TITLE: Semiproper but not proper
QUESTION [13 upvotes]: Assume V=L.  Is there a semi-proper notion of forcing that is not proper?
Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.

REPLY [7 votes]: The following is proved as Claim 2.1 in Shelah's paper ``Forcing axiom failure for any $\lambda>\aleph_1$''

Theorem. There is a forcing notion of size $2^{\aleph_2}$ which is not proper but
  is $\{\aleph_1\}$-semi proper.

In fact Shelah proves more: for every regular $κ>ℵ_1$ there is a forcing notion $P$ of cardinality $2^κ$ such that there is a stationary set $S⊆[2^κ]^{ℵ_0}$ not preserved by $P$ and if $χ>2^κ$, $p∈P∈N≺(H(χ),∈), N$ countable, then there is $q∈P$ stronger than $p$ and $q$ forces that $N∩κ$ is an initial segment of $N[G]∩κ.$