The Keisler–Shelah theorem for $\mathsf{QmbC}$ through semantical atomization

Logic Journal of the IGPL 28 (5):912-935 (2020)
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In this paper, we consider some contributions to the model theory of the logic of formal inconsistency $\mathsf{QmbC}$ as a reply to Walter Carnielli, Marcelo Coniglio, Rodrigo Podiacki and Tarcísio Rodrigues’ call for a ‘wider model theory.’ This call demands that we align the practices and techniques of model theory for logics of formal inconsistency as closely as possible with those employed in classical model theory. The key result is a proof that the Keisler–Shelah isomorphism theorem holds for $\mathsf{QmbC}$, i.e. that the strong elementary equivalence of two $\mathsf{QmbC}$ models $\mathfrak{A}$ and $\mathfrak{B}$ is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between $\mathsf{QmbC}$ models, explicitly prove Łoś’ theorem and introduce a useful technique of model-theoretic ‘atomization’ in which the satisfaction sets of non-deterministically evaluated formulae are associated with new predicates. Finally, we consider some of the extensions of $\mathsf{QmbC}$, explicitly showing that Keisler–Shelah holds for $\mathsf{QCi}$ and suggesting that it holds of extensions like $\mathsf{QCila}$ and $\mathsf{QCia}$ as well.



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Thomas Ferguson
City University of New York

References found in this work

Book Reviews. [REVIEW]Wilfrid Hodges - 1997 - Studia Logica 64 (1):133-149.
Defining LFIs and LFUs in extensions of infectious logics.Szmuc Damian Enrique - 2016 - Journal of Applied Non-Classical Logics 26 (4):286-314.
Notes on the Model Theory of DeMorgan Logics.Thomas Macaulay Ferguson - 2012 - Notre Dame Journal of Formal Logic 53 (1):113-132.

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