Add to ORKGThis paper continues my earlier work, which showed there is a broad family of propositional many valued logics that have a strict/tolerant counterpart. Here we generalize those results from propositional to a range of both modal and quantified many valued logics, providing strict/tolerant counterparts for all. This paper is not self-contained; some results from earlier papers are called on, and are not reproved here. The key new machinery added to earlier work, allowing modalities and quantifiers to be handled in similar ways, is the central use of bilattices that are function spaces, and more generally lattices that are function spaces. Two versions of the central proofs are considered, one at length and the other i...