Backradiation resolved?

Why is it that the greenhouse discussion doesn't seem to get anywhere? I believe that all parties are partly to blame for this. The people who could give an authoritative account of the effect are hiding and instead leaving the job to politicians like Al Gore and Lord Monckton. Since neither of these two know what the greenhouse effect is, they redefine it to be the Thyndall gas effect. However, in a recent exchange, Lindzen now claims that the Thyndall gas effect of Al Gore is not quite the same Thyndall gas effect as that Lord Monckton supports. I agree with Lindzen on two points though, the discussion has become quite bizarre and I think greenhouse skeptics should take much more care checking the relevant sources before confronting climatologists. Two of the major obstacles that I see in front of me are the following:

1. If the greenhouse hypothesis is not true then the "atmospheric problem" remains unsolved.

2. People have a very hard time managing "back-radiation".

The first obstacle could be dealt with principles well known to lawyers but apparently unknown to scientists, namely that it is the responsibility of the prosecutor to prove the guilt of the defendant, not the other way around. Hence, you can be sharp when scrutinizing the greenhouse hypothesis while beeing tentative when presenting your own ideas. There are promising observations and alternative theories, but they are not complete and we shouldn't pretend that they are. 

The second obstacle possibly stems from the frustration and numbness that can follow from having watched too many cartoons. I think, however, that it could be tackled with a little bit of mathematics, but not too much. There is a problem facing the greenhouse skeptics: 

If there is indeed some kind of thermodynamic "atmospheric effect", is it then completely out of the question that part of this effect is carried out by IR-radiation?

Let me restate the simplest mathematical representation of the basic radiative equilibrium that I have found in the litterature:

The first of these expresses the ground temperature as a function of solar heating Fs and an absorption coefficient a0. The second equation describes the lapse rate, but notice that the latter formula does not depend on Fs. Let me now constrast this with the classic heat equation:

Ok, so what has this got to do with back-radiation? Well, the heat equation can be motivated with a "back-radiation" argument. Imagine that you have two plates radiating against each other proportional to its respective temperature with proportionality constant k. Then the net heat transfer from hot to cold will be simply the temperature difference times k. Hence, when you discuss back-radiation you must specify what you are actually aiming for. Is it that the heat equation is wrong, that the derivation of the heat equation is wrong or is it that the boundary condition (first formula) is misapplied. I belong to the latter category. Instead of fixing the surface temperature at a particular value (also known as a Dirichlet boundary condition) it seems to me much more natural to impose an open boundary condition at the surface (also known as a Neumann boundary condition).

Is this what the back-radiation debate is really about?