Catálogo de publicaciones - libros

A is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family of first-order LFIs (which includes da Costa’s original system , as well as thousands of other logics). We show that our semantics is effective and modular, and we use this effectiveness to derive some important properties of logics in this family.

In this note we provide a straightforward translation for sets of formulas and for existential formulas s.t. expresses “ is derivable constructively from iff it is derivable at all”.

We introduce the class of Symmetric MV-algebras. Such algebras have a suitable behavior with respect to a family of MV-polynomials. It turns out that the class of Symmetric MV-algebras can be characterized as the class of MV-algebras having homomorphic image in the variety generated by a single MV-chain with  + 1 elements, where  = 1 or is a prime number. Also, using symmetric MV-algebras, we provide a new characterization of the above mentioned varieties.

It is shown that a conservative expansion of infinite valued Łukasiewicz logic by new connectives univocally determined by their axioms does not necessarily have a complete semantics in the real interval [0,1]. However, such extensions are always complete with respect to valuations in a family of MV-chains. Rational Łukasiewicz logic being the largest one that has a complete semantics in [0,1]. In addition, this logic does not admit expansions by axiomatic implicit connectives that are not already explicit. Similar results are obtained for -valued Łukasiewicz logic and for the logic of abelian lattice ordered groups. These and related results are obtained by the study of compatible operations implicitly defined by identities in the varieties of MV-algebras and abelian ℓ-groups; the pertaining algebraic results having independent interest.

An outline of the history of the algebras corresponding to Łukasiewicz many-valued logic from the pioneering work by G. Moisil in 1940 until D. Mundici’s work in 1986.

A characteristic feature of quantum computation is the use of logical operations. These correspond to that are mathematically represented by unitary operators defined on convenient Hilbert spaces. Two questions arise: 1) to what extent is quantum computation bound to the use of reversible logical operations? 2) How to identify the logical operations that admit a by means of appropriate gates? We introduce the notion of of a binary function defined on the real interval [0,1], and we prove that for any binary Boolean function there exists a unique fuzzy extension admitting a quantum computational simulation. As a consequence, the Łukasiewicz conjunction and disjunction do not admit a quantum computational simulation.

Generalizations of the Cantor–Bernstein theorem have been proved for different types of algebras, starting from -complete orthomodular lattices and -complete MV-algebras and continuing with more general structures, including (pseudo) effect algebras and (pseudo) BCK-algebras. E.g., for -complete MV-algebras a version of the Cantor–Bernstein theorem has been proved which assumes that the bounds of isomorphic intervals are boolean.

There is another direction of research which has been paid less attention. We ask which algebras satisfy the Cantor–Bernstein theorem in the same form as for -complete boolean algebras, without any additional assumption. In the case of orthomodular lattices, it has been proved that this class is rather large. E.g., every orthomodular lattice can be embedded as a subalgebra or expressed as an epimorphic image of a member of this class. On the other hand, also the complement of this class is large in the same sense. We study the analogous question for MV-algebras and we find out interesting examples of MV-algebras which possess or do not possess this property. This contributes to the investigations of the scope of validity of the Cantor–Bernstein theorem in its original form.

We try to make a distinction between the idea of representing and that of interpreting a mathematical structure. We present a slight generalization of Di Nola’s Representation Theorem as to incorporate this point of view. Furthermore, we examine some preservation and functorial aspects of the Boolean power construction.

Mundici’s functor establishes a categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a strong unit. In this short note we present a first step towards the generalization of such a relationship when we replace MV-algebras by weaker structures obtained by dropping the divisibility condition. These structures are the so-called involutive monoidal t-norm based algebras, IMTL-algebras for short. In this paper we restrict ourselves to linearly ordered IMTL-algebras, for which we show a one-to-one correspondence with a kind of ordered grupoid-like structures with a strong unit. A key feature is that the associativity property in such a new structure related to a IMTL-chain is lost as soon the IMTL-chain is no longer a MV-chain and the strong unit used in Mundici’s functor is required here to have stronger properties. Moreover we define a functor between the category of such structures and the category of IMTL algebras that is a generalization of Mundici’s functor and, restricted to their linearly ordered objects, a categorical equivalence.