A Note on Rough Concepts Logic

Abstract

In papers [4,5] Pawlak introduced the notion of a rough set and approximation space. In [6] Pawlak formulated some concept of rough logic. The notion of the approximate truth was considered by many philosophers and logicians and in the last time by computer scientists. This was motivated by some research in artificial intelligence as for example expert systems, approximate reasoning methods and information system with imprecise information.

The concept of rough logic introduced in [6] based on the notion of approximate truth determined by rough sets. Following these ideas Rasiowa and Skowron in [7] proposed the apropriate first order logic for concepts of rough definability. We denote this logic by L _R . In [9] Szczerba proposed some logic with additional quantifier as rough concepts logic. We denote this logic by L(Q _R ). The aim of this paper is a comparizing of these two logics with respect to their expressive power and giving some propositions of some modificated versions of rough concepts logics.

We use more or less standard notation. By [a] _R we denote the equivalence class of the element a with respect to the equivalence relation R. We write $\mathit{L}⩽{\mathit{L}}^{\prime }$ if expressive power of the logic L is weaker then t.he expressive power of the logic ${\mathit{L}}^{\prime }$ (i.e. each class of models definable by a sentence from L is also definable by a sentence from ${\mathit{L}}^{\prime }$ ). If $\mathit{L}⩽{\mathit{L}}^{\prime }$ and ${\mathit{L}}^{\prime }⩽\mathit{L}$ then we say that L and ${\mathit{L}}^{\prime }$ are equivalent and denote this by $\mathit{L}\equiv {\mathit{L}}^{\prime }$ . If $\mathit{L}⩽{\mathit{L}}^{\prime }$ but $\mathit{L}\not\equiv {\mathit{L}}^{\prime }$ then we write $\mathit{L}\mathrm{<}{\mathit{L}}^{\prime }$ .