The “paradox of the heap” seems at first like a trick, a brainteaser that must have some clever catch. But it reveals itself, as it defies easy understanding, to be a philosophical problem. You might approach it as a puzzle, only to end up devising a solution so deep that it would challenge our thinking about language, knowledge and the nature of reality. By the time of her death from brain cancer in July at 48, Delia Graff Fara, a philosopher at Princeton, had done just that.

Start with a heap of sand. If you remove a single grain, it remains a heap. Repeat this process enough times, however, and you have a heap of sand that contains, say, one grain. This is absurd: One grain is not a heap. Something has gone wrong, but it is not obvious what. Either there is a precise number of grains at which point a heap becomes a nonheap, or there is no such thing as a heap, or classical logic is flawed (perhaps it is only ever sort of true that something is a heap). Which bullet to bite?

This paradox, which originated with the ancient Greeks, is troubling because it is ubiquitous. It applies not just to being a heap but also to being tall, or red, or bald, or soft — or any other gradient-like property. When Fara began working on this paradox as a graduate student at the Massachusetts Institute of Technology in the 1990s, philosophers had come to view it as an instance of a larger problem: vagueness. We want to take seriously our talk of hot and cold weather, bald and full-haired men, day and night, but the boundaries that distinguish such things can seem blurry to the point of incoherence.

Fara was unsatisfied with the solutions on offer. Some philosophers argued that vagueness was a form of ignorance: that there is a precise number of grains separating a heap from a nonheap, but we don’t know what it is. Others argued that vagueness was a result of semantic indecision: that there are lots of possible things we could mean by “heap,” each of which would establish a precise number of grains for heap-hood, but we haven’t taken the trouble to specify that meaning. Still others, looking to avoid a sharp distinction between heaps and nonheaps, sought to develop nonclassical or “fuzzy” logics, which experimented with degrees of truth.

These solutions were advanced with argumentative ingenuity and technical sophistication, often in ways that minimized their oddities. But Fara, herself a skilled practitioner of logic and formal semantics, was looking for a solution that addressed a common-sense question: If there is a precise point at which a heap becomes a nonheap — and unless you want to abandon natural language or classical logic, it seems there must be — why are we so inclined to think otherwise?

Fara’s theory, which she presented in a 2000 paper called “Shifting Sands,” had an answer. She argued that vagueness was an expression of our ever-changing purposes: that there is a precise point at which a heap becomes a nonheap, but it “shifts around” as our objectives do. In fact, because the act of considering two comparable heaps accentuates their similarity, “the boundary can never be where we are looking.” No wonder we think it doesn’t exist.

Imagine that a gym teacher has hastily divided a large class of students into two groups according to height. If you enter the gym, you will have no trouble declaring one group the tall students and the other the short ones. But had you been presented with the undivided class and asked to say where the tallness boundary was, you would have despaired of an answer. Tallness is not just a matter of height, Fara concluded. As with all such properties, what gets to be tall is also shaped by our interests at a given moment.

“Shifting Sands” became an influential and highly cited paper, not least because it had implications for other areas of philosophy. It was also notable for being written by a young woman in a field that skews heavily toward men, both demographically and in terms of works cited. Concerned about such imbalances, Fara devoted considerable effort to helping women in the profession.

When it came to racial diversity, an area in which philosophy is similarly lopsided, Fara was also a champion, if more circumspect. Her mother, who raised Fara as a single parent in New York, was African-American; her father, who died when she was a child, was of Irish and Jewish descent. Because of her appearance, Fara was often assumed to be white or queried clumsily about where she was “from.” She told her husband she hoped to avoid being defined by her race. But as much as she let misperceptions slide, or answered politely that she was “born in Queens,” it was an issue she could never fully escape.

At the end of her life, Fara became interested in the philosophy of race. She wondered if her ideas about vagueness — about how seemingly independent properties of the world are infused with human interests — might be relevant. It would have been more blurry boundaries to think about.