The theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.
"Ranked partial structures." J. Symbolic Logic 68 (4) 1109 - 1144, December 2003. https://doi.org/10.2178/jsl/1067620176