A characterization of Continuous Logic by using quantale-valued logics

TL;DR

This paper generalizes Continuous Logic by using quantale-valued distances, establishing key model-theoretic properties under certain conditions, and exploring new logics beyond the classical assumptions.

Contribution

It introduces a broad class of quantale-valued logics extending Continuous Logic and proves fundamental theorems within this framework, relaxing previous restrictions.

Findings

Proves a version of the Tarski-Vaught test in the new setting

Establishes a version of Łoś's Theorem for quantale-valued logics

Identifies conditions under which classical model-theoretic properties hold

Abstract

In this paper, we propose a generalization of Continuous Logic ([BBHU08]) where the distances take values in suitable co-quantales (in the way as it was proposed in [Fla97]). By assuming suitable conditions (e.g., being co-divisible, co-Girard and a V-domain), we provide, as test questions, a proof of a version of the Tarski-Vaught test (Proposition 4.2) and {\L}o\'s Theorem (Theorem 5.27) in our setting. Iovino proved in [Iov01] that there is no any logic extending (equivalent logics to) Continuous Logic satisfying both Countable Tarski-Vaught chain Theorem and Compactness Theorem. Since [0, 1] satisfies all of the assumptions given above, we get new logics by dropping any of those assumptions.