A temporal semantics for Nilpotent Minimum logic

In [Ban97] a connection among rough sets (in particular, pre-rough algebras) and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95, EG01]), in which the logic of every instant is given by {\L}3: a completeness theorem will be shown. This is the prosecution of the work initiated in [AGM08] and [ABM09], in which the authors construct a temporal semantics for the many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table

Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.Comment: 16 page

Information completeness in Nelson algebras of rough sets induced by quasiorders

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.Comment: 15 page