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))