Philosophy
The Logic of Vague Predicates and the Sorites Paradox
Quick fact
The sorites paradox, also known as the 'little-by-little' argument, can be run with any vague predicate, and it forces us to choose between three seemingly obvious premises: the base premise, the inductive premise, and the bivalence of classical logic – yet all three cannot be true together.
Why this is interesting
Imagine a single grain of sand – clearly not a heap. Add one more, and it's still not a heap. So when does it suddenly become a heap? The sorites paradox shows that this simple question breaks classical logic.
Read the full explanation
Understanding The Logic of Vague Predicates and the Sorites Paradox
Vague predicates are words like 'tall', 'heap', 'bald', or 'old' that have no exact boundary. In classical logic, every statement is either true or false – this is the principle of bivalence. But for a vague predicate, there is no definite point where 'not tall' becomes 'tall'. The sorites paradox arises from this fuzziness. Start with the clearly true statement that a person with 0 hairs is bald. Add one hair at a time – if a person with n hairs is bald, then a person with n+1 hairs is also bald (because adding one hair can't make a bald person not bald). By repeating this, we conclude that someone with a million hairs is bald – which is clearly false. The paradox shows a conflict between our intuition that vagueness is real and classical logic's demand for sharp truth values.
A deeper explanation
The deep mechanism behind the sorites paradox is the combination of three assumptions: (1) the base premise (e.g., 0 hairs is bald), (2) the inductive premise (if n hairs is bald, then n+1 hairs is bald), and (3) the law of excluded middle – every statement is either true or false. Classical logic accepts all three, but they lead to a contradiction. There are several ways out. One is to deny the inductive premise: there must be a sharp boundary, but we don't know where – this is epistemicism. Another is supervaluationism, which says that vague statements are true if they are true under every sharpening of the predicate – but the inductive premise is not true under every sharpening, so it fails. A third is many-valued logic, which assigns degrees of truth – e.g., a statement can be 0.8 true – and the inductive premise then has a truth value less than 1, breaking the chain. Each response preserves some part of classical logic but gives up something else. The paradox matters because it challenges the idea that formal logic can perfectly capture natural language, and it pushes us to develop richer logical systems that can handle uncertainty and approximation.