Varieties of topological quasi-Boolean algebras in the vicinity of pre-rough algebras [28, 29] are expanded to residuated algebraic structures by introducing a new implication operation and its residual in these structures. Sequent calculi for some classes of residuated algebraic structures are established. These sequent calculi have the strong finite model property which yields the decidability of the word problem for corresponding classes of algebraic structures.

Over the last two decades substantial advances have been made in our understanding of diagrammatic logics. Many of these logics have the expressiveness of monadic first-order logic, sometimes with equality, and are equipped with sound and complete inference rules. A particular challenge is the representation of negated statements. This paper addresses the problem of how to represent negated statements involving constants, thus asserting the absence of specific individuals, in the context of Euler-diagram-based logics. Our first contribution is to explore the potential benefits of explicitly representing absence using constants, in terms of clutter reduction, and to highlight ontological issues that arise. We go on to define a measure of clutter arising from constants. By defining a set of semantics-preserving inference rules, we are able to algorithmically minimize diagram clutter, in part made possible by the inclusion of absence. Consequently, information about individuals can be represented in a minimally cluttered way. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper is concerned with some issues connected with the foundations of rough set theory. Particularly the problem of definition of a rough set is discussed.

Fuzzy Logic a la Pavelka has been reintroduced here in terms of consequence relation instead of consequence operator. Metalogical notions like, consistency and inconsistency are proposed as graded notions. The relationship between consequence relation and inconsistency, both fuzzy here, is studied. Another metalogical notion viz., equivalence of two sets of formulae that originates from Tarski is defined and investigated in this context when the premise is a fuzzy set.

A topological space is a “space”, where “near” makes sense; it is formally defined by the Topological Neighborhood System (TNS). Here, we explore the concept of “conflict” by the system of “Anti-TNS”; by that we mean a “mathematical structure” that consists of a set of “punctured” neighborhoods, namely, the center point p of all neighborhoods of TNS has been removed. “Conflicts” are important concepts in computer security. The primary results is the axiomatization of ATNS. The main results are surprising: The set of the axioms is the same as that of topological spaces. Similar results for pretopological spaces also are obtained.

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.

The notion of graded consequence and some other metalogical notions like consistency, inconsistency, tautologihood and theoremhood to which grades have been introduced earlier by us are reviewed in the context of generalized operators. Some preliminary results regarding the relation between the notion of graded consequence and the notion of graded inconsistency are proved. The method of axiomatization is reconsidered in this general situation.

A semantics of identity inspired by rough set modeling of uncertainty is proposed and the underlying substitutivity principles are presented. A survey of some theories of identity is given, in particular, an interpretation of identity in E-unificatio

While studying rough equality within the framework of the modal system S5 , an algebraic structure called rough algebra [1], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that

In this paper, 'identity' is eventually considered to be determined by a specified collection of 'properties' represented extensionally by subsets. Generally, the collection may contain fuzzy subsets representing vague properties, thus rendering vagueness to the concept of identity. Saturatedness condition, which in essence is a portion of Leibniz's principle of identity, has been dealt with in some detail. A hierarchy of identities is obtained. This hierarchy also indicates some relationship between the concepts 'identity' and 'property'. Some applications of the results to the modelling of indistinguishability in information systems have been indicated.