  We present quantitative logics with two-step semantics based on the framework
of quantitative logics introduced by Arenas et al. (2020) and the two-step
semantics defined in the context of weighted logics by Gastin & Monmege (2018).
We show that some of the fragments of our logics augmented with a least fixed
point operator capture interesting classes of counting problems. Specifically,
we answer an open question in the area of descriptive complexity of counting
problems by providing logical characterizations of two subclasses of #P, namely
SpanL and TotP, that play a significant role in the study of approximable
counting problems. Moreover, we define logics that capture FPSPACE and
SpanPSPACE, which are counting versions of PSPACE.
