The Keisler–Shelah theorem for $\mathsf{QmbC}$ through semantical atomization. (29th November 2018)

• Record Type: Journal Article
• Authors: Ferguson, Thomas Macaulay
• Is Part Of: Logic journal of the IGPL. Volume 28:Number 5(2020)
• Page Start: 912
• Page End: 935
• Publication Date: 2018-11-29
• Subjects: Keisler-Shelah theorem -- logics of formal inconsistency -- QmbC model theory -- Los' theorem; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1093/jigpal/jzy067
• Languages: English
• ISSNs: 1367-0751

Abstract

Abstract: 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 (LFIs) 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.