Michael and Angela have just turned fifty-five. They know two people who have died in the past few years—one from cancer, another in a car accident. It occurs to them that they should make a plan for their kids. They have some money in the bank. Suppose they were both killed in a plane crash—what would happen to it?

They have four children, who range in age from their late teens to their late twenties. Chloe, the oldest, is a math wiz with a coding job at Google; she hopes to start her own company soon. Will, who has a degree in social work, is paying off his student debt while working at a halfway house for recovering addicts. The twins, James and Alexis, are both in college. James, a perpetually stoned underachiever, is convinced that he can make it as a YouTuber. (He’s already been suspended twice, for on-campus pranks.) Alexis, who hopes to become a poet, has a congenital condition that could leave her blind by middle age.

At first, Michael and Angela plan to divide their money equally. Then they start to think about it. Chloe is on the fast track to remunerative Silicon Valley success; Will is burdened by debt in his quest to help the vulnerable. If James were to come into an inheritance, he’d likely grow even lazier, spending it on streetwear and edibles; Alexis, with her medical situation, might need help later in life. Maybe, Michael and Angela think, it doesn’t make sense to divide the money into equal portions after all. Something more sophisticated might be required. What matters to them is that their children flourish equally, and this might mean giving the kids unequal amounts—an unappealing prospect.

The philosopher Ronald Dworkin considered this type of parental conundrum in an essay called “What Is Equality?,” from 1981. The parents in such a family, he wrote, confront a trade-off between two worthy egalitarian goals. One goal, “equality of resources,” might be achieved by dividing the inheritance evenly, but it has the downside of failing to recognize important differences among the parties involved. Another goal, “equality of welfare,” tries to take account of those differences by means of twisty calculations. Take the first path, and you willfully ignore meaningful facts about your children. Take the second, and you risk dividing the inheritance both unevenly and incorrectly.

In 2014, the Pew Research Center asked Americans to rank the “greatest dangers in the world.” A plurality put inequality first, ahead of “religious and ethnic hatred,” nuclear weapons, and environmental degradation. And yet people don’t agree about what, exactly, “equality” means. In the past year, for example, New York City residents have found themselves in a debate over the city’s élite public high schools, such as Stuyvesant and Bronx Science. Some ethnicities are vastly overrepresented at the schools, while others are dramatically underrepresented. What to do? One side argues that the city should guarantee procedural equality: it should insure that all students and families are equally informed about and encouraged to study for the entrance exam. The other side argues for a more direct, representation-based form of equality: it would jettison the exam, adopting a new admissions system designed to produce student bodies reflective of the city’s demography. Both groups pursue worthy egalitarian goals, but each approach runs against the other. Because people and their circumstances differ, there is, Dworkin writes, a trade-off between treating people equally and treating them “as equals.”

The complexities of egalitarianism are especially frustrating because inequalities are so easy to grasp. C.E.O.s, on average, make almost three hundred times what their employees make; billionaire donors shape our politics; automation favors owners over workers; urban economies grow while rural areas stagnate; the best health care goes to the richest. Across the political spectrum, we grieve the loss of what Alexis de Tocqueville called the “general equality of conditions,” which, with the grievous exception of slavery, once shaped American society. It’s not just about money. Tocqueville, writing in 1835, noted that our “ordinary practices of life” were egalitarian, too: we behaved as if there weren’t many differences among us. Today, there are “premiere” lines for popcorn at the movies and five tiers of Uber; we still struggle to address obvious inequalities of all kinds based on race, gender, sexual orientation, and other aspects of identity. Inequality is everywhere, and unignorable. We’ve diagnosed the disease. Why can’t we agree on a cure?

In January of 2015, Jeremy Waldron, a political philosopher at New York University’s School of Law, delivered a series of lectures at the University of Edinburgh on the fundamental nature of human equality. He began by provoking his audience. “Look around you,” he said, “and look at the differences between you.” The crowd included the old and the young, men and women, the beautiful and the ugly, the rich and the poor, the healthy and the infirm, the high-status and the low. In theory, Waldron said, the audience could contain “soldiers as well as civilians, fugitives and convicts as well as law-abiding citizens, homeless people as well as property owners”—even “bankrupts, infants, lunatics,” all with different legal rights.

In a book based on those lectures, “One Another’s Equals: The Basis of Human Equality,” Waldron points out that people are also marked by differences of skill, experience, creativity, and virtue. Given such consequential differences, he asks, in what sense are people “equal”? Waldron believes in our fundamental equality; as a philosopher, however, he wants to know why he believes in it.

According to the Declaration of Independence, it is “self-evident” that all men are created equal. But, from a certain perspective, it’s our inequality that’s self-evident. A decade ago, the writer Deborah Solomon asked Donald Trump what he thought of the idea that “all men are created equal.” “It’s not true,” Trump reportedly said. “Some people are born very smart. Some people are born not so smart. Some people are born very beautiful, and some people are not, so you can’t say they’re all created equal.” Trump acknowledged that everyone is entitled to equal treatment under the law but concluded that “All men are created equal” is “a very confusing phrase to a lot of people.” More than twenty per cent of Americans, according to a 2015 poll, agree: they believe that the statement “All men are created equal” is false.

In Waldron’s view, though, it’s not a binary choice; it’s possible to see people as equal and unequal simultaneously. A society can sort its members into various categories—lawful and criminal, brilliant and not—while also allowing some principle of basic equality to circumscribe its judgments and, in some contexts, override them. Egalitarians like Dworkin and Waldron call this principle “deep equality.” It’s because of deep equality that even those people who acquire additional, justified worth through their actions—heroes, senators, pop stars—can still be considered fundamentally no better than anyone else. By the same token, Waldron says, deep equality insures that even the most heinous murderer can be seen as a member of the human race, “with all the worth and status that this implies.” Deep equality—among other principles—ought to tell us that it’s wrong to sequester the small children of migrants in squalid prisons, whatever their legal status. Waldron wants to find its source.

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In the course of his search, he explores centuries of intellectual history. Many thinkers, from Cicero to Locke, have argued that our ability to reason is what makes us equals. (But isn’t this ability itself unequally distributed?) Other thinkers, including Immanuel Kant, have cited our moral sense. (But doesn’t this restrict equality to the virtuous?) Some philosophers, such as Jeremy Bentham, have suggested that it’s our capacity to suffer that equalizes us. (But then, many animals suffer, too.) Others have nominated our capacity to love. (But what about selfish, hard-hearted people?) It would be helpful, on a practical level, if there were a well-defined basis for our deep equality. Such a basis might guide our thinking. If deep equality turned out to be based on our ability to suffer, for example, then Michael and Angela might feel better about giving their daughter Alexis, who risks blindness, more money than her siblings. But Waldron finds none of these arguments totally persuasive.