Dialectics of Counting and Measures of Rough Theories

New concepts of rough natural number systems, recently introduced by the present author, are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and measures of mutual consistency of multiple models of knowledge. In this research paper, the explicit dependence on the axiomatic theory of granules of \cite{AM99} is reduced and more results on the measures and representation of the numbers are proved.

💡 Research Summary

The paper addresses a fundamental limitation of classical Rough Set Theory (RST) – its heavy reliance on a fixed notion of granules (or “atoms”) to define lower and upper approximations, which becomes inadequate when dealing with more complex relational structures, tolerance relations, or cover‑based models. To overcome this, the author introduces a new algebraic framework called Rough Y‑Systems (RYS+), together with a “rough natural number” calculus that enables a dialectical form of counting for indiscernible objects.

RYS+ is defined as a tuple ⟨S, W, P, (l_i){i=1}^n, (u_i){i=1}^n, +, ·, ∼⟩ where S is the universe, W⊆S a distinguished sub‑universe, P a binary “part‑of” relation (reflexive, antisymmetric, and transitive), and each l_i, u_i are surjective maps from S onto W representing lower and upper approximation operators. The framework also supplies derived relational operators such as overlap (O), underlap (U), proper part (P), overcross (X), and proper overlap, together with set‑like sum (+) and product (·) defined via description operators of first‑order predicate logic. Unlike classical set algebra, these operations are allowed to be only weakly associative and weakly commutative, reflecting the inherent vagueness of the underlying objects.

The central methodological innovation is “dialectical counting”. Traditional counting assumes that every element is distinct and can be assigned a unique natural number. In many rough contexts, however, objects are only partially discernible; groups of indiscernible elements should be treated as a single counting unit. The author therefore constructs a rough natural number system in which each equivalence class (or granule) contributes a single “rough unit”. The arithmetic of these units respects the weak algebraic laws of RYS+, enabling the definition of cardinalities that are not merely integers but ordered pairs (or more complex structures) encoding both lower and upper information.

With this machinery, the paper revisits two core RST measures: inclusion functions and knowledge‑dependency indices. Classical inclusion is a binary predicate (A⊆B) that yields 0 or 1. In the new setting, inclusion becomes a graded quantity: for sets A and B, the rough inclusion degree is defined as
 α(A,B) = |{x∈A | P(x,B)}| / |A|,
where P(x,B) denotes that x is a part of some element of B. This yields a value in