Yes, but fans only move hot air.
I'll take my placebo, it keeps me happy.
It is, still air holds heat better than moving air, this is one of the reasons air from a fan will "feel" cooler than it did standing.
Exactly what ubersonic posted.
Here is the scientific answer
Fans do not produce cold air, they produce the sensation of colder air. There are two different phenomena at play depending on the air temperature.
If the air temperature is less than ~24C, then the reason why the human body perceives it as colder with the fan on has to do with the air boundary layer: that is a thin layer of slow-moving air a few millimetres thick all over your exposed skin (this layer also gets thicker where you have body hair). This air layer effectively insulates your body. When it's windy, or when there is a fan on, this thin layer of slow-moving air is disrupted and your insulation is literally blown off by this fast-moving air, and you feel colder.
If the air temperature is above ~24C, your body increasingly needs to sweat to control its temperature. When you sweat, the water on your skin will evaporate: the heat required for evaporation is extremely large, so a small amount of water does a tremendous amount of bodily cooling and this is a very effective way of staying cool. Unfortunately, in such conditions the boundary layer described in the previous paragraph is an impediment to your body's attempts at cooling down, because the air next to your skin isn't moving. When there is a fan on, that boundary layer is disrupted and lots of fresh air comes in and evaporates water on your skin much faster, and consequently you feel much cooler.
Think when you are doing sports in hot weather e.g.hard running or cycling.
When you are moving you feel cooler than when you stop, even though the air temp is the same.