Since the Supreme Court reinstated the death penalty in the United States, in 1976, Texas has been responsible for more than a third of the country’s executions—five hundred and thirty-eight out of a thousand four hundred and forty. The most egregious reason is the state’s unique and grudging approach in cases where the defendant claims intellectual disability.

In 2002, in Atkins v. Virginia, the Supreme Court reached the decision that, no matter how heinous the crime, an intellectually disabled person cannot be sentenced to death. Disabilities of reasoning, judgment, and control of impulses, the Court said, do not allow a person to “act with the level of moral culpability that characterizes the most serious adult criminal conduct.” Because offenders with intellectual disabilities are less blameworthy, the Court said, imposing the death penalty contributes neither to deterrence of capital crimes nor to retribution for them, and so it causes “purposeless and needless” pain and is cruel and unusual punishment.

The Court recognized that there was “serious disagreement” about which offenders were intellectually disabled. “Not all people who claim to be mentally retarded will be so impaired as to fall within the range of mentally retarded offenders about whom there is a national consensus,” the majority opinion said. (“Intellectual disability” has replaced “mental retardation” as the favored term.) The Court anticipated a variety of approaches to enforcing its prohibition, and left to the states “the task of developing appropriate ways to enforce the constitutional restriction upon its execution of sentences.”

Most states with the death penalty rely on a combination of intelligence testing and clinical assessment to confirm that a defendant has severe intellectual disabilities. In 2004, the Texas Court of Criminal Appeals, the state’s highest criminal court, created its own definition of intellectual disability, in a case called Ex Parte Jose Garcia Briseno. In the Briseno opinion, the C.C.A. said that reliance on clinical testing is “exceedingly subjective.” The court’s responsibility, it said, was “to define that level and degree of mental retardation at which a consensus of Texas citizens would agree that a person should be exempted from the death penalty.” The court decided it was possible to be intellectually disabled according to medical and scientific standards, which apply to no more than three per cent of Americans, yet not disabled enough to be exempt from execution in Texas.

The Texas approach to intellectual disability is so different from national standards that, according to the American Bar Association, the state has regularly sentenced to death “defendants with intellectual disabilities whom other jurisdictions almost certainly would have recognized as exempt.” Jordan Steiker, a professor at the University of Texas Law School, and Richard Burr, the lawyer who represented Jose Briseno before the C.C.A., estimate that Texas has executed thirty to forty people with strong claims of intellectual disability, and that between thirty and forty of the two hundred and forty-two people on the state’s death row have similarly strong claims to exemption. This week in the Washington Post, Steiker and his sister, Carol Steiker, a professor at Harvard Law School, wrote that Texas “focuses on questions that no medical professional would deem appropriate in diagnosing intellectual disability, such as whether an offender’s family and friends thought he had intellectual disability.” They continued, “Instead of relying on the same approach to intellectual disability that Texas uses in every other context (such as placement in special education or eligibility for disability benefits), the court sought to redefine the condition in the capital context so that only offenders who meet crude stereotypes about intellectual disability are shielded from execution.”

On Tuesday, the Supreme Court heard oral arguments in Moore v. Texas, about whether the state is violating the Constitution by prohibiting judges from using current medical standards in deciding whether a defendant, Bobby James Moore, is exempt from capital punishment. Moore, now fifty-seven, has been on death row for thirty-seven years for his part in a failed supermarket robbery in Houston, in which he shot and killed a sales associate. (Moore has said the shooting was accidental.)

In 1995, a federal district court granted Moore a new sentencing hearing after the court found that his lawyers had “grossly mishandled the representation of Moore and violated their oath as members of the bar with astonishing frequency” by, among other ways, failing to present any mitigating evidence, including of the defendant’s impaired mental development and functioning. In 2001, he was sentenced to death again, after a jury determined that there was not sufficient mitigating evidence to warrant a sentence of life imprisonment.

While Moore was waiting for the C.C.A. to review that sentence, the Supreme Court issued its ruling in Atkins, and Moore’s lawyer petitioned the state trial court for another hearing. In that hearing, in 2014, the judge found that Moore was intellectually disabled, and recommended that the C.C.A. grant his claim. The C.C.A. rejected that recommendation, because that court “erred by disregarding our case law” in defining intellectual disability. The appeals court said, “We conclude that, at this juncture, the legal test we established in Briseno remains adequately ‘informed by the medical community’s diagnostic framework.’ ” Last June, the Supreme Court granted Moore’s request to hear his case, which will likely determine whether Moore lives or dies, and whether many others in Texas will live or die as well.

Moore is, by any reasonable account, intellectually disabled. He failed first grade twice, and was promoted to second grade only because his school thought he should be with children close to his age. In fifth grade, as one of a group of African-American students assigned to a largely Hispanic school, he was hit in the head with a chain and a brick, which left his whole head swollen and very possibly caused a traumatic brain injury. He has that problem, according to Shawanda Williams-Anderson, a neuropsychologist, and Robert P. Borda, a clinical neuropsychologist, each of whom examined him in 2013. Borda gave Moore a Tinkertoy test in which he was instructed to “make something.” A score below seven generally equates with the inability to live independently. Borda testified about Moore, “He had a score of one, which is the lowest score I’ve ever recorded, and I’ve done a lot of testing of brain-injury people.”

Throughout elementary and middle school, Moore failed to grasp academic skills; he continued to get social promotions until he failed all his subjects in ninth grade and dropped out. His father, who was generally abusive, beat him for being “stupid.” When Moore was thirteen, he could not tell time, the days of the week, the months of the year, the seasons, or the standards of measure. He also met each of the basic requirements for intellectual disability. On seven I.Q. tests taken between 1971 and 1989, his average score was 70.66, which reflects significant cognitive impairment. (In Atkins, the Supreme Court noted that between one and three per cent of the population has an I.Q. lower than seventy-five.) Borda concluded that Moore has “a significant Intellectual and Developmental Deficiency, and by any current standards should be considered to have functioned within the Mentally Deficient (or Mentally Retarded) range for all of his teen and adult life.”

The Texas Court of Criminal Appeals is infamous for rejecting the vast majority of death-sentence appeals in the state. The best-known example is its denial, in 1995, of an appeal by Calvin Jerold Burdine on the ground of ineffective counsel, because Burdine’s lawyer had slept through long stretches of the defendant’s trial. It wasn’t clear to the C.C.A. that the lawyer was asleep during the important parts of the trial, so the court upheld Burdine’s death sentence. A few years later, when a federal district judge ordered Texas to release the defendant or give him a new trial, the judge stated the obvious: “Sleeping counsel is equivalent to no counsel at all.” The state missed the deadline for the trial, so the judge ordered Texas to release Burdine. He wrote, “Throughout its brief, the state seeks to minimize its failure to comply with established procedural rules as ‘excusable neglect.’ However, a similar procedural error by defense counsel in a capital case could result in a defendant's execution.”