A Model-Theoretic Analysis of Fidel-Structures for mbC

Abstract

In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to analyze the class of mbC-structures. Thus, substructures, union of chains, direct products, direct limits, congruences and quotient structures can be analyzed under this perspective. In particular, a Birkhoff-like representation theorem for mbC-structures as subdirect poducts in terms of subdirectly irreducible mbC-structures is obtained by adapting a general result for first-order structures due to Caicedo. Moreover, a characterization of all the subdirectly irreducible mbC-structures is also given. An alternative decomposition theorem is obtained by using the notions of weak substructure and weak isomorphism considered by Fidel for C n-structures.

A simple decision procedure for da Costa's Cn logics by Restricted Nmatrix semantics

“…From now on, only non-trivial Boolean algebras will be considered. 13 The Boolean complement in a Boolean algebra will be denoted by ∼. (1) By iterating clause (V 5) of Definition 6.1, we obtain the following:…”

“…In [13] was proposed the study or Fidel structures as being first-order (Tarskian) structures modeling certain (Horn) axioms, allowing so to study them within the rich framework of model theory. From the observation we made at Subsection 5.1, Fidel structures could be alternatively analyzed from the perspective of category theory, specifically within the category of multialgebras.…”

“…Moreover, there is a functor from the category of Boolean algebras to the category of swap structures -a full subcategory of the category of multialgebras over Σ • (see[14]). 13 A Boolean algebra is non-trivial if 0 = 1, which is equivalent to say that it has at least two elements. As it was done with B2, the Boolean operation corresponding to each binary connective # of Σ will be also written as #.…”

Symmetric operators on modal pseudocomplemented De Morgan algebras

“…The former is a non-determinism via multialgebras. In the paper [11], we see that the formulas of first-order logic have a disconnect with his corresponding interpretations, producing a technical difficulty to give proofs. However, the latter, Fidel's non-determinism strongly uses the algebraic fragment of the system and formulas with negation have a valoration belonging to a certain algebra.…”