Information Completeness in Nelson Algebras of Rough Sets Induced by Quasiorders

Abstract

Abstract. 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. Motivation: Mixing classical and non-classical logicsMixing logical behaviours is a more and more investigated topic in logic. For instance, labelled deductive systems by D. M. Gabbay [5] are used at this aim, and the "stoup" mechanism introduced by J-Y. Girard in [6] makes intuitionistic and classical deductions interact.In 1989, P. A. Miglioli with his co-authors [16] introduced a constructive logic with strong negation, called effective logic zero and denoted by E 0 , containing a modal operator T such that for any formula α of E 0 , T(α) means that α is classically valid. More precisely, given a Hilbert-style calculus for constructive logic with strong negation (CLSN), also called Nelson logic [18], the rules for T are (∼α →⊥) → T(α) and (α →⊥) → ∼T(α), where ∼ denotes the strong negation. One obtains that α is valid in classical logic (CL) if and only if T(α) is provable in E 0 . Therefore, T acts as an intuitionistic double negation ¬¬ which, in view of the Gödel-Glivenko theorem, is able to grasp classical validity in the intuitionistic propositional calculus (INT) by stating that ⊢ CL α if and only if ⊢ INT ¬¬α.However, T fulfils additional distinct features. Firstly, CLSN is equipped with a weak negation ¬, defined similarly to the intuitionistic negation. But, the combinations ¬¬, ∼¬, or ¬∼ are not able to cope with classical tautologies (see [23], for example). Secondly, consider the Kuroda formula ∀x ¬¬α(x) → ¬¬∀x α(x). As noted in [16], it is an example of the divergence between double negation and an operator intended to represent classical truth, because the formula ∀x T(α(x)) → T(∀x α(x))

Involutive right-residuated l-groupoids

“…the so-called Nelson-identity. Nelson algebras are the algebraic counterparts of the constructive logic with strong negation (see [18,19]). Spinks and Veroff proved [22] that to any Nelson algebra N = (A, ∨, ∧, →, ∼ 0, 1) corresponds an integral commutative residuated lattice L(N ) = (A, ∨, ∧, * , ⇒, 0, 1).…”

Rough Sets and Other Mathematics: Ten Research Programs

“…When a bi-Heyting algebra? Is it possible a characterization of these cases and, moreover, a rough set, hence informational, interpretation as it is done in [7] for finite algebras and particular infinite cases (see …”

“…Not only they are useful in data-mining (cf. [5]), but in this case the construction of rough set systems assumes an unexpected amazing meaning (see [7]) (on the topic, see also [14]). In fact, if P is a partial order upper bounded by a set M of maximal elements (always if it is finite), then for all m ∈ M, ↑ m is a singleton definable set.…”

Monteiro Spaces and Rough Sets Determined by Quasiorder Relations: Models for Nelson algebras

“…Rough sets induced by quasiorders are in the focus of current interest; see [11][12][13]16], for example. Let us denote by ℘(U ) the power set of U , that is, the set of all subsets of U .…”