It doesn't make sense to ask for a "formal argument" about something like that. You can't write a closed form equation for a system that big, with that many things going on. Any "pure logic" argument you use can't be validated; you have no way to know you've taken everything into account. Trying to do that sort of thing is what got Aristotle into trouble.
There are large simulation models that act a lot like the climate. They are of course validated against the actual climate, and have been improving. If you introduce more heat into those models, you get more extreme weather. Actually the real climate seems to be more sensitive to change than the models, probably because people were "conservatively" tuning the models to always act like the observed climate regardless of "minor" perturbations.
I don't think simulation is "formal". I do think it's pretty convincing.
Is the entropy of our weather going up as a result of climate change
I am not sure that's a well-formed question. Chaos and extreme excursions aren't really entropy. But throwing in a lot of heat definitely increases the number of accessible states...
are there other metrics that point to increased extreme temperatures?
Aren't the actual measured extreme temperatures a "metric of extreme temperatures"? What more do you want, exactly?
Also, why are you asking about this on Less Wrong rather than someplace where there are actual experts?
Also, why are you asking about this on Less Wrong rather than someplace where there are actual experts?
Do you have a recommendation for a venue where to ask the question?
That said, I do expect LessWrong to be a place where there are people who thought critically about claims and while climate change isn't widely seen as an existential risk in this community we do have enough people who think it matters enough to have a clear view of the state of the evidence.
I don't think simulation is "formal". I do think it's pretty convincing.
I don't see why rigorous si...
I think the analysis basically derives from modeling weather as something like a normal distribution around a mean (climate). If the mean of the distribution increases, the probability mass on a fixed region above the mean increases, often dramatically. See this post for a deeper dive on this phenomenon: https://putanumonit.com/2015/11/10/003-soccer1/
I don’t think anyone is arguing that the variance of temperatures is increasing, or at least that’s not what people usually mean. There are second order effects with things like shifting El Niño, but no one I’ve heard thinks they’re going to make rare cold weather events more likely to a dangerous extent.
The title of the question is "extreme weather events" which (in my mind) makes me think of things like hurricanes and droughts and floods. The text of the question is more specifically "extreme temperatures". The latter seems straightforward: as in Not Relevant's answer, higher mean temperature leads (other things equal) to extreme (on an absolute not relative scale) high temperature events happening more often, and to extreme low temperature events happening less often, and I think that's what we see and that's what everyone expects. The former topic (have hurricanes, droughts, floods, etc. been getting more frequent / extreme? are they expected to in the future?) is a more interesting question that's not super-obvious from first principles. I recall that it's a controversial topic within the field, or at least it used to be, I haven't been following the debate.
I think some of this also hinges on the definition of an extreme event. For instance I don't think of it as a drought when a desert doesn't get any rain, that's just normal weather for a desert. I think climate change is expected to alter the pattern of rainfall, so there could be "events which are normal where they used to happen, but extreme in the places they now happen".
In popular press, I frequently read claims according to which climate change not only leads to increased average temperatures but also claims that it leads to more extreme temperatures. Those claims are however usually not made in a formal mathematical way.
Have people made those claims in a formal way that holds up to scrutiny? I could imagine measures such as entropy to measure how extreme weather happens to be. Is the entropy of our weather going up as a result of climate change or are there other metrics that point to increased extreme temperatures?