This story first appeared in the New York Times on February 1, 2015.

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In the churning over the refusal of some parents to immunize their children against certain diseases, a venerable Latin phrase may prove useful: Post hoc, ergo propter hoc. It means, “After this, therefore because of this.” In plainer language: Event B follows Event A, so B must be the direct result of A. It is a classic fallacy in logic.

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It is also a trap into which many Americans have fallen. That is the consensus among health professionals trying to contain recent spurts of infectious diseases that they had believed were forever in the country’s rearview mirror. They worry that too many people are not getting their children vaccinated, out of a conviction that inoculations are risky.

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Some parents feel certain that vaccines can lead to autism, if only because there have been instances when a child got a shot and then became autistic. Post hoc, ergo propter hoc. Making that connection between the two events, most health experts say, is as fallacious in the world of medicine as it is in the field of logic.

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An outbreak of measles several weeks ago at Disneyland in Southern California focused minds and deepened concerns. It was as if the amusement park had become the tragic kingdom. Dozens of measles cases have spread across California. Arizona and other nearby states reported their own eruptions of this nasty illness, which officialdom had pronounced essentially eradicated in this country as recently as 2000.

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But it is back. In 2014, there were 644 cases in 27 states, according to the Centers for Disease Control and Prevention. Should the pace set in January continue, the numbers could go still higher in 2015. While no one is known to have died in the new outbreaks, the lethal possibilities cannot be shrugged off. If the past is a guide, one or two of every 1,000 infected people will not survive.

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To explore how matters reached this pass, Retro Report, a series of video documentaries studying major news stories of the past and their consequences, offers this special episode. It turns on a seminal moment in anti-vaccination resistance. This was an announcement in 1998 by a British doctor who said he had found a relationship between the M.M.R. vaccine—measles, mumps, rubella—and the onset of autism.

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Typically, the M.M.R. shot is given to infants at about 12 months and again at age 5 or 6. This doctor, Andrew Wakefield, wrote that his study of 12 children showed that the three vaccines taken together could alter immune systems, causing intestinal woes that then reach, and damage, the brain. In fairly short order, his findings were widely rejected as—not to put too fine a point on it—bunk. Dozens of epidemiological studies found no merit to his work, which was based on a tiny sample. The British Medical Journal went so far as to call his research “fraudulent.” The British journal Lancet, which originally published Dr. Wakefield’s paper, retracted it. The British medical authorities stripped him of his license.

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Nonetheless, despite his being held in disgrace, the vaccine-autism link has continued to be accepted on faith by some. Among the more prominently outspoken is Jenny McCarthy, a former television host and Playboy Playmate, who has linked her son’s autism to his vaccination: He got the shot, and then he was not O.K. Post hoc, etc.

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Steadily, as time passed, clusters of resistance to inoculation bubbled up. While the nationwide rate of vaccination against childhood diseases has stayed at 90 percent or higher, the percentage in some parts of the country has fallen well below that mark. Often enough, these are places whose residents tend to be well off and well educated, with parents seeking exemptions from vaccinations for religious or other personal reasons.

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At the heart of the matter is a concept known as herd immunity. It means that the overall national rate of vaccination is not the only significant gauge. The rate in each community must also be kept high to ensure that pretty much everyone will be protected against sudden disease, including those who have not been immunized. A solid display of herd immunity reduces the likelihood in a given city or town that an infected person will even brush up against, let alone endanger, someone who could be vulnerable, like a 9-year-old whose parents rejected inoculations, or a baby too young for the M.M.R. shot. Health professionals say that a vaccination rate of about 95 percent is needed to effectively protect a community. Fall much below that level and trouble can begin.

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Mass vaccinations have been described by the C.D.C. as among the “10 great public health achievements” of the 20th century, one that had prevented tens of thousands of deaths in the United States. Yet diseases once presumed to have been kept reasonably in check are bouncing back. Whooping cough is one example. Measles draws especially close attention because it is highly infectious. Someone who has it can sneeze in a room, and the virus will linger in the air for two hours. Any unvaccinated person who enters that room risks becoming infected and, of course, can then spread it further. Disneyland proved a case in point. The measles outbreak there showed that it is indeed a small world, after all.

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What motivates vaccine-averse parents? One factor may be the very success of the vaccines. Several generations of Americans lack their parents’ and grandparents’ visceral fear of polio, for example. For those people, “you might as well be protecting against aliens—these are things they’ve never seen,” said Seth Mnookin, who teaches science writing at the Massachusetts Institute of Technology and is the author of The Panic Virus, a 2011 book on vaccinations and their opponents.

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Mr. Mnookin, interviewed by Retro Report, said skepticism about inoculations is “one of those issues that seem to grab people across the political spectrum.” It goes arm in arm with a pervasive mistrust of many national institutions: the government that says vaccinations are essential, news organizations that echo the point, pharmaceutical companies that make money on vaccines, scientists who have hardly been shown to be error-free.

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Then, too, Mr. Mnookin said, scientists don’t always do themselves favors in their choice of language. They tend to shun absolutes, and lean more toward constructions on the order of: There is no vaccine-autism link “to the best of our knowledge” or “as far as we know.” Those sorts of qualifiers leave room for doubters to question how much the lab guys do, in fact, know.

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Thus far, the Disneyland measles outbreak has failed to deter the more fervent anti-vaccine skeptics. “Hype.” That is how the flurry of concern in California and elsewhere was described by Barbara Loe Fisher, president of the National Vaccine Information Center, an organization that takes a dim view of vaccinations. The hype, Ms. Fisher said in a Jan. 28 post on her group’s website, “has more to do with covering up vaccine failures and propping up the dissolving myth of vaccine acquired herd immunity than it does about protecting the public health.” Clearly, she remained untroubled that most health professionals regard her views as belonging somewhere in Fantasyland.

AT ISSUE: SOURCES FOR DEVELOPING A CAUSE-AND-EFFECT ARGUMENT