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 products 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.
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- 1.
For a good introductory article on Paraconsistency see Priest et al. (2016) and the references therein.
- 2.
The general case in which this conditions is dropped is briefly analyzed by Priest in (2014).
- 3.
- 4.
In difference with several authors, we admit the trivial one-element Boolean algebra, see Remark 10.39.
- 5.
This means that \(A \subseteq A'\), \(\mathbf{0}^{\mathcal {A}}=\mathbf{0}^{\mathcal {A}'}\), \(\mathbf{1}^{\mathcal {A}}=\mathbf{1}^{\mathcal {A}'}\) and, for every \(a,b \in A\): \(a \#^\mathcal {A}b = a \#^{\mathcal {A}'}b\) and \({-}^\mathcal {A}a= {-}^{\mathcal {A}'}a\) for \(\# \in \{\sqcap , \sqcup \}\). Note that we write \(s^\mathcal {A}\) instead of \(s^\mathcal {E}\) when s correspond to a symbol of the subsignature \(\Theta _{BA}\) of \(\Theta \).
- 6.
Since in Model Theory the equality predicate \(\approx \) is considered as a predicate symbol which is always interpreted as the standard equality, the injectivity of an embedding is a consequence of the definition.
- 7.
The propositional consequence relation \(\Vdash _\mathbf{K}^{\mathbf{mbC}}\) generated by a class K of mbC-structures should not be confused with the first-order consequence relation \(\models _\mathbf{K}\) defined over first-order \(\Theta \)-sentences as follows: \(\Upsilon \models _\mathbf{K} \sigma \) if, for every \(\mathcal {E}\in \mathbf{K}\), it holds: \(\mathcal {E}\models \sigma \) whenever \(\mathcal {E}\models \varrho \) for every \(\varrho \in \Upsilon \).
- 8.
In order to define \(\bar{v}\) by induction on the complexity of \(\gamma \), the complexity measure on \(For(\Sigma )\) must be defined in a way such that \({\circ }\beta \) has a complexity degree strictly greater than that of \(\lnot \beta \), for every \(\beta \in For(\Sigma )\).
- 9.
That is, \(\theta \) is an equivalence relation which is preserved by the operations of the Boolean algebra \(\mathcal{A}\).
- 10.
Observe that, in particular, \(h:\mathcal {A}\rightarrow \mathcal {A}'\) is a Boolean homomorphism.
- 11.
In addition, also as a consequence of Chang and Keisler (2012, Proposition 6.2.2), it follows that the class F mbC is closed under reduced products; in particular, it is closed under ultraproducts.
- 12.
As usual, if I and Z are two sets, then \(Z^I\) denotes the set of mappings from I to Z.
- 13.
Such homomorphisms are called isomorphisms in Fidel (1977, Definition 6).
- 14.
Observe that the interpretation of function symbols and constants is well-defined since f is a homomorphism of first-order structures, hence it is an algebraic homomorphism.
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Acknowledgements
We are grateful to X. Caicedo and to the anonymous referees for their remarks, suggestions, and criticisms, which helped to improve the paper. The first author was financially supported by an individual research grant from CNPq, Brazil (308524/2014-4). The second author acknowledges support from a postdoctoral grant from FAPESP, Brazil (2016/21928-0).
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Coniglio, M.E., Figallo-Orellano, A. (2019). A Model-Theoretic Analysis of Fidel-Structures for mbC. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_10
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