In Can Baskent and Thomas Ferguson (ed.), Graham Priest on Dialetheism and Paraconsistency. Springer. pp. 189-216 (2020)

Marcelo E. Coniglio
University of Campinas
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 Cn-structures.
Keywords Fidel structures  Logics of formal inconsistency  Birkhoff decomposition theorem
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Transfinite Numbers in Paraconsistent Set Theory.Zach Weber - 2010 - Review of Symbolic Logic 3 (1):71-92.
A Semantical Analysis of the Calculi Cn.Newton C. A. da Costa - 1977 - Notre Dame Journal of Formal Logic 18:621.
A Note on Naive Set Theory in ${\Rm LP}$.Greg Restall - 1992 - Notre Dame Journal of Formal Logic 33 (3):422-432.

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