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That was some Twitter exchange Monday between Texas Republican Senator Ted Cruz and new Chicago Mayor Lori Lightfoot! Not only was it scorching (especially on Lightfoot’s part), but it was crucially important for clarifying a major problem with how Americans have been debating the issue of gun violence and what to do about it.

The problem concerns how to classify the kind of gun violence that has plagued low-income neighborhoods in big cities like Chicago for so long, and therefore how best to reduce it, and here’s why Cruz emerges as a clear winner.

Right after a Labor Day weekend in the Windy City that saw 41 shootings that resulted in seven deaths, Cruz took to social media to tweet

Gun control doesn’t work. Look at Chicago. Disarming law-abiding citizens isn’t the answer. Stopping violent criminals—prosecuting & getting them off the street—BEFORE they commit more violent crimes is the most effective way to reduce murder rates. Let’s protect our citizens.”

Lightfoot was incensed. Her response:

60% of illegal firearms recovered in Chicago come from outside IL—mostly from states dominated by coward Republicans like you who refuse to enact commonsense gun legislation. Keep our name out of your mouth.”

And she backed up her claim with a graphic.  (See this post for both tweets.)

But here‘s what Lightfoot overlooked: Let’s grant her apparent assumption that the share of these out-of-state guns that have been seized in the city roughly matches their share of Chicago’s total illegal gun supply. Let’s also grant her apparent assumption that better gun laws could actually reduce this supply meaningfully. Even so, it would still be a humongous stretch to conclude that Chicago would become significantly more peaceful.

Just look at these numbers: Chicago’s 2.71 million population came to just over 21 percent of the Illinois total as of last year. But according to the latest (2016) figures, Chicago’s homicide rate of 27.7 per 100,000 residents was 355 percent higher than Illinois’ homicide rate of 7.8 percent per 100,000 residents.

Even more striking: In 2016, 997 murders took place in Illinois that year. Of those, more than 76 percent (762) occurred in Chicago. That is, the number of murders in the city was nearly four times greater than what you’d expect if such violent crimes happened uniformly throughout the state. If out-of-state guns were the main problem, you’d expect their effects to be spread much more evenly, if not perfectly evenly.

What the Lightfoot-Cruz debate boils down to is the former’s claim that Chicago’s main gun violence-related problem has relatively little to do with Chicago, and the latter’s claim that something about Chicago matters critically – including in terms of attracting the out-of-state guns responsible for such an outsized share of Illinois murders. The data not only clearly vindicate Cruz. They powerfully remind that the term “gun violence” nowadays is too often used in America to describe a wide variety of behaviors, and that many of them aren’t remotely likely to be solved solely or mainly with tighter gun laws.