There is a peculiar consequence of this, pointed out by Cosma Shalizi. Suppose we have a deterministic physical system S, and we observe this system carefully over time. We are steadily gaining information about its microstates, and therefore by this definition, its entropy should be decreasing.

You might say, "the system isn't closed, because it is being observed." But consider the system "S plus the observer." Saying that entropy is nondecreasing over time seems to require that the observer is in doubt about its own microstates. What does that mean?