  In 1988, Ivlev proposed four-valued non-deterministic semantics for modal
logics in which the alethic T axiom holds good. Unfortunately, no completeness
was proved. In previous work, we proved completeness for some Ivlev systems and
extended his hierarchy, proposing weaker six-valued systems in which the T
axiom was replaced by the deontic D axiom. Here, we eliminate both axioms,
proposing even weaker systems with eight values. Besides, we prove completeness
for those new systems. It is natural to ask if a characterization by finite
ordinary (deterministic) logical matrices would be possible for all those
systems. We will show that finite deterministic matrices do not characterize
any of them.
