Approximating Propositional Calculi by Finite-valued Logics

Bernays introduced a method for proving underivability results in propositional calculi by truth tables. In general, this motivates an investigations of how to find, given a propositional logic, a finite-valued logic which has as few tautologies as possible, but which has all the valid formulas of the given logic as tautologies. It is investigated how far this method can be carried using (1) one or (2) an infinite sequence of finite-valued logics. It is shown that the best candidate matrices for (1) can be computed from a calculus, and how sequences for (2) can be found for certain classes of logics (including, in particular, logics characterized by Kripke semantics).