What W5LZ is trying to say is that if you were to compare a perfect omnidirectional antenna with one that is directional (assuming they are both being fed the exact same amount of power) you see that as the directional antenna's pattern gets stronger in one direction, there is a lack of strength in other directions. If that wasn't the case, the antenna wouldn't be directional.
I believe this has to do with basic conservation of energy. If you were to feed a theoretical, perfect, lossless, omnidirectional antenna in free space 100W for example, the field generated consists of that 100W of power spread throughout the field, concentrated strongest at the antenna and growing weaker with distance.
If you take a theoretical, perfect, lossless, directional antenna or array and feed it 100W, the field it produces will also consist of that same 100W, only now it will be concentrated in some particular direction more than others. The only way for more energy to be focused in one direction is to take energy away from the other directions. Anything else violates the laws of physics because it means energy is appearing spontaneously from nothing. There's only so much energy to go around, 100W worth for this discussion, so to put more in one direction means you must take away from the others. This also applies to reception of signals, only then we're talking about sensitivity in a certain direction rather than outgoing energy.
I've spoken previously on this forum about the fact that I work with lasers, and lasers are actually a great example of what we're discussing here.
In a laser, the reason that the beam can be so powerful is because it's concentrated in a very small area and all in one precise direction. In laser terms, 500mW is quite a bit of power, easily capable of burning flesh, setting fire to wood, and engraving plastics and many other solid substances (that's 1/2 of a Watt). If you took the energy from a 60W light bulb and found a way to concentrate it in one precise direction like a laser beam it would burn a hole through steel with no difficulty. This equates to "gain". In order for the energy to be concentrated in one direction, energy must be removed from the other directions and redirected.
I agree with eagle's perfect idea. However, the co-phase setup that Grim is talking about, is typically far from perfect at best. For one thing, if you had a co-phase setup installed with everything as perfect as possible, you would still have some increased loss...due to transformation by the harness, and a very low starting impedance at the antennas. IMO, this would tend to minimize any improved gain advantage that others have noted.
I didn't fully understand Grim's analogy earlier using three meters, but if I had understood, I would probably disagree...even without my having tested such a setup myself. Maybe Grim would describe for us what he did with a little more detail?
IMO, theoretically a well designed and installed co-phase setup should produce similar results, as some described here. There may be some improved gain in one direction, but the process also adds more loss with the co-phase type system. This tends to equalize results...as compared to a single antenna that is also designed and installed well.