Omitting uncountable types, and the strength of $[0,1]$-valued logics

Abstract

We study $[0,1]$-valued logics that are closed under the Łukasiewicz-Pavelka connectives; our primary examples are the the continuous logic framework of Ben Yaacov and Usvyatsov \cite{Ben-Yaacov-Usvyatsov:2010} and the Łukasziewicz-Pavelka logic itself. The main result of the paper is a characterization of these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.