But while a correlation does not prove cause-and-effect, a lack of correlation -- or more properly, a negative correlation -- can DISprove cause-and-effect.
Only in a closed system, unless you presume to have knowledge of the grand unifying theorem, and can thus explain every action in the universe.
Example: something -- all evidence points to one animal -- has been killing your chickens. You suspect the neighbor's dog. So you start keeping tabs on when the dog is let out, and when it is in the house. It turns out, after examination, that whatever it is has been killing your chickens when the dog was locked up in the house. There is no dispute... it is indisputable that the dog wasn't there when the chickens died. This negative correlation between the dog being out and dead chickens has DISproved your theory that the dog was killing the chickens.
Or, which is the recurring problem of the debate, there are two dogs, meaning that while your specific dog didn't kill the chickens, the biological family dog (Canis I believe) is responsible for the increased chicken mortality in the area. This is actually the same example as you first provide, with the rum and minister, except you have obfuscated the scenario.
It gets a bit more complicated when the numbers go up but the same principle still holds. If your theory is that X causes Y, and you find a negative correlation, for example X goes up while Y goes down, you have DISproved that X causes Y. Otherwise, barring other outside influences, you would have (no dispute) observed that Y went up as X went up. Anything else contradicts your theory.
I like how "barring other outside influences" is mentioned only in passing here, while it is considered the key disrupting factor in scientific statistical analysis, something a lot of very smart people spend a lot of time on accounting for and avoiding.
And X going up with Y going down only works when X is the entire environment. If X is merely a part of the environment (as in both of your examples) it proves that X is 1: negatively correlated to Y, or 2: X is not correlated to Y but something(s) else is, or 3: X is correlated to Y while something else is stronger negatively correlated to Y. Given that these three points can be proven without any analysis, it does not seem the statistical addition shed much light on the facts.
And in the gun-control debate, we have in fact had ample time and opportunity to control for other factors. And it is extremely important to note that try as we might, we have found no other causal factors that apply to the situation. Yet even so, as X (per-capita gun ownership and frequency of carry) has gone up, Y (violent crime of all sorts) has continued to go down. Therefore: X does not cause Y. Q.E.D.
I love this. Astra Zeneca, Pfizer, Merck, GSK, and many, many other pharmaceutical companies are spending billions and billions of dollars on trying to control for all the factors of a single human being, and yet are unable to do so for approximately 19 of every 20 candidates that go to phase 3 trials. And here, all along, you ( though stated as "we" by which I guess you are referring to other paid members of NRA) have somehow managed to control for all factors of every human being in the United States. That is an impressive feat. Even more impressive, you have managed to reduce this incredible superhuman complexity to just two features, X and Y. Not even FOX News can boil the world down so succinctly. Well done.
It isn't an opinion. It's as scientific as it gets.
I shall leave the refutation of this part as an exercise to the reader.
