Persona: Rivieccio, Umberto
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Rivieccio
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Publicación Intuitionistic modal algebras(Springer Nature, 2023-09-15) Celani, Sergio A.; Rivieccio, UmbertoRecent research on algebraic models of quasi-Nelson logic has brought new attention to a number of classes of algebras which result from enriching (subreducts of) Heyting algebras with a special modal operator, known in the literature as a nucleus. Among these various algebraic structures, for which we employ the umbrella term intuitionistic modal algebras, some have been studied since at least the 1970s, usually within the framework of topology and sheaf theory. Others may seem more exotic, for their primitive operations arise from algebraic terms of the intuitionistic modal language which have not been previously considered. We shall for instance investigate the variety of weak implicative semilattices, whose members are (non-necessarily distributive) meet semilattices endowed with a nucleus and an implication operation which is not a relative pseudo-complement but satisfies the postulates of Celani and Jansana’s strict implication. For each of these new classes of algebras we establish a representation and a topological duality which generalize the known ones for Heyting algebras enriched with a nucleus.Publicación Representation of De Morgan and (Semi-)Kleene Lattices(Springer, 2020-04-02) Rivieccio, Umberto; CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), BrasilTwist-structure representation theorems are established for De Morgan and Kleene lattices. While the former result relies essentially on the quasivariety of De Morgan lattices being finitely generated, the representation for Kleene lattices does not and can be extended to more general algebras. In particular, one can drop the double negation identity (involutivity). The resulting class of algebras, named semi-Kleene lattices by analogy with Sankappanavar’s semi-De Morgan lattices, is shown to be representable through a twist-structure construction inspired by the Cornish–Fowler duality for Kleene lattices. Quasi-Kleene lattices, a subvariety of semi-Kleene, are also defined and investigated, showing that they are precisely the implication-free subreducts of the recently introduced class of quasi-Nelson lattices.Publicación Fragments of Quasi-Nelson: The Algebraizable Core(Oxford University Press, 2022-10-01) Rivieccio, Umberto; CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), BrasilThis is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. -algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.Publicación Quasi-Nelson algebras and fragments(Cambridge University Press, 2021-05-11) Rivieccio, Umberto; Jansana, RamonThe variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.Publicación Prelinearity in (quasi-)Nelson logic(ELSEVIER, 2022-09-20) Flaminio, Tommaso; Rivieccio, Umberto; programa Horizonte 2020 (H2020) de la Comisión Europea; Ministerio de Ciencia, Innovación y Universidades (MCIN), a través de la Agencia Estatal de Investigación (AEI), en el marco del Plan Estatal de Investigación Científica y Técnica y de InnovaciónThe algebraic theory of quasi-Nelson logic, a non-involutive generalization of Nelson's constructive logic with strong negation, has been shown to be surprisingly rich in a series of recent papers. In the present paper we bring quasi-Nelson logic into the fuzzy setting by adding the prelinearity axiom to it. We observe that the resulting system is an extension of the well-known Weak Nilpotent Minimum logic, as well as a rotation logic in the sense of recent work by P. Aglianò and S. Ugolini. We characterize the algebraic models of prelinear quasi-Nelson logic as twist-structures over Gödel algebras endowed with a nucleus operator and use the insight thus gained to look at subvarieties corresponding to extensions of well-known fuzzy systems. Our study of the quasi-Nelson negation in a prelinear setting also allows us to show that the variety of prelinear quasi-Nelson algebras is generated by a single standard algebra, thus obtaining a single chain completeness theorem for the logic.Publicación The Value of the One Value: Exactly True Logic revisited(Springer, 2023-08-07) Kapsner, Andreas; Rivieccio, Umberto; Ministry of Science and Innovation of SpainIn this paper we re-assess the philosophical foundation of Exactly True Logic (ETL), a competing variant of First Degree Entailment (FDE). In order to do this, we first rebut an argument against it. As the argument appears in an interviewwith Nuel Belnap himself, one of the fathers of FDE, we believe its provenance to be such that it needs to be taken seriously.We submit, however, that the argument ultimately fails, and that ETL cannot easily be dismissed.We then proceed to give an overview of the research that was inspired by this logic over the last decade, thus providing further motivation for the study of ETL and, more generally, of FDE-related logics that result from semantical analyses alternative to Belnap’s canonical one. We focus, in particular, on philosophical questions that these developments raise.Publicación Fragments of quasi-Nelson: residuation(Taylor & Francis, 2023-05-03) Rivieccio, Umberto; Ministry of Science and Innovation of SpainQuasi-Nelson logic (QNL) was recently introduced as a common generalisation of intuitionistic logic and Nelson's constructive logic with strong negation. Viewed as a substructural logic, QNL is the axiomatic extension of the Full Lambek Calculus with Exchange and Weakening by the Nelson axiom, and its algebraic counterpart is a variety of residuated lattices called quasi-Nelson algebras. Nelson's logic, in turn, may be obtained as the axiomatic extension of QNL by the double negation (or involutivity) axiom, and intuitionistic logic as the extension of QNL by the contraction axiom. A recent series of papers by the author and collaborators initiated the study of fragments of QNL, which correspond to subreducts of quasi-Nelson algebras. In the present paper we focus on fragments that contain the connectives forming a residuated pair (the monoid conjunction and the so-called strong Nelson implication), these being the most interesting ones from a substructural logic perspective. We provide quasi-equational (whenever possible, equational) axiomatisations for the corresponding classes of algebras, obtain twist representations for them, study their congruence properties and take a look at a few notable subvarieties. Our results specialise to the involutive case, yielding characterisations of the corresponding fragments of Nelson's logic and their algebraic counterparts.Publicación Bilattice logic of epistemic actions and knowledge(ELSEVIER, 2020-06-01) Bakhtiari, Zeinab; Ditmarsch, Hans van; Rivieccio, UmbertoBaltag, Moss, and Solecki proposed an expansion of classical modal logic, called logic of epistemic actions and knowledge (EAK), in which one can reason about knowledge and change of knowledge. Kurz and Palmigiano showed how duality theory provides a flexible framework for modeling such epistemic changes, allowing one to develop dynamic epistemic logics on a weaker propositional basis than classical logic (for example an intuitionistic basis). In this paper we show how the techniques of Kurz and Palmigiano can be further extended to define and axiomatize a bilattice logic of epistemic actions and knowledge (BEAK). Our propositional basis is a modal expansion of the well-known four-valued logic of Belnap and Dunn, which is a system designed for handling inconsistent as well as potentially conflicting information. These features, we believe, make our framework particularly promising from a computer science perspective.