Some classes of topological quasi-Boolean algebras, including algebraic structures related with r... more Some classes of topological quasi-Boolean algebras, including algebraic structures related with rough sets, are enriched with residuated and adjoint pairs. The strong finite model property for these classes of algebraic structures is established. The decidability of equational theories of these classes of algebras is derived from the finite model property.
ABSTRACT The article analyses prevalent definitions of rough sets from the foundational and mathe... more ABSTRACT The article analyses prevalent definitions of rough sets from the foundational and mathematical perspectives. In particular, the issue of language dependency in the definitions, and implications of the definitions on the issue of vagueness are discussed in detail.
Rough set theory has seen nearly two decades of research on both foundations and on diverse appli... more Rough set theory has seen nearly two decades of research on both foundations and on diverse applications. A substantial part of the work done on the theory has been devoted to the study of its algebraic aspects. 'Rough algebras' now abound, and have been shown to be instances of various algebraic structures, both well-established and relatively new, e.g., quasi-Boolean, Stone, double Stone, Nelson, Lukasiewicz algebras, on the one hand, and topological quasi-Boolean, prerough and rough algebras, on the other. More interestingly and importantly, some of these latter algebras find a new dimension (interpretation) through representations as rough structures. An attempt is made here to present the various relationships and to discuss the representation results.
ABSTRACT A dialogue is an 'activity' by a pair of agents to arrive at some kind o... more ABSTRACT A dialogue is an 'activity' by a pair of agents to arrive at some kind of understanding over a concept/belief/piece of information etc. represented by a subset (the extension) in some universe of discourse. The universe is partitioned into two different sets of granules (equivalence classes) representing the perceptions of the agents. So, there are two approximation spaces at the beginning. A third approximation space arises out of superimposition of the two partitions. A dialogue is a finite process of gradual enhancement of the two base subsets of the agents, in their 'common' approximation space. Through this process, various kinds of overlap may emerge between the two final subsets. A first introduction of the idea of a dialogue in rough context was made in [6]. This paper further develops the notion and focusses upon the study of the above-mentioned overlaps in a systematic manner. Given two sets A and B in an approximation space, there are nine possible inclusion relations among the sets lo(A),A, up(A), lo(B),B and up(B) where lo and up denote the lower and upper approximation operators respectively. There are five resulting equivalence classes and the quotient set forms a lattice by implication ordering. That is, of the nine relations, only five are independent and they form an implication or entailment lattice. Starting with this basic lattice other implication lattices are formed. Relationship of these lattices with the various overlap conditions between the final pair of sets arrived at after a dialogue is studied. Finally, examples are given, one of which relates dialogues in rough context with rough belief revision [3] - in a line similar to the approach of [5].
In this paper the relationship between rough set theory and modal logic has been discussed. Pawla... more In this paper the relationship between rough set theory and modal logic has been discussed. Pawlakian rough set theory has obvious connection with modal logic system \(S_5\). With the introduction of various other lower and upper approximation operators, other modal systems come into picture. Besides, the possibility of new modal systems also crop up. Some of these issues are focused here.
This chapter comprises the basic concepts of the theory. It includes the syntactic, semantic and ... more This chapter comprises the basic concepts of the theory. It includes the syntactic, semantic and axiomatic notions of graded consequence, and their interrelations.
In this paper we study some basic algebraic structures of rough algebras. We proved that the clas... more In this paper we study some basic algebraic structures of rough algebras. We proved that the class of topological quasi-Boolean algebra 5s (tqBa5s) has the finite embeddability property (FEP). Further we also extend this result to some related classes of algebras.
Navya-Nyāya, “The New Reasoning”, is a formal philosophical logic developed in India from the 11t... more Navya-Nyāya, “The New Reasoning”, is a formal philosophical logic developed in India from the 11th to the 17th centuries CE, and which builds on the older traditions of Nyāya and Vaiśeṣika. Not surprisingly, Navya-Nyāya is fundamentally different from classical Western logic and from the meanings ascribed to traditional logical diagrams. For instance, although it is not entirely correct to describe Navya-Nyāya as extensional or intensional, it has an intensional flavour: abstractions are built up from concrete individuals of which we know only their possession, or not, of certain properties. In this paper we look at the implications of these semantics for the use of logical diagrams in Navya-Nyāya. We survey the use of diagrams in modern studies of Navya-Nyāya, notable examples having been produced by Wada, Das and Ganeri. We use notions of well-matchedness, iconicity and Cheng’s recent framework to analyse the effectiveness of the notations in the context of their intended purposes.
Some classes of topological quasi-Boolean algebras, including algebraic structures related with r... more Some classes of topological quasi-Boolean algebras, including algebraic structures related with rough sets, are enriched with residuated and adjoint pairs. The strong finite model property for these classes of algebraic structures is established. The decidability of equational theories of these classes of algebras is derived from the finite model property.
ABSTRACT The article analyses prevalent definitions of rough sets from the foundational and mathe... more ABSTRACT The article analyses prevalent definitions of rough sets from the foundational and mathematical perspectives. In particular, the issue of language dependency in the definitions, and implications of the definitions on the issue of vagueness are discussed in detail.
Rough set theory has seen nearly two decades of research on both foundations and on diverse appli... more Rough set theory has seen nearly two decades of research on both foundations and on diverse applications. A substantial part of the work done on the theory has been devoted to the study of its algebraic aspects. 'Rough algebras' now abound, and have been shown to be instances of various algebraic structures, both well-established and relatively new, e.g., quasi-Boolean, Stone, double Stone, Nelson, Lukasiewicz algebras, on the one hand, and topological quasi-Boolean, prerough and rough algebras, on the other. More interestingly and importantly, some of these latter algebras find a new dimension (interpretation) through representations as rough structures. An attempt is made here to present the various relationships and to discuss the representation results.
ABSTRACT A dialogue is an 'activity' by a pair of agents to arrive at some kind o... more ABSTRACT A dialogue is an 'activity' by a pair of agents to arrive at some kind of understanding over a concept/belief/piece of information etc. represented by a subset (the extension) in some universe of discourse. The universe is partitioned into two different sets of granules (equivalence classes) representing the perceptions of the agents. So, there are two approximation spaces at the beginning. A third approximation space arises out of superimposition of the two partitions. A dialogue is a finite process of gradual enhancement of the two base subsets of the agents, in their 'common' approximation space. Through this process, various kinds of overlap may emerge between the two final subsets. A first introduction of the idea of a dialogue in rough context was made in [6]. This paper further develops the notion and focusses upon the study of the above-mentioned overlaps in a systematic manner. Given two sets A and B in an approximation space, there are nine possible inclusion relations among the sets lo(A),A, up(A), lo(B),B and up(B) where lo and up denote the lower and upper approximation operators respectively. There are five resulting equivalence classes and the quotient set forms a lattice by implication ordering. That is, of the nine relations, only five are independent and they form an implication or entailment lattice. Starting with this basic lattice other implication lattices are formed. Relationship of these lattices with the various overlap conditions between the final pair of sets arrived at after a dialogue is studied. Finally, examples are given, one of which relates dialogues in rough context with rough belief revision [3] - in a line similar to the approach of [5].
In this paper the relationship between rough set theory and modal logic has been discussed. Pawla... more In this paper the relationship between rough set theory and modal logic has been discussed. Pawlakian rough set theory has obvious connection with modal logic system \(S_5\). With the introduction of various other lower and upper approximation operators, other modal systems come into picture. Besides, the possibility of new modal systems also crop up. Some of these issues are focused here.
This chapter comprises the basic concepts of the theory. It includes the syntactic, semantic and ... more This chapter comprises the basic concepts of the theory. It includes the syntactic, semantic and axiomatic notions of graded consequence, and their interrelations.
In this paper we study some basic algebraic structures of rough algebras. We proved that the clas... more In this paper we study some basic algebraic structures of rough algebras. We proved that the class of topological quasi-Boolean algebra 5s (tqBa5s) has the finite embeddability property (FEP). Further we also extend this result to some related classes of algebras.
Navya-Nyāya, “The New Reasoning”, is a formal philosophical logic developed in India from the 11t... more Navya-Nyāya, “The New Reasoning”, is a formal philosophical logic developed in India from the 11th to the 17th centuries CE, and which builds on the older traditions of Nyāya and Vaiśeṣika. Not surprisingly, Navya-Nyāya is fundamentally different from classical Western logic and from the meanings ascribed to traditional logical diagrams. For instance, although it is not entirely correct to describe Navya-Nyāya as extensional or intensional, it has an intensional flavour: abstractions are built up from concrete individuals of which we know only their possession, or not, of certain properties. In this paper we look at the implications of these semantics for the use of logical diagrams in Navya-Nyāya. We survey the use of diagrams in modern studies of Navya-Nyāya, notable examples having been produced by Wada, Das and Ganeri. We use notions of well-matchedness, iconicity and Cheng’s recent framework to analyse the effectiveness of the notations in the context of their intended purposes.
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