Generalized Rough Sets (Preclusivity Fuzzy-Intuitionistic (BZ) Lattices)

The standard Pawlak approach to rough set theory, as an approximation space consisting of a universe U and an equivalence (indiscernibility) relation R U x U, can be equivalently described by the induced preclusivity ("discernibility") relation U x U \ R, which is irreflexive and symmetric.We generalize the notion of approximation space as a pair consisting of a universe U and a discernibility or preclusivity (irreflexive and symmetric) relation, not necessarily induced from an equivalence relation. In this case the "elementary" sets are not mutually disjoint, but all the theory of generalized rough sets can be developed in analogy with the standard Pawlak approach. On the power set of the universe, the algebraic structure of the quasi fuzzy-intuitionistic "classical" (BZ) lattice is introduced and the sets of all "closed" and of all "open" definable sets with the associated complete (in general nondistributive) ortholattice structures are singled out.The rough approximation of any fixed subset of the universe is the pair consisting of the best "open" approximation from the bottom and the best "closed" approximation from the top. The properties of this generalized rough approximation mapping are studied in the context of quasi-BZ lattice structures of "closed-open" ordered pairs (the "algebraic logic" of generalized rough set theory), comparing the results with the standard Pawlak approach. A particular weak form of rough representation is also studied.     

Algebras of Intervals and a Logic of Conditional Assertions

Intervals in boolean algebras enter into the study of conditional assertions (or events) in two ways: directly, either from intuitive arguments or from Goodman, Nguyen and Walker's representation theorem, as suitable mathematical entities to bear conditional probabilities, or indirectly, via a representation theorem for the family of algebras associated with de Finetti's three-valued logic of conditional assertions/events. Further representation theorems forge a connection with rough sets. The representation theorems and an equivalent of the boolean prime ideal theorem yield an algebraic completeness theorem for the three-valued logic. This in turn leads to a Henkin-style completeness theorem. Adequacy with respect to a family of Kripke models for de Finetti's logic, ukasiewicz's three-valued logic and Priest's Logic of Paradox is demonstrated. The extension to first-order yields a short proof of adequacy for Körner's logic of inexact predicates.     

Every Finitely Reducible Logic has the Finite Model Property with Respect to the Class of ♦-Formulae

In this paper a unified framework for dealing with a broad family of propositional multimodal logics is developed. The key tools for presentation of the logics are the notions of closure relation operation and monotonous relation operation. The two classes of logics: FiRe-logics (finitely reducible logics) and LaFiRe-logics (FiRe-logics with local agreement of accessibility relations) are introduced within the proposed framework. Further classes of logics can be handled indirectly by means of suitable translations. It is shown that the logics from these classes have the finite model property with respect to the class of -formulae, i.e. each -formula has a -model iff it has a finite -model. Roughly speaking, a -formula is logically equivalent to a formula in negative normal form without occurrences of modal operators with necessity force. In the proof we introduce a substantial modification of Claudio Cerrato's filtration technique that has been originally designed for graded modal logics. The main core of the proof consists in building adequate restrictions of models while preserving the semantics of the operators used to build terms indexing the modal operators.     

Varieties of Monadic Heyting Algebras. Part I

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].