  We show that the proportion of permutations of length $n$ with principal
M\"{o}bius function equal to zero, $Z(n)$, is asymptotically bounded below by
0.3995. If a permutation $\pi$ contains two intervals of length 2, where one
interval is an ascent and the other a descent, then we show that the value of
the principal M\"{o}bius function $\mu [1, \pi]$ is zero, and we use this
result to find the lower bound for $Z(n)$. We also show that if a permutation
$\phi$ has certain properties, then any permutation $\pi$ which contains an
interval order-isomorphic to $\phi$ has $\mu[1, \pi] = 0$.
