Remarks on special lattices and related constructive logics with strong negation.

Pagliani, Piero

doi:10.1305/ndjfl/1093635588

Cited by 10 publications

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“…However, its stable part, that is, the part which is closed under uniform substitution, coincides with a well-known maximal intermediate constructive logic, namely Medvedev's logic, a faithful interpretation of the intuitionistic logical principles (see [14,15]). One year later, P. Pagliani [19] was able to exhibit an algebraic model for E 0 . It turned out that these models are a special kind of Nelson algebras, called effective lattices.…”

Section: Motivation: Mixing Classical and Non-classical Logicsmentioning

confidence: 99%

“…In [19], Pagliani introduced effective lattices. They are special type of Nelson algebras determined by Glivenko congruences on Heyting algebras, that is, for any Heyting algebra H and its Glivenko congruence Γ, the corresponding effective lattice is the Nelson algebra N Γ (H).…”

Section: Preliminaries: Heyting Algebras Nelson Algebras and Effectiv...mentioning

confidence: 99%

“…Note that for all x ∈ H, x Γ 1 if and only if x is dense. This means that N Γ (H) consists of the pairs (a, b) such that a ∧ b = 0 and a ∨ b is dense (see Remark 2 in [26] and Proposition 9 of [19]). Additionally, it is proved in [19] that effective lattices are models for the logic E 0 , with T defined by T((a, b)) = (a * * , b * * ).…”

Section: Preliminaries: Heyting Algebras Nelson Algebras and Effectiv...mentioning

confidence: 99%

“…This means that N Γ (H) consists of the pairs (a, b) such that a ∧ b = 0 and a ∨ b is dense (see Remark 2 in [26] and Proposition 9 of [19]). Additionally, it is proved in [19] that effective lattices are models for the logic E 0 , with T defined by T((a, b)) = (a * * , b * * ). Note that each Heyting algebra defines exactly one effective lattice, and that all effective lattices are determined this way.…”

Section: Preliminaries: Heyting Algebras Nelson Algebras and Effectiv...mentioning

confidence: 99%

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Information Completeness in Nelson Algebras of Rough Sets Induced by Quasiorders

Järvinen¹,

Pagliani²,

Radeleczki

2012

Stud Logica

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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))

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Section: Motivation: Mixing Classical and Non-classical Logicsmentioning

confidence: 99%

Section: Preliminaries: Heyting Algebras Nelson Algebras and Effectiv...mentioning

confidence: 99%