In this talk, based on joint work with Thomas Ferguson and Paweł Pawłowski, we develop a formal framework for what we call Suszko-style reductions, drawing on the theory of restricted non-deterministic matrices (RNmatrices). Originally introduced by Coniglio and Toledo—but anticipated by several earlier contributions—these reductions generalize Suszko’s classic result that every Tarskian consequence relation admits a two-valued semantics. We extend this paradigm in two directions: first, to the multiple-conclusion setting of Scott consequence relations; second, to non-Tarskian logics by suitably adjusting the RNmatrix structure. We discuss the broader significance of these results, as well as the theoretical gains and trade-offs involved in pursuing such reductions within a non-deterministic semantic framework.