  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.
