Americans desire two seemingly contradictory qualities in their presidents: They want them to be sharp enough to make the world’s most difficult decisions, but old enough to have the life experience befitting the nation’s highest office.

President Trump’s detractors have raised concerns that he doesn’t quite fit that first category. There are already some publicly available reasons to worry about Trump’s health: Besides being the oldest president ever elected, he’s overweight, appears to subsist on a junk-food-heavy diet, and avoids exercise.

Pointing to the confusing grammar of Trumpian utterances like, “... there is no collusion between certainly myself and my campaign, but I can always speak for myself—and the Russians, zero,” STAT News asked neurologists to review Trump’s speech. They found he is now using simpler words and sentences than he did in the 1980s, which could be a sign of cognitive decline.

The latest speculation that Trump’s mind might be deteriorating came this week, when the president appeared to slur his words during a speech about Israel. (It was dry mouth, his spokesman said.)

At the daily White House press briefing Thursday, press secretary Sarah Huckabee Sanders said Trump “does have a physical scheduled for the first part of next year, the full physical that most presidents go through. That will take place at Walter Reed [National Military Medical Center], and those records will be released by the doctor following that.”

If it turns out that Trump’s health is less than “astonishingly excellent,” as his doctor claimed during the campaign, he’d be in good company. Nearly half the presidents have had significant illnesses while in office, according to the Los Angeles Times, and one study suggested that nearly a third of the country’s first 37 presidents—through Nixon—had a mental illness.

Some have speculated that President Reagan began showing signs of Alzheimer’s long before he left office. One study found that, with time, he began to use more nonspecific and repetitive words. Still, he emerged from his final physical shouting, “Clean bill of health!” Some researchers also think the aftereffects of anesthesia from cancer surgery hampered his decision-making abilities in the lead-up to the 1985 Iran-Contra scandal.

It’s true that recent presidents and candidates have shared their health statuses publicly, but the level of the disclosures has varied. President Obama’s last physical listed his vital statistics and noted that he chews nicotine gum on occasion.

In 1992, The New York Times described Bill Clinton as “less forthcoming about his health than any presidential nominee in the last 20 years” because, while three doctors had written letters attesting to his health, he didn’t give interviews on the topic and declined to make his doctor available to reporters.

Meanwhile, a much more detailed 2005 Times story noted that President Bush had recently lost eight pounds, and that when he “ran for 26 minutes and 20 seconds, his heart rate reached 183 beats a minute, showing no abnormalities.”

Then again, “we’ve had very sick men in office, and no one has known,” said Rose McDermott, a Brown University professor who has researched the health—or lack thereof—of past presidents. They were helped along by galling levels of secrecy.

“Grover Cleveland hid his surgery for jaw cancer, going so far as to have the operation done on a boat in New York Harbor,” said Nicole Hemmer, a media-history professor at the University of Virginia. “And of course the public was not informed of the full extent of Woodrow Wilson's debilitating stroke in 1919.”

In 1944, several doctors claimed publicly that President Roosevelt, who was running for reelection, was in good health, even as one of them privately said he doubted the president would live another four years. Indeed, he died just a few months later.

President Kennedy had a well-concealed case of Addison’s disease, a hormonal disorder, as McDermott has written. He treated it, in part, with injections of amphetamines and steroids from Max Jacobson, a doctor whose nickname was “Dr. Feelgood.” His regular endocrinologist cautioned him that Jacobson’s formulas are “not for responsible individuals who at any split second may have to decide the fate of the universe.”

It was only in the 1970s, after Watergate, that there was a sense the presidency should be more transparent, including when it comes to health matters, said John Rogan, a law professor at Fordham University.

Still, there’s no legal requirement that the president get a physical, or that the physical include neurological exams. There’s no requirement that he disclose all of the physical’s contents to the public. Presidents, like everyone else, are protected by medical-privacy laws.

Past presidents’ medical exams have sometimes reported that they are “fit for duty.” That’s a term of art—there’s no quantifiable bar of good health that “fit for duty” implies.

Even if the president’s doctor told him he was suffering from a condition that might impair his functioning in office, like the early stages of dementia, it’s unclear what would happen next. The White House doctor would probably tell an aide or advisor about Trump’s diagnosis, but he also might not. If such a diagnosis were publicly disclosed, the Twenty-Fifth Amendment leaves it to the Cabinet and Vice President Pence to decide whether Trump is unfit to serve. The Cabinet could opt to do nothing. If the Cabinet determines that he’s incapacitated, Trump could contest its finding. If Congress fails to approve the Cabinet’s decision​, it would put the president back in power​.

A Fordham Law School clinic recently recommended adding a full-time psychologist to the White House Medical Unit, the medical personnel who already serve the president. But, the article notes, presidents and their staff would probably resist that move, since it would invite rumors that the president needs psychological help.

Others have recommended creating a panel of doctors who would evaluate the president every year. But who would sit on the panel? Two Republicans and two Democrats, and you might have a stalemate as to whether the president’s verbal slip means he is unfit. Two Republicans and one Democrat, or vice versa, and it would look biased.

“So, even if he had Alzheimer’s, we might never know?” I asked McDermott.

“We’d probably know,” she said. “After he dies.”