Commonly referred to as “cherry-picking,” this fallacy occurs when someone uses an argument that contains some element of truth while selectively omitting important details in order to deceive or misrepresent the facts to support a false conclusion. The stronger the omitted evidence, the more grievous the fallacy. You also commit this "cherry-picking fallacy" when you search for a pattern or evidence that fits your beliefs as opposed to adjusting your beliefs to fit the evidence.
There are several great examples of this fallacy. A classic example is that of marksman drawing his "targets" after shooting, thus making it appear as though he is a good shot. Resumes are also a good example of cherry-picking, which is why employers seek additional recommendations and often require interviews. Politics are rife with half-truths to convince the unsuspecting listener: it is foolish to vote for a candidate because he says he is Christian without taking into consideration his actual behavior.
Examples:
Climate change deniers who ignore large-scale patterns of climate, biological, and ecological data in favor of outliers and anecdotes that support their own conclusions (same with anti-vaxers).
"You should not trust Peter with your children. I once saw him smack a child with his open hand." (Where did he smack him? On the back because he was choking? Context is important.)
"I'm a really good driver. In the past thirty years, I have gotten only four speeding tickets." (What other kind of tickets has he gotten? How long has he been driving?)
After being stopped for drunk driving, the inebriated driver proclaims "I only had a couple of beers" in slurred speech. (What did he drink other than beers? How big were they?)
The classic story about blind men and an elephant: Each blind man touches a different part of the elephant and reaches a different conclusion about the nature of the elephant; while each man's experience of the elephant is accurate, none of them have a full understanding of the nature of the beast. One may be touching the tail and believe that the elephant is long and thin, another may be touching the belly and say that it is round and big.
“The truth is that Ciudad Juárez, Mexico is one of the world's fastest growing cities and can boast of a young, ambitious and hard-working population, mild winters, a dry and sunny climate, low cost medical and dental care, a multitude of churches and places of worship, delicious local cuisine and a swinging nightclub scene. Taken together, all these facts clearly prove that Juarez is one of the world’s most desirable places for young families to live, work and raise a family.”
The makers of Sugarette Candy Drinks point to research showing that of the five countries where Sugarette drinks sell the most units, three of them are in the top ten healthiest countries on Earth, therefore Sugarette drinks are healthy. (When, in fact, the two ten healthiest countries also are the wealthiest and can afford these drinks. This is an example of both a half-truth fallacy and a correlation/causation fallacy.)
Tara: I hear you got a one-week deadline extension for that GIS map that was due on the Chief's desk this morning. How did you wrangle that, considering we are already a week behind schedule for submitting our team's project?
Reno: I just told him that I was putting the finishing touches on it but our computer system crashed this weekend.
Tara: Well yea, but you weren't even here at the office this weekend when it crashed. The rest of us were here working, but you had taken off to New York with your girlfriend. You didn't even work on it at all this weekend did you?
Reno: No, but the Chief didn't need to know that did he? And he had better not find out.
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