  We focus on the persistence principle over weak interpretability logic. Our
object of study is the logic obtained by adding the persistence principle to
weak interpretability logic from several perspectives. Firstly, we prove that
this logic enjoys a weak version of the fixed point property. Secondly, we
introduce a system of sequent calculus and prove the cut-elimination theorem
for it. As a consequence, we prove that the logic enjoys the Craig
interpolation property. Thirdly, we show that the logic is the natural basis of
a generalization of simplified Veltman semantics, and prove that it has the
finite frame property with respect to that semantics. Finally, we prove that it
is sound and complete with respect to some appropriate arithmetical semantics.
