Which choices of truth tables and notions of consequence for two logics \(\textsf{L}_1\) and \(\textsf{L}_2\) ensure the satisfaction of the following split interpolation property: if two formulas \(\phi \) and \(\psi \) share at least one propositional atom and \(\phi \) classically entails \(\psi \) , then there is a formula \(\chi \) that shares all its propositional atoms with both \(\phi \) and \(\psi \) , such that \(\phi \) entails \(\chi \) in \(\textsf{L}_1\) and \(\chi \) entails \(\psi \) in \(\textsf{L}_2\) ? We identify the cases in which this property holds for any pair of propositional logics based on the same three-valued Boolean normal monotonic scheme for connectives and two monotonic consequence relations. Since the resulting logics are subclassical, every instance of this property constitutes a particular refinement of Craig’s deductive interpolation theorem, as it entails the latter and further restricts the range of possible interpolants.
