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 Implicative twist-structures(Springer Alemania, 2014-09-03) Rivieccio, UmbertoThe twist-structure construction is used to represent algebras related to non-classical logics (e.g., Nelson algebras, bilattices) as a special kind of power of better-known algebraic structures (distributive lattices, Heyting algebras). We study a specific type of twist-structure (called implicative twist-structure) obtained as a power of a generalized Boolean algebra, focusing on the implication-negation fragment of the usual algebraic language of twist-structures. We prove that implicative twist-structures form a variety which is semisimple, congruence-distributive, finitely generated, and has equationally definable principal congruences. We characterize the congruences of each algebra in the variety in terms of the congruences of the associated generalized Boolean algebra. We classify and axiomatize the subvarieties of implicative twist-structures. We define a corresponding logic and prove that it is algebraizable with respect to our variety.Publicación Four-valued modal logic: Kripke semantics and duality(IEEE Xplore, 2017-02) Rivieccio, Umberto; Jung, Achim; Jansana, RamonWe introduce a family of modal expansions of Belnap–Dunn four-valued logic and related systems, and interpret them in many-valued Kripke structures. Using algebraic logic techniques and topological duality for modal algebras, and generalizing the so-called twist-structure representation, we axiomatize by means of Hilbert-style calculi the least modal logic over the four-element Belnap lattice and some of its axiomatic extensions. We study the algebraic models of these systems, relating them to the algebraic semantics of classical multi-modal logic. This link allows us to prove that both local and global consequence of the least four-valued modal logic enjoy the finite model property and are therefore decidable.Publicación Bilattice Logic Properly Displayed(Elsevier, 2019-05-15) Greco, Giuseppe; Fei Liang,; Palmigiano,Alessandra; Rivieccio, UmbertoWe introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.Publicación An infinity of super-Belnap logics(Taylor & Francis, 2012) Rivieccio, UmbertoWe look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical matrix. We show that the last logic of the chain is not finitely axiomatisable.Publicación Residuated bilattices(Springer, 2012) Jansana, Ramón; Rivieccio, UmbertoWe introduce a new product bilattice construction that generalizes the well-known one for interlaced bilattices and others that were developed more recently, allowing to obtain a bilattice with two residuated pairs as a certain kind of power of an arbitrary residuated lattice. We prove that the class of bilattices thus obtained is a variety, give a finite axiomatization for it and characterize the congruences of its members in terms of those of their lattice factors. Finally, we show how to employ our product construction to define first-order definable classes of bilattices corresponding to any first-order definable subclass of residuated lattices.Publicación Modal twist-structures over residuated lattices(Oxford University Press, 2014-06) Ono, Hiroakira; Rivieccio, UmbertoWe introduce a class of algebras, called twist-structures, whose members are built as special squares of an arbitrary residuated lattice. We show how our construction relates to and encompasses results obtained by several authors on the algebraic semantics of non-classical logics. We define a logic that corresponds to our twist-structures and show how to expand it with modal operators, obtaining a paraconsistent many-valued modal logic that generalizes existing work on modal expansions of both Belnap–Dunn logic and paraconsistent Nelson logic.Publicación Dualities for modal N4-lattices(Oxford University Press, 2014-08) Jansana, Ramon; Rivieccio, UmbertoWe introduce a new Priestley-style topological duality for N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to S. Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that we obtain a simple description for our topological structures, which allows us to extend the duality to other algebraic structures such as N4-lattices with monotonic modal operators, and also to provide a neighbourhood semantics for the non-normal modal logic corresponding to these algebras.Publicación Nothing but the truth(Springer, 2013) Pietz, Andreas; Rivieccio, UmbertoA curious feature of Belnap’s “useful four-valued logic”, also known as first-degree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic.Publicación Neutrosophic logics: prospects and problems(Elsevier, 2008) Rivieccio, UmbertoNeutrosophy has been introduced some years ago by Florentin Smarandache as a new branch of philosophy dealing with “the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra”. A variety of new theories have been developed on the basic principles of neutrosophy: among them is neutrosophic logics, a family of many-valued systems that can be regarded as a generalization of fuzzy logics. In this paper we present a critical introduction to neutrosophic logics, focusing on the problem of defining suitable neutrosophic propositional connectives and discussing the relationship between neutrosophic logics and other well-known frameworks for reasoning with uncertainty and vagueness, such as (intuitionistic and interval-valued) fuzzy systems and Belnap’s logic.