.- Local Validity, Grothendieck Topologies and Rough Sets.- Approximation and Algebraic Logic.- A Logico-Philosophic Excursus.- Frames (Part II).- The Modal Logic of Rough Sets.- Modality and Knowledge.- Modalities and Relations.- Modalities, Topologies and Algebras.- The Propositional Modal Logic of Rough Sets.- Frames (Part III).- Mathematical Toolkits.

Leseproben

"
If we assume that one of the two categorisations, say E, must provide the conceptual explanation of the other, E, then it should be assumed that E is charged with some founding role in our gnoseological dynamics. Although the choice of E is an ""a priori"", depending for instance on the meaning of the parameters in A, once we have decided that E has to provide the basic categorisation, E is given a privileged role. Such a role imposes a particular direction to the gnoseological dynamics so that E cannot be influenced by E.

For instance, A might be a set of decisions about members of U (for instance, decisions for investments in a group of regions) while A might be a set of evaluations related to those decisions (for example, socioeconomic evaluations about those regions). In this case we should like, for instance, to check the consistency of our decisions, with respect to A in order to compute the best consistent decisions, or we would like to know the reliability of possible extensions of a decision to other members of U. This is the usual approach in Rough Set Analysis and we call it monological approach, because of the directedness from a privileged categorisation towards the other.

But one can also think of E and E as categorisations induced by the perception of two distinct gnoseological subjects, S and S. In this case we cannot privilege one of them. On the contrary we should be interested in questions like: how do S and S interact? What are the conditions for such an interaction being possible? How can E and E be re-formed so that the subjects S and S may arrive at a consensus on a claim like ""x is X"", where X is not a sharply defined property (or, equivalently, X is not the extension of a property)? How can E and E merge together? In other words, now we do not have a privileged direction forcing one categorisation to be subordinated by the other. Therefore this reading is called dialogical approach.

This approach is a novelty and at in embryo stage within the specialized literature but we mention it in order to underline the philosophical and mathematical differences from the monological approach (some details about the so-called Dynamic spaces are shown in Part III). The monological variant is clearly an instance of the dialogical approach.

In accordance with it, one of the two Indiscernibility Spaces, say E, will act as background while E becomes the set of perceptions that must be focused in contrast to this background. To some extent the monological approach has a metaphysical flavour since in this approach E plays the role of ""object"" while E plays the role of ""subject"". In this framework, we call E an ""internal categorisation"" and E an ""external categorisation"".

Nevertheless we shall be primarily concerned with the monological approach since researches in this direction has almost reached a status of well-established scientific ""corpus"". Moreover, the monological approach is able to supply the basis for understanding the problems behind the dialogical approach."

Über Autor(innen)

Piero Pagliani: After studies in Logic, Philosophy of Science, Linguistic, Mathematics and Computer Science, Piero Pagliani graduated in Philosophy at the State University of Milan. Italy. After serving large IT companies, at present Piero Pagliani is a knowledge management consultant. A researcher in the field of algebraic logic and expert systems, he is the author of a number of scientific papers and co-author of three books on Rough Set Theory published by Physica-Verlag. Piero Pagliani had been a lecturer in Model Theory at the Faculty of Computer Science of the University of Rome "La Sapienza" and gave talks and lectures in China, France, Germany, Japan, Poland, Romania, Canada, USA and India, where he collaborates with the Calcutta Logic Circle and the Department of Pure Mathematics of the University of Calcutta.Mihir K. Chakraborty: Received Masters Degree in Pure Mathematics from the university of Calcutta and Ph. D. in Functional Analysis from the University of Kalyani, West Bengal, India. Tought in several universities. At present Professor of Pure Mathematics, University of Calcutta. Held visiting assignments in variuos institutions and universities of India and abroad. Founder of Calcutta logic circle and mebber of many academic bodies of India and abroad e.g. the editorial board of TRS. Published about 100 articles in international journals. Research interest : Logic, Rough Set Theory, Fuzzy set Theory, Many valued Logics, Non-monotonic Logics, Algebraic Logic ,Functional Analysis, Foundation and philosophy of mathematics and science.