Some great points on being seen in general--especially from Hydrated--here, but really, the problem with thick fog is that even the brightest lights get sucked up by the fog, and visibility doesn't happen soon enough. No one can see in the fog.

I think heavy fog is the most dangerous situation on the road, especially for cyclists, because we are overtaken very quickly by cars, popping out the gray with very little time for the drivers to notice, assess, and react. We have very little option but to rely on the quick reactions of drivers. That's a scary scenario to me.

I would recommend choosing routes carefully, and probably avoiding roads altogether unless they have a wide shoulder or bike lane. You'll have to be very vigilant and smart to stay safe in the fog.