Earlier quoted context omitted.
Well yeah, but they have to remove all the other factors like heat, wind and rain, which are correlated to air pollution. There is likely less violent crimes on rainy days, just because more people stay home. To do this properly, they would have to do a controlled experiment with people in rooms with or without air pollution. And measure how sanguine they get or something of that nature.
They obvious control for these factors. They actually report that the effects are stronger at lower temperature.
Sometimes you can use common sense to infer a causal relationship (violent crime linked to emergency room visits), but it seems like a leap here, especially as they're on a bit of a fishing expedition with multiple types of pollution and multiple types of crime.