Imagine a pile of sand. A single grain is removed… then another… and another… When does the pile of sand cease to be? Maybe we’ll never know, and it doesn’t matter: get to know the sorites paradox.
Imagine yourself as a pile of sand. Carefully remove a single grain. Do you still have a lot?
Almost everyone will say yes. After all, a grain cannot, by itself, transform a heap into a “non-heap”.
Now apply the same reasoning over and over again: remove one more grain, and another, and another. At every step, we are still facing a lot. But as the number of grains is finite, the process inevitably ends in something that no longer makes sense to call a “heap”: three grains, two, one, and, finally, none.
The chain of inferences leads to an absurd result, suggesting that there is something wrong with the initial principle. But accepting that sometimes removing a single grain is enough for there to no longer be a pile seems strange. How can one grain make such a difference?
This problem has been known since ancient times sorites paradox — from the Greek sōritēs, “mount”. It’s a classic example of how vague language can generate contradictions when pushed to the limit.
If we knew the exact number of grains that a pile of sand has, the puzzle would cease to exist. The problem is that there is no such clear border.
In everyday life, this rarely bothers. We use the word based on impressions and contexts. But the vagueness is no longer just a linguistic detail: for example, if a municipality accused someone of having dumped “a pile of sand” in a public space, a fine could ultimately depend on what counts as a “heap” or not.
The discussion eventually takes on other dimensions. “When does a person begin to exist?” one might also ask. And in the opposite direction, in the process of brain death, when a person ceases to exist? Can someone become poor by losing a penny? Can you become tall by growing a millimeter? Everything that is subject to vagueness has no concrete answer.
Fuzzy logic
An influential response to the problem of vagueness has been to fuzzy logic, which rejects bivalence (which says that each statement is either true or false) and proposes a continuum of “degrees of truth”: a statement can be completely true, completely false, or somewhere in between.
Thus, as the grains are removed from the heap, the phrase “there is a heap” would progressively become “less true”without a sudden jump between truth and falsehood.
But fuzzy logic, by making the truth more flexible, ends up compromising key principles of classical logic, on which standard mathematics depends. One example is the excluded middle principle: “either there are a lot or there aren’t”. For fuzzy logic, when “there is a lot” is only half true, then “either there is a lot or there isn’t” would also be only half true, explains Timothy Williamson, who teaches Logic at the University of Oxford, no .
Imagine two piles of sand, duplicated grain by grain, one on the right and one on the left. A grain is always removed from one side and the corresponding grain from the other, maintaining perfect duplication at each stage. Under these conditions, it seems indisputable that: if there is a hill on the right, there is a hill on the left, and vice versa.
However, according to fuzzy logic, there will be a time when “there is a bunch on the right” is half true and half false, and the same on the left side. Applying the rules, the compound sentence “there is a hill on the right and there is not a hill on the left” would also end up receiving an intermediate value, as if we should be torn between accepting and rejecting it. This is, for Williamson, unacceptable, because the phrase implies a difference between the two sides, when, by hypothesis, there is no difference. Fuzzy logic fails to capture the subtleties of vagueness.
But perhaps the logic is not broken. Classical logic, with bivalence and excluded middle, is simple, powerful and widely tested. The problem would be elsewhere: in the knowledge. There may be a factual truth (an exact point at which, when you remove a grain, there is no longer a pile) without us being able to recognize what that point is. The obstacle is epistemological: we don’t know where the border is, even if it exists.
Vague words like “lot” are used so elastically that trying to fix their limits with precision may not have a solid basis. And although language is a human construct, that doesn’t mean it’s transparent: we often know there’s a lot; we often know that there is not; and sometimes we don’t know, and perhaps that is inevitable, the professor believes.
