A quick search of the net indicates that a "switching decoder" is a normal decoder featuring one or more relays. Other than that I cannot find any proper information regarding how one would work.
If you're just concerned with the single transistor, remember that a logic 0 is also 0 volts, while a logic 1 is a positive voltage - usually 3 to 5 volts, depending on the logic family used.
In the case of a transistor, it is going to be in a common emitter arrangement, so the voltage on the collector will be 0 when the transistor is on - a logic 0. When the transistor is off, the collector will have the full voltage on it, for a logic 1. It just switches between those two states.
I too am unfamiliar with the term "switching decoder". Perhaps it is just a matter of terminology.
If you are referring to a specific device then it would be helpful if you can provide a part number and manufacturer for the device so that we will have a place from which to begin assisting you with your inquiry.
If you are referring to a process or technique then it would be helpful if you could elaborate.
Well one thing that comes to mind is four 4-input AND gates, one for each board. The board will be turned on IF the corresponding switch is closed AND the other three switches are open. Same logic for each board. I suppose you could do it with discrete transistors the way Diode Transistor Logic was done 45 years ago or so.
eggtree, are you aware of how logic gates work? If so, then you simply need to view the data sheet for your ICs, which will show you what functions they perform and how to connect them.
Which brings up a question I thought of a while back...*goes to make new post*