I thought the argument was about the effect on temperature from the various absorption spectra of different chemicals as their ratios are changed within a volume. The basics of it are pretty solid. We know that there is an effect because that is dictated by the physics. You can even test it at home with a few sealed fishtanks.

The government (at least, the US government) does not want to push this agenda. One key data point dating back to G.W. Bush is, NASA/NOAA actually built a satellite to measure one of the most important unknown variables in Earth's climate ("albedo" -- basically how "white" the surface of the Earth is when seen from space) but it has been stuck in a warehouse because the US administrations didn't want to launch it. It is finally getting to launch at the end of this month; but it was completed in 2001.

Individual scientists might be able to "prove" arbitrary points in the sense of baffling people with things they don't understand until they accept the scientist as authority. They could perhaps even "prove" in the sense of cooking up statistics to support an arbitrary point. But we're not talking about an individual scientist talking to an individual person.

What's instead happened is that a bunch of disparate scientists for disparate reasons have called attention to one central historical question (the trend of global average temperature); and a whole scientific research program has been created to investigate it involving lots of scientists at lots of institutions in lots of countries. That research program has confidently been able to conclude that those early scientists were wrong -- they predicted global cooling precipitating a new impending ice age -- and that instead we are going in precisely the opposite direction, and that the major causal force emerging in the models is a forcing term from CO2-emissions which have spiked dramatically.

This research program has also been able to identify several risk factors for runaway warming, which would be just as bad as runaway cooling. Slight warming and slight cooling are also both bad, mostly because the Earth is huge. So, I'm from a physics background and the basic problem is what we'd call "relaxation to equilibrium." If you imagine a massive boulder which is stationary and you dig a hole next to it so that it rolls somewhat to the side, there are a couple of concerns. Only one of those is whether there's an appreciable slope which will cause the boulder to roll uncontrollably downhill -- the "runaway warming" idea. But you can still get injured even if it doesn't roll very far, if you're in its way when it rolls -- and the bigger the boulder, the more momentum it will have to do that damage.

That's sort of a two-minute way to understand why slight warming/cooling can enhance tropical storms and send parts of the polar vortex out over the US midwest: the Earth has a new equilibrium point for all sorts of climate variables which were functions of the global temperature, and as things relax to this new equilibrium they will potentially cause large weather upsets simply because the Earth is big.