The usual semantics for the modal systems $T, S4$, and $S5$ assumes that the set of possible worlds contains at least one member. Recently versions of these modal systems have been developed in which this assumption is dropped. The systems discussed here are obtained by slightly weakening the liberated versions of $T$ and $S4$. The semantics does not assume the existence of possible worlds, and the accessibility relation between worlds is only required to be quasi-reflexive instead of reflexive. Completeness and independence results are established.