The sorites paradox
One grain of sand is not a heap. Adding one grain never turns a non-heap into a heap. So, by that logic, a million grains are not a heap either.
- Reading time
- 5 min
- Difficulty
- Intermediate
- Branch
- Logic
- First posed
- Eubulides of Miletus, 4th century BCE
The puzzle
Take a heap of sand: a million grains. Remove one grain. It is still a heap; one grain cannot make the difference between a heap and a non-heap. Remove another. Still a heap. Keep going.
Each step seems obviously true: if n grains make a heap, then n minus 1 grains make a heap too. But follow the steps far enough and you reach a single grain, or zero grains, and conclude that they make a heap. That is absurd.
The argument can run the other way too. One grain is not a heap. Adding a single grain never turns a non-heap into a heap. So two grains are not a heap, nor three, nor a million.
The puzzle is credited to Eubulides of Miletus, a contemporary of Aristotle. Its name comes from the Greek soros, meaning heap. It applies to almost every word we use: tall, bald, rich, old, red, adult, alive. Each has clear cases and clear non-cases, with no sharp line between them. This feature is called vagueness.
Why it matters
Vagueness is everywhere in language, and in law and ethics. At what age is someone mature enough to vote? How many weeks into a pregnancy does a fetus acquire moral status? When does a collection of cells become a person? Legal systems draw sharp lines, at 18 years or a certain blood alcohol level, but those lines seem to cut through something continuous.
The sorites also challenges logic itself. It uses only the simplest reasoning, repeated many times, and reaches a false conclusion. Something has to give: the premises, the logic, or our idea of truth.
Key terms
- VaguenessThe property of words like 'bald', 'heap' and 'tall' that have borderline cases and no obvious sharp boundary.
- Borderline caseA case where it is unclear whether a vague word applies, such as whether a man with a certain number of hairs is bald.
- Principle of bivalenceThe principle that every statement is either true or false, with no third option. Vagueness and the liar paradox put it under pressure.
- SupervaluationismA theory of vagueness: a vague sentence is true if it comes out true on every acceptable way of making the vague word precise, false if false on all, and neither otherwise.
- EpistemicismThe view that vague words have perfectly sharp boundaries that we simply cannot know. There is an exact number of hairs at which a man becomes bald.
Proposed solutions
Epistemicism: there is a sharp line
There is an exact number of grains at which a heap stops being a heap. We just cannot know where it is.
Supervaluationism: true on every sharpening
A vague statement is true if it would be true however we made the word precise, false if false on every precisification, and neither in between.
Degrees of truth
Truth comes in degrees. '1,000 grains is a heap' might be 0.7 true. Each step lowers the truth only slightly, and the paradox dissolves.
Contextualism
The boundary of a vague word shifts with context. Whenever we consider two adjacent cases, our standards adjust so that both get the same verdict.
Each solution, tested
Epistemicism: there is a sharp line
- How it resolves the paradox
- Timothy Williamson argues that classical logic should be kept: every statement is true or false. So 'n grains make a heap' is false for some smallest n, and the inductive premise fails at exactly that point. We cannot discover the line because the meaning of 'heap' is fixed by our overall usage in ways too complex for us to track. Our ignorance explains why vague words seem to lack boundaries.
- Why it doesn’t fully work
- It seems incredible that 'heap' has a precise boundary, say at 4,327 grains, that no one ever decided and no one could ever discover. What fact could make it 4,327 rather than 4,328?
- What defenders say
- Williamson replies that this incredulity is not an argument, and that the alternatives require giving up classical logic, which has a much higher cost.
Supervaluationism: true on every sharpening
- How it resolves the paradox
- We could sharpen 'heap' in many acceptable ways: at 100 grains, at 120, at 150. A million grains is a heap on all of them; one grain is a heap on none. For a borderline number, the answer varies. 'There is a sharp boundary' comes out true, since every sharpening has one, but no particular boundary is true. This keeps much of classical logic while explaining why there is no fact about where the line is.
- Why it doesn’t fully work
- It makes 'there is a number n such that n grains are a heap but n minus 1 are not' true, while denying that any particular n makes it true. Critics find this as hard to accept as epistemicism.
Degrees of truth
- How it resolves the paradox
- This fits how vague words actually work: things can be more or less red, more or less tall. Fuzzy logic, developed by Lotfi Zadeh in 1965 and widely used in engineering, gives a precise mathematical treatment. Each conditional 'if n is a heap, so is n minus 1' is very nearly true, and many nearly-true steps can add up to a false conclusion.
- Why it doesn’t fully work
- It replaces one sharp boundary with infinitely many: exactly where does a heap become 0.5 true? This is called the problem of higher-order vagueness.
Contextualism
- How it resolves the paradox
- Delia Graff Fara and others argue that whenever you compare two nearly identical cases, the context makes them look the same, so the boundary moves elsewhere, somewhere you are not looking. That is why every step seems safe, though some step somewhere crosses a line.
- Why it doesn’t fully work
- This explains why we cannot find the boundary, but critics question whether it fully explains why the premises seem true, or tells us which conclusion is correct.
Who argued what
- Epistemicism: there is a sharp line Timothy Williamson, Roy Sorensen
- Supervaluationism: true on every sharpening Kit Fine, Rosanna Keefe
- Degrees of truth Lotfi Zadeh, Nicholas J. J. Smith
- Contextualism Delia Graff Fara, Diana Raffman
Where you stand
Where does the vagueness come from?
In surveys
Choose one to see how people answer.
Variations
The bald man
A man with 100,000 hairs is not bald. Removing one hair never makes a non-bald man bald. So a man with no hair is not bald. The same structure applies to anything that changes gradually.
The colour strip
A strip of 100 colour patches runs from red to orange, each patch indistinguishable from its neighbours to the naked eye. If one patch is red, and the next looks exactly the same, surely it is red too. Yet the last patch is orange. Here the vagueness seems built into perception itself.
The ship of Theseus
Replacing one plank never turns one ship into another. Yet replacing all of them seems to. The Ship of Theseus is a sorites about identity.
Go deeper
Vagueness in law
Law uses sharp numerical cut-offs, ages, speed limits, filing deadlines, precisely because ordinary vague terms invite endless argument. But it also deliberately uses vague standards like 'reasonable care' and 'without undue delay', which let judges respond to particular circumstances. Legal philosophers debate when precision is fairer and when vagueness is wiser.
Related ideas
Further reading and sources
- EncyclopediaStanford Encyclopedia of Philosophy: Sorites Paradox
- WikipediaSorites paradox
- SurveyPhilPapers Survey, 2020 · question: Vagueness; about 1,785 professional philosophers; accept or lean toward
- BookVagueness, Timothy Williamson
- BookParadoxes, R. M. Sainsbury
Open questions
- Is there a precise number of grains at which sand becomes a heap?
- Can a statement be partly true?
- If the boundary of 'adult' is arbitrary, is it fair to draw it anyway?