The present paper investigates information systems stemming from rough set theory against the backdrop of the Ellsberg Paradox. In the early 1960s, D. Ellsberg introduced and discussed a type of informational uncertainty/ambiguity that could not be adequately modelled by the concept of measurable risk. Twenty years later, Z. Pawlak developed rough set theory—a mathematical framework aimed at modelling and managing different types of informational uncertainty that may be found in data analysis (including various forms of imprecision, vagueness, or indecision). However, as in economics in the 1960s, by uncertainty it has commonly been meant, in default of any other alternative interpretation, a sort of measurable risk. In the present study, we would like to fill this research gap and examine D. Ellsberg’s distinction between measurable risk and unmeasurable ambiguity in the context of various forms of information systems related to rough set theory. It turns out that this distinction brings an interesting reformulation of rough sets and reveals hidden facets of this theory. Specifically, we investigate two versions of ambiguity that can be derived from the Ellsberg paradox and relate them to the row and column oriented representations of information systems.

The concepts of a consequence relation and operation, though very abstract and theoretical, may be related to specific categories of information systems (i.e. mathematical frontends of data tables); as it has been demonstrated by D. Vakarelov, there exist correspondence between Pawlak information systems and Scott as well as Tarski consequence operations. This line of research goes (via representation) from abstract concepts to data. In this paper we would like to take the opposite direction: from data (via construction) to consequence relations. The main emphasis is laid here not on general categories of consequence relations (e.g. Scott or Tarski ones) but on concrete operators that can be retrieved from information systems (e.g. different examples of Scott consequence). To this end, we employ Galois connections and adjunctions (en masse called Galois mappings) and study the consequence relations that can be built via these maps. The main novelty of our research comes from the investigation of consequence relations induced by adjunctions rather than monotone Galois connections, which have been the main subject of studies so far. Surprisingly, the operations obtained from adjunctions possess a number of counter-intuitive properties, which (in turn) request some intelligible interpretations. And this is our next objective: to make sense of these consequence relations in the context of information processing.

Data tables provide the standard means of representation of qualitative or quantitative information about objects of interest. They also form a starting point for the task of information processing: an operation of passing from raw data or information to semantically processed knowledge. The fundamental issue here is the question about the meaning of data: What do entries in the data table actually tell us about objects? It entails another question: How should the meaning be further processed? The primary aim of the article is an attempt to answer these two questions. To this end we are going to employ conceptual scales from formal concept analysis, an important theory of data processing introduced by Rudolf Wille, and apply them to data tables so as to obtain multivalued information systems, which were introduced and developed within the conceptual framework of rough set theory by Zdzisław Pawlak and Ewa Orłowska. Our main idea is to regard multivalued information systems as the semantics or meaning of the original tables. This idea allows us to describe and combine classical rough set theory, dominance-based rough set approach, and formal concept analysis within the single framework of multivalued information systems, which is rich enough to cope with a number of semantical nuances that may occur during the process of data analysis.

Pattern structures were introduced by Ganter and Kuznetsov in the framework of formal concept analysis (FCA) as a mean to direct analysis of objects having complex descriptions, e.g., descriptions presented in the form of graphs instead of a set of properties. Pattern structures actually generalise /replace the original FCA representation of the initial information about objects, that is, formal contexts (which form a special type of data tables); as a consequence, pattern structures are regarded in FCA as given (in some sense a priori to the analysis) rather than built (a posteriori) from data. The main goal of this paper is twofold: firstly, we would like to export the idea of pattern structures to and consistently with the framework/methodology of rough set theory (RST); secondly, we want to derive pattern structures from simple data tables rather than to regard them as the initial information about objects. To this end we present and discuss two methods of generating non-trivial pattern structures from simple information systems/tables. Both methods are inspired by near set theory, which is a methodology theoretically close to rough set theory, but developed in the topological settings of (descriptive) nearness of sets. Interestingly, these methods bear formal connections to other ideas from RST such as generalised decisions or symbolic value grouping.

Technically put, a metaphor is a conceptual mapping between two domains, which allows one to better understand the target domain; as Lakoff and Núñes put it, the main function of a metaphor is to allow us to reason about relatively abstract domains using the inferential structure of relatively concrete domains. In the paper we would like to apply this idea of framing one domain through conceptual settings of another domain to rough set theory (RST). The main goal is to construe rough sets in terms of the following mathematical metaphor: RST is a modular set-arithmetic. That is, we would like to map/project modular arithmetic onto rough sets, and, as a consequence, to redefine the fundamental concepts/objects of RST. Specifically, we introduce new topological operators (which play a similar role as remainders in modular arithmetic), discuss their formal properties, and finally apply them to the problem of vagueness (which has been intertwined with RST since the 1980's).

Near sets were introduced by J.F. Peters in 2007 in the context and within the conceptual framework of rough sets, which were initiated by Z. Pawlak in the early 1980s. However, due to further evolution and development, near set theory has become an independent field of study. For this reason, nowadays, the relationships between near set theory and rough set theory are not easy to spot. In this short paper we would like to re-define near sets and to re-think their foundations and relationships to/bearing on rough sets. To this end we translate the basic concepts of near set theory into the framework of modal logic, which has already been successfully applied to rough sets. The concept of nearness of sets, however, was originally defined globally (that is, with respect to the whole underlying space), but modal logic is intrinsically local: the logical value of a formula is computed with respect to a single point and its neighbourhood. Our approach to near sets is local in the very same sense: we are concerned with nearness of sets seen from the perspective of a single point. Interestingly, this local perspective brings together rough set theory and near set theory, revealing their deep theoretical connections. Therefore, what we offer is a modal and algebraic "shared history" of the two theories at issue.

Data tables provide a convenient means of representation of descriptive information about objects. They serve also as standard input for data analysis tools or theories. In this paper we focus our attention upon the special class of data tables, namely multivalued information systems introduced by Z. Pawlak and E. Orłowska in the early 80s. The main idea presented in the paper is to interpret multivalued information systems as semantically processed single valued data tables. This interpretation allows us to describe classical rough set theory, dominance-based rough set theory, and formal concept analysis within the framework of multivalued information systems.

The paper studies the rough granular computing paradigm within the conceptual settings of multi-modal logic. The main idea is to express a generalised approximation space (U; I; κ), where U is the universe of objects, I is an uncertainty function, and κ is a rough inclusion function, in terms of bi nary relations, and then to consider the corresponding modal operators. The new modal structure obtained in this way is rich enough to define closure and interior operators corresponding to the classical rough approximation operators and their well-known uni-modal generalisations. In contrast to the standard modal interpretation of rough set approximations, in the new settings one can directly deal with information granules and their properties, which is crucial for granular computing paradigm. More precisely, we are provided with means of describing features of objects and information granules, as well as inclusion degrees between granules.

Building on our diverse research traditions in the study of reasoning, language and communication, the Polish School of Argumentation integrates various disciplines and institutions across Poland in which scholars are dedicated to understanding the phenomenon of the force of argument. Our primary goal is to craft a methodological programme and establish organisational infrastructure: this is the first key step in facilitating and fostering our research movement, which joins people with a common research focus, complementary skills and an enthusiasm to work together. This statement—the Manifesto—lays the foundations for the research programme of the Polish School of Argumentation.

Rough inclusion functions are mappings considered in rough set theory with which one can measure the degree of inclusion of a set (information granule) in a set (information granule) in line with rough mereology. On the other hand, similarity indices are mappings used in cluster analysis with which one can compare clusterings, and clustering methods with respect to similarity. In this article we show that a large number of similarity indices, known from the literature, can be generated by three simple rough inclusion functions, the standard rough inclusion function included.

Sprawozdanie z konferencji naukowej zorganizowanej przez Polskie Towarzystwo Matematyczne. Oddzial w Bialymstocki, Wydzial Ekonomii i Zarządzania Uniwersytetu w Bialymstoku, Wydzial Matematyki i Informatyki Uniwersytetu w Bialymstoku oraz Wydzial Informatyki Politechniki Bialostockiej, 1-4 lipca 2014 r.

Sprawozdanie z konferencji naukowej zorganizowanej przez Polskie Towarzystwo Matematyczne. Oddzial w Bialymstocki, Wydzial Ekonomii i Zarządzania Uniwersytetu w Bialymstoku, Wydzial Matematyki i Informatyki Uniwersytetu w Bialymstoku oraz Wydzial Informatyki Politechniki Bialostockiej, 1-4 lipca 2014 r.

The paper addresses the problem of concept formation (in other words, knowledge granulation) in the framework of rough set theory. The proper treatment of this problem requires taking into account both the dynamics of the universe and different scales at which concepts may be formed. These both aspects have been already separately discussed in rough set theory, with special emphasis put upon the Granular Computing paradigm as a suitable framework to deal with different scales of description. Following the example of the game Life, construed by Hawking as a simple means of explaining the process of concept formation in science, we shall describe a corresponding dynamics in Pawlak information systems.

Representation theory is a branch of mathematics whose original purpose was to represent information about abstract algebraic structures by means of methods of linear algebra usually, by linear transformations and matrices. G.-C. Rota in his famous Foundations defined a representation of a locally finite partially ordered set locally finite poset P in terms of a module over a ring $\mathbb{A}$, which can further be extended by the addition of a convolution operation to an associative $\mathbb{A}$-algebra called an incidence algebra of P. He applied this construction to solve a number of important problems in combinatorics. Our goal in this paper is to discuss the concept of an incidence algebra as a representation of a Pawlak information system. We shall analyse both incidence algebras and information systems in the context of granular computing, a paradigm which has recently received a lot of attention in computer science. We discuss therefore the concept of an incidence algebra on two levels: the level of objects which form a preordered set and the level of information granules which form a poset. Since incidence algebras induced on these two levels are Morita equivalent, we may focus our attention on the incidence algebra of information granules. We take the lattice of closed ideals of this algebra, where the maximal elements serve as a representation of information granules. The poset of maximal closed ideals obtained in this way is isomorphic to the set of information granules of the Pawlak information system equipped with a natural information order.

The paper addresses the problem of concept formation knowledge granulation in the settings of rough set theory. The original version of rough set theory implicitly accommodates a lot of well-established philosophical assumptions about concept formation as presented by A. Rand. However, as suggested by S. Hawking and L. Mlodinow, one has also to consider the dynamics of the universe of objects and different scales at which concepts may be formed. These both aspects have already been discussed separately in rough set theory. Different forms of dynamics have been addressed explicitly --especially the case of extending the universe by new objects; in contrast, different scales of description have been addressed implicitly, mainly within the Granular Computing GrC paradigm. Following the example of Life, the famous game invented by J. Conway, we describe the corresponding dynamics in Pawlak information systems using a GrC driven methodology. Having dynamics discussed, we address the problem of concept formation at zoom-out scales of description. To this end, we build Scott systems as information systems describing the universe at a coarser scale than the original scale of Pawlak systems. We regard these systems as a special type of classifications, which have already been studied in the context of rough sets by A. Skowron et al.

In this article we present three inclusion functions which characterise the nearness relation between finite sets of objects defined in line with J. F. Peters, A. Skowron, and J. Stepaniuk [26]. By means of these functions we extend the notion of nearness to the graded case where one can measure the degree to which one set is near to another one.

Representation theory is a branch of mathematics whose original purpose was to represent information about abstract algebraic structures by means of methods of linear algebra (usually, by linear transformations and matrices). Rota in his famous “Foundations” defined a representation of a locally finite partially ordered set (poset) P in terms of a module over a ring A, which can be further extended to an associative A-algebra called incidence algebra of P . He applied this construction to solve a number of important problems in combinatorics. Our goal in this paper is to apply Rota’s construction of incidence algebras to (arbitrary) Pawlak information systems. To be more precise, we analyse both incidence algebras and information systems in the context of granular computing. Therefore, starting from objects and an indiscernibility relation, we focus our attention upon information granules (i.e. equivalence classes) and a corresponding incidence algebra; finally, we discuss a lattice of closed ideals of this algebra (whose maximal elements serve as a representation of information granules). In this way we obtain a (partially ordered) set of maximal (closed) ideals which is isomorphic to the set of information granules of a Pawlak information system (also equipped with a natural information order).

In this article we continue searching for functions which might be used as measures of inclusion of information granules in information granules. Starting with a 3-valued logic having an adequate logical matrix, we show how to derive a corresponding graded inclusion function. We report on the results of examination of several best known 3-valued logics in this respect. We also give some basic properties of the inclusion functions obtained. (A preliminary version of this article was presented at the Workshop on Concurrency, Specification and Programming (CS&P'2010) held in Helenenau b. Berlin, Germany [9]. Many thanks to the anonymous referees for useful comments and to Marcin Wolski for help with word processing. The author kindly acknowledges the usage of Paul Taylor's TeX macros for drawing commutative diagrams. The research has been supported by grants N N516 368334 and N N516 077837 from Ministry of Science and Higher Education of the Republic of Poland.)

In this paper we keep on discussing satisfiability of conditions by objects when information about the situation considered, including objects of some sort and concepts comprised of them, is incomplete. Our approach to satisfiability is that of concept modelling and we have a rough granular view on the problem. Objects considered are known partially, in terms of values of attributes of Pawlak information systems. An additional knowledge (domain knowledge) is assumed to be available. We choose descriptor languages for Pawlak information systems as specification languages in which we will express conditions about objects and concepts.

In this article we consider approximation spaces where instead of rough inclusion functions, the so-called weak quasi-rough inclusion functions (weak q-RIFs) are used to measure the degree of inclusion of a set in a set. Theoretical properties of rough approximation operators based on such inclusion measures are discussed.