March 18, 2024
Kai Macann
The Sorites Paradox
How many grains of sand comprise a heap?
Introduction
How many grains of sand make a heap? The sorites paradox, attributed to Eubulides of Miletus, is a thought experiment with many variations that started with this deceptively simple question.
Let's use some real numbers as an example. If 10,000 grains of sand is a heap, then 9,999 must still be a heap. If 9,999 is a heap, then 9,998 must be. Following this logic, if two grains are a heap, then one grain is still a heap.
While the first premises seem true, the final premise is obviously incorrect. When do the premises stop being true? At what line does the heap stop being a heap? Another example of a sorites case is the bald man variation.
The logic of both these paradoxes makes sense. Each increment is imperceptible, however it's obvious that one grain shouldn't be considered a heap, any more than a man with 100,000 hairs should be considered bald. Clearly, there is an error in logic.
Is there a definitive line between a heap and a non-heap?Nihilistic View
The simplest—though most radical—response to the sorites paradox is to reject its premises entirely. A single grain of sand is not a heap, but neither are 10,000 grains. Likewise, having 100,000 hairs does not make someone bald, just as having only one hair does not.
This nihilistic view challenges the validity of vague predicates—terms that give rise to ambiguous “borderline” cases where it’s unclear whether the predicate applies—by arguing that such predicates cannot meaningfully describe anything.
For instance, a number is either prime or it is not; there is no room for ambiguity. In contrast, concepts like “bald” or “heap” are inherently subjective, rendering them incapable of holding definitive meaning. This of course is undesirable to many theorists, and practically very hard to live by.
Does the nihilistic view seem correct?Epistemic View
The “epistemic view” offers an alternative approach, suggesting that there are indeed absolute thresholds for vague predicates—a specific point where someone becomes bald or a collection of sand becomes a heap—we simply don't know where these points lie.
According to this view, the initial premise of the sorites paradox holds, 10,000 grains of sand do constitute a heap, however at some precise point the truth shifts, and what was once a heap ceases to be one. For instance, a man with 124 hairs would still be bald. But if the cutoff were 125 hairs, then at that point, the man would no longer be considered bald.
Many people find the epistemic view difficult to accept because the exact threshold is unknown and seems impossible to determine. Even if we could determine it, the line seems arbitrary—why should 125 hairs be the cutoff point? why not 200?
Does the epistemic view seem right?Truth-Value Gap
A less extreme theory is the simple truth-value gap theory, which states that when it comes to baldness it is true to say that some people are bald, false to say that others are, but "undefined" for all cases in between.
It is true that a man with 124 hairs is bald. It is "kind of" true that a man with 125 hairs is bald. It is false that a man with 250 hairs is bald. In which case anything greater than the first and lesser than the second is a third category of undefined.
Simple truth-value gap theory remains unappealing to many theorists because it still relies on rigid boundaries. Why should having 124 hairs count as definitely bald, but 125 hairs only “kind of” bald?
This leads to the deeper problem of higher-order vagueness—the point at which something stops being clear-cut and becomes vague is, itself, vague. It feels counterintuitive that a single hair could mark the difference between being clearly bald and ambiguously bald.
Does the simple truth-value gap theory seem right?Supervaluationism
Supervaluationism argues that a sentence is true or false only if it holds across all reasonable interpretations. If a statement is true across all interpretations, it is referred to as "super-true", if it is false across interpretations it is referred to as "super-false".
Statements like "53 is a prime number" are super-true because being a "prime number" has only one definition and 53 meets it. For vague terms however, such as “bald” or “heap” which have many subjective definitions, super-truth or super-falsity are unattainable. Instead, it hinges on your own definition.
For example, according to supervaluationism if you define a heap as 1,000 grains then it is true that 1,000 grains is a heap. If you define a heap as 2,000 grains then it is false that 1,000 grains is a heap. It is super-false that one grain is a heap, and super-true that 10,000 grains are a heap.
However, supervaluationism is also criticised for higher-order vagueness. Additionally, it’s difficult in practice to determine whether a statement satisfies all reasonable interpretations or not, and even harder to define what qualifies as a reasonable interpretation in the first place.
Does supervaluationism seem right?Degrees of Truth
The final theory, known as “degrees of truth,” suggests that while we may accept the premises of an argument, we can still reject the conclusion. This theory challenges the idea that statements are strictly true or false, proposing instead that truth exists on a continuum.
Consider a typical premise from a sorites argument: “If a person with one hair is bald, then a person with two hairs is also bald.” In this case, the truth of the ‘if’ part is slightly stronger than the ‘then’ part.
With each small change—like adding one more hair—the statement becomes less true. This gradual shift explains how the sorites paradox allows us to move step by step from something that is almost entirely true to something that is nearly false.
Does the degrees of truth view seem right?Real-World Impact
The sorites paradox illustrates the importance of logic and critical thinking in addressing counterintuitive dilemmas where intuition alone may fall short.
This reasoning extends to complex issues such as determining when a fetus should be regarded as a person, or distinguishing between “normal” and in need of a diagnosis in the medical field. Ultimately, grappling with paradoxes such as this allows us to address the more difficult, real-world questions with greater clarity and nuance. But given the endless ambiguity of language, is this paradox ultimately solvable, or is the exact nature of truth inherently unknowable?