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Thomas comments on Global warming is a better test of irrationality that theism - Less Wrong
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Don't worry for downvotes. I downvote everything I disagree. How else would you know it? I am also frequently downvoted, and I find some pride in that.
But go back to the topic! You say it's possible that despite of the fact that T(y)<T(y-20), where CO2(y)>CO2(y-20) - the function still grows? Uhm ... I am not that certain. It is at least bumpy.
This I find particularly fishy. Half the amount of CO2 10 times! Then it will be less than 1ppm CO2 in the atmosphere. Do you expect 30 degrees cooling?
I doubt it.
But do half it 40 times. Some molecules of CO2 are still with us. Do you expect 120 degrees cooler outside?
Don't do that.
Let's say an object is tethered to a wall by a coil of metal. A displacement of 1 cm gives a 1 N restoring force and a displacement of 2 cm gives a 2 N restoring force. It's foolish to think the coil would give a 100000 N restoring force at 1 km; it would just break! So we can't treat the coil as a spring in the small displacement regime?
The elasticity of metals are all well known. You can get the data all over the internet.
Where can I get the same data about this "sensitivity of doubling" for various gasses and atmospheres? Also regarding to planet's star?
Suppose someone said that world population is increasing by one billion per decade. And then you said, that's impossible, because the population would have been negative in 1900... All that "argument" shows is that the trend in question couldn't be true across all time; it doesn't refute the contention that that is what's happening now. In the same way, what you just said only shows that doubling the CO2 can't lead to 3 degrees increase for all possible concentrations of CO2. It has no bearing on whether this is true for the range of concentrations that matter to us.
But in fact that is the expected result. "... the planet's effective temperature ... is about −18 °C, about 33°C below the actual surface temperature of about 14 °C. The mechanism that produces this difference between the actual surface temperature and the effective temperature is due to the atmosphere and is known as the greenhouse effect." CO2 is directly responsible for only about a third of this, water vapor does most of the job, but the water vapor is the dependent variable.
Reality is complicated because water vapor causes cooling (as clouds) as well as warming, and there is room to argue about how these effects vary, but the arguments you present aren't enough.