Maybe "logic board" is not the best term to use, although I thought that's what the mechanic said! The Haynes manual shows a "Heater and A/C Control" board in the electrical schematic that covers the A/C system, so that must be what he was referring to. It has some solid state electronic components in it, so in a sense it is a logic board in that it processes user inputs and is supposed to provide certain outputs if it is working properly. Apparently these things are pretty expensive; it might be possible to get one from a salvage yard more cheaply but I think I'm going to go the cheap and redneck route and wire in a switch to control the A/C compressor.