If this is your standard for entropy increase, then surely it shouldn't matter what you are observing. If you are observing a refrigerator, your brain is pumping just as much heat into the environment as when you are observing spontaneous equilibriation. Yet in the case of the refrigerator we usually say, in contrast to the two-gas system I described, that it is entropy decreasing. But if the basis for calling the two-gas system entropy increasing is the heat output by the observer's brain, why doesn't the refrigerator qualify as entropy increasing as well?
Or are you disagreeing that we would (or should) call a two-gas system exhibiting spontaneous thermal equilibriation an entropy-increasing system? I think I'm not getting your view exactly right.
This is the semantic problem that you dismissed. When I talk about the refrigerator, it's clear that I mean to draw an imaginary boundary around the refrigerator only and pretend for a second that that is all there is anywhere. Then the entropy is decreasing. If I talk about the process by which I acquired that knowledge, then I have to expand my imaginary boundary to include the source of the photons that bounced off the refrigerator, for instance, and the waste heat my brain produced to acquire this knowledge. That process, the acquiring of the knowledge...
Link to the Question
I haven't gotten an answer on this yet and I set up a bounty; I figured I'd link it here too in case any stats/physics people care to take a crack at it.