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OK. Say, for sake of argument, that CO2 is a major contributor to global warming - is it a 90% contributor, a 60% contributor, or what? We have no idea. The models are neither accurate enough nor proof.
We have built massive hot urban megapoloises, cut down masses of trees, expanded deserts; are you saying that these have no effect? Are you saying that the million of other factors, which we are totally ignorant of, and which have changed in the last 50 years, have no effect.
The statement that CO2 causes Global warming may be good politics - it is bad science. We do NOT have enough proof.
Continental drift was a new, somewhat radical hypothesis when I was first in college. It gained credibility as it became more understood, more was learned about ocean bottoms and rift zones, and it was put to the test. Additional data supported the hypothesis and it gradually became accepted.
The way consensus developed with continental drift is similar to how it has developed with global warming. The consensus initially was very skeptical, it didn't start with a consensus. The consensus built as one test after another further supported the hypothesis and each test strengthened rather than contradicted the case.
He changed his mind, of course, and long before he died. He says the objections to continental drift were based on mechanism - how can continents move like that? When a mechanism was presented - plate tectonics - objections melted away. Mainly because there was so much evidence.
From time to time, decisions have to be made. When a paper is submitted to a scientific journal, the editorial board has to decide whether to publish it. Somebody might come up with some device that uses some unusual effects to achieve some useful function, and various potential investors have to decide whether to commit resources. A physician and patient need to decide on a course of treatment.
My point here is that science thrives on diversity and disagreement. As far as science itself goes, there is never a need to force any kind of winner-take-all final battle. It's when action is required that we get stuck making a decision, casting our vote, placing our bet, reaching a verdict, etc. But all this is outside science - really it is some kind of political process.
Nobody ever gets to rely on the level of certainty involved in a mathematical proof when making decisions about how best to act in the real world. The scientific evidence might be quite strong in support of direction A, but it isn't hard to find situations in history where it turned out that the strong scientific argument was based on fundamental errors and in fact direction B was the much better path.
If you're the person making the decision, and it's just your personal business, you get to pick the theory you like. If there seems to be a wide scientific consensus one way or the other... well, sometimes there is a fortune to be made by bucking the trend!
If, however, the decision involves lots of people, e.g. an editorial board or a legislative body, then usually the decision making process is codified to some degree, and usually involves some kind of consensus. Wasn't it the Indiana state legislature that passed a law that the ratio of a circle's circumference to its diameter was 22:7? Codified consensus doesn't have to respect mathematical proof!
"CO2 causes Global warming" may not reach the level of scientific consensus that would make you comfortable as the basis for a highly consequential decision, but I sure don't think "CO2 does not cause Global warming" has any better consensus behind it!
It's a fine pickle, really!
The whole climate or geochemical system of which CO2 forms one part - that is complex to an utterly overwhelming degree. To what extent there is a significant warming trend and to what extent human burning fossil fuels causes that... it's too politically charged for any decently clear scientific verdict to emerge, but any effective argument will have to be far more involved than your 1-2-3 syllogism.
Here's an analogy to see how your reasoning might not hold. Do you know what a trim tab is, on a ship's rudder? To get the ship to turn right, you basically turn the trim tab to the left. If the rudder were fixed to the hull, the ship would indeed slowly turn left. But because the rudder is movable, the trim tab just pushes the rudder over, and the ship turns right.
So, for example, while CO2 itself has a greenhouse effect, maybe it also affects plant growth somehow in a way that in turn affects water vapor, and that indirect effect could be much stronger that the direct greenhouse effect from the CO2. That is the kind of alternative theory about how burning fossil fuels affects the climate that climate scientists have to work through and eliminate.
It seems like the evidence, after a considerable amount of scientific work to eliminate such alternatives, does point to a connection between fossil fuel burning and a global warming trend. But the connection is much more difficult to establish than the simple syllogism you propose.
There is a danger that folks who doubt the burning-warming link will hear such a syllogism and might get the idea that the scientific argument is that flimsy, which would encourage their skepticism. Of course, a certain amount of skepticism is still appropriate. But not that much! So I would encourage you not to induce that inappropriately high level of skepticism! Which is why I am trying to help you understand the flimsiness of the syllogism.