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

TL;DR

This paper characterizes certain $[0,1]$-valued logics, including continuous logic and Łukasiewicz-Pavelka logic, through an extension of the omitting types theorem applicable to uncountable languages.

Contribution

It provides a model-theoretic characterization of $[0,1]$-valued logics closed under Łukasiewicz-Pavelka connectives, extending the omitting types theorem to uncountable languages.

Findings

Characterization of $[0,1]$-valued logics via model-theoretic properties

Extension of the omitting types theorem to uncountable languages

Application to continuous logic and Łukasiewicz-Pavelka logic

Abstract

We study $[0,1]$-valued logics that are closed under the {\L}ukasiewicz-Pavelka connectives; our primary examples are the the continuous logic framework of Ben Yaacov and Usvyatsov \cite{Ben-Yaacov-Usvyatsov:2010} and the {\L}ukasziewicz-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.