Abstract
In this article, we propose 3-valued semantics of the logics compatible with Stone and dual Stone algebras. We show that these logics can be considered as 3-valued by establishing soundness and completeness results. We also establish rough set semantics of these logics where the third value can be interpreted as not certain but possible.
Highlights
Introduction and rough setsRough set theory, introduced by Pawlak [3, 4] as a tool to deal with uncertainty in an informationIn 1940, Moisil introduced 3-valued Lukasiewicz alge- system
In this article, we propose 3-valued seman- In [2], Kumar and Banerjee answered this question aftics of the logics compatible with Stone and dual Stone firmatively in the case of Kleene algebras
We show that these logics can be considered duced a logic LK for Kleene algebras, which is sound as 3-valued by establishing soundness and completeness and complete with respect to a 3-valued consequence results
Summary
Stone algebras (lattices) were introduced by Gratzer and Schmidt [15], and have been extensively studied in literature ([16,17,18,19], cf. [20]). 2. A dual Stone algebra DS = (DS, ∨, ∧, ¬, 0, 1) is embedded into 2I × 3J¬, for some index sets I and J. In particular if B is a Boolean algebra, the Stone algebra B∼[2] and dual Stone algebra B¬[2] can be embedded into 3I∼ and 3J¬ respectively, for appropriate index sets I and J. Notation 3 Let JL denote the set of all completely join irreducible elements of L, and J(x) := {a ∈ JL : a ≤ x}, for any x ∈ L. In [2] we characterized the completely join irreducible elements of lattices 3I and B[2], where B is a complete atomic Boolean algebra. X ∈ {a, 1}, the following element in 3I
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