  Many-valued logics in general, and fuzzy logics in particular, usually focus
on a notion of consequence based on preservation of full truth, typical
represented by the value 1 in the semantics given the real unit interval [0,1].
In a recent paper (\emph{Foundations of Reasoning with Uncertainty via
Real-valued Logics}, arXiv:2008.02429v2, 2021), Ronald Fagin, Ryan Riegel, and
Alexander Gray have introduced a new paradigm that allows to deal with
inferences in propositional real-valued logics based on multi-dimensional
sentences that allow to prescribe any truth-values, not just 1, for the
premises and conclusion of a given entailment. In this paper, we extend their
work to the first-order (as well as modal) logic of multi-dimensional
sentences. We give axiomatic systems and prove corresponding completeness
theorems, first assuming that the structures are defined over a fixed domain,
and later for the logics of varying domains. As a by-product, we also obtain a
0-1 law for finitely-valued versions of these logics.
