Definition
The sorites paradox — from the Greek soros, meaning heap — is the puzzle of the heap. A million grains of sand make a heap. Removing a single grain from a heap surely leaves a heap. Apply that step repeatedly and you are eventually left with one grain, or none — which is plainly not a heap. Yet every step looked unimpeachable.
The same argument runs with any vague predicate. One hair more or less cannot turn a non-bald man bald; one penny cannot turn a rich person poor. Step by step, the chain marches from a true premise to a false conclusion.
Why it matters
How it works
The paradox has three ingredients: a true base case (a million grains is a heap), an inductive step (removing one grain preserves heaphood), and repeated application. Classical logic chains these together and delivers the absurd conclusion. Since the conclusion must be rejected, one of the other ingredients must go.
Each theory of vagueness blocks the argument differently. The epistemicist says the inductive step is simply false — there is a precise last grain that counts as a heap, though we cannot know which. The fuzzy logician says each step degrades the truth value slightly, so the conclusion is far less true than the premises and the chain leaks truth as it runs. The supervaluationist says no particular step is the one that fails, yet the inductive premise still cannot be straightforwardly true. The sorites endures because no response is cost-free.