We study the quantum logics which satisfy the Riesz interpolation
property. We call them the RIP logics. We observe that the class of
RIP logics is considerable large~-- it contains all lattice quantum
logics and, also, many (infinite) non-lattice ones. We then find out
that each RIP logic can be enlarged to an RIP logic with a
preassigned center.  We continue, showing that the ``nearly" Boolean
RIP logics must be Boolean algebras. In a somewhat surprising
contrast to this, we finally show that the attempt for the
$\sigma$-complete formulation of this result fails: We show by
constructing an example that there is a non-Boolean nearly Boolean
$\sigma$-RIP logic. As a result, there are interesting $\sigma$-RIP
logics which are intrinsically close to Boolean $\sigma$-algebras.
