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most people cannot really grasp what the phrase 'emergency' means when we're talking about the climate
Probably because "emergency" in this context doesn't have the ordinary plain language meaning they are used to. I think the activists who have decided to use "emergency" and "crisis" as desperate emotive appeals to action may have misjudged the effect. If people think they are being lied to and manipulated, they are likely to dismiss the valid points being made as well as the overblown propaganda.
gbj_tester-
I agree. The obvious problem is that there has been concern about anthropogenic global warming for at least six decades (probably more), always accompanied by dire warnings that either haven't come true as acutely as was projected at the time, or that are completely removed from people's everyday reality and, as you say, direct experience. Obviously, manipulation of public opinion by powerful fossil fuel-linked interests plays a major role, but those other factors, too. (I do think action should have been taken decades ago and I don't particularly relish the coming test of humanity's skill at brinkmanship.)
Oliver Schick
I think the real problem is that most people (and I'll include myself in this) cannot really grasp what the phrase 'emergency' means when we're talking about the climate. Humans are pretty poor at reacting to things until they are directly affected by them.