The Keisler–Shelah theorem for QmbC through semantical atomization. (29th November 2018) - Ferguson, Thomas Macaulay
Abstract: In this paper, we consider some contributions to the model theory of the logic of formal inconsistency 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 QmbC, i.e. that the strong elementary equivalence of two QmbC models A and B is equivalent to them having strongly isomorphic ultrapowers. As intermediate steps, we introduce some notions of model-theoretic equivalence between 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 QmbC, explicitly showing that Keisler–Shelah holds for QCi and suggesting that it holds of extensions like QCila and QCia as well.