backfire effect
n. The strengthening of a person‘s belief in a false idea by presenting evidence against that idea.

Example Citations:
In 2006, the political scientists Brendan Nyhan and Jason Reifler identified a phenomenon called the “backfire effect.” They showed that efforts to debunk inaccurate political information can leave people more convinced that false information is true than they would have been otherwise.
—Maggie Koerth-Baker, “Why Rational People Buy Into Conspiracy Theories,” The New York Times, May 21, 2013

Once something is added to your collection of beliefs, you protect it from harm. You do it instinctively and unconsciously when confronted with attitude-inconsistent information. Just as confirmation bias shields you when you actively seek information, the backfire effect defends you when the information seeks you, when it blindsides you.
—David McRaney, “The Backfire Effect,” You Are Not So Smart, June 10, 2011

Earliest Citation:
A similar “backfire effect” also influenced conservatives told about Bush administration assertions that tax cuts increase federal revenue. One group was offered a refutation by prominent economists that included current and former Bush administration officials. About 35 percent of conservatives told about the Bush claim believed it; 67 percent of those provided with both assertion and refutation believed that tax cuts increase revenue.
—Shankar Vedantam, “The Power of Political Misinformation,” The Washington Post, September 15, 2008

Notes:
The more general sense of backfire effect — when an attempt to influence someone or something in a particular way has the opposite effect — has been around for a while:

That leads, naturally, to a discussion of the Burger King campaign, which depends in large part on a survey purporting to show that more consumers prefer Burger King’s Whopper than McDonald’s Big Mac or Wendy’s Single....

“I think there may be a backfire effect that Burger King hadn’t counted on,” Rushlow says. “A number of people are perverse enough that they don’t like to be told what they prefer.”
—Marton Merzer, “Why so ads attack the rivals by name?,” The Miami Herald, October 18, 1982

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