Philosophy
The Logic of Vagueness and Supervaluationism
Quick fact
Supervaluationism, developed mainly by Bas van Fraassen and Kit Fine, treats a sentence like 'This is red' as true only if it is true under every acceptable way of making the vague term precise. This lets it avoid the sorites paradox while still preserving the law of excluded middle.
Why this is interesting
Have you ever wondered whether a person with 100 hairs is bald? What about 1,000? At what exact number does 'bald' become false? This puzzle of vague language seems to break classical logic—but supervaluationism finds a way out.
Read the full explanation
Understanding The Logic of Vagueness and Supervaluationism
Classical logic assumes every statement is either true or false. But vague words like 'tall' or 'heap' have fuzzy boundaries. For instance, there is no single height that turns a 'tall' person into a 'not tall' person. Supervaluationism handles this by imagining all the possible, precise ways we could define a vague term. These are called 'precisifications.' A sentence is 'supertrue' if it comes out true on every precisification, 'superfalse' if false on all, and a borderline case if it is true on some and false on others. For such borderline cases, no truth value is assigned—there is a 'gap'—but the sentence still obeys classical logic within each precisification. This clever manoeuvre lets us reject the precision that causes the sorites paradox without abandoning the law of excluded middle.
A deeper explanation
The key insight is that supervaluationism shifts the truth condition from actual precision to invariant truth across all acceptable sharpenings. Consider the statement 'Either he is bald or he is not bald.' In classical logic, this is a tautology. In supervaluationism, it is supertrue because for any precisification, one disjunct is true. However, the individual disjuncts—'He is bald' and 'He is not bald'—are neither supertrue nor superfalse for a borderline case; they lack a determinate truth value. This is why the law of excluded middle is preserved even though bivalence fails. The sorites paradox is blocked because the premise 'For any n, if a man with n hairs is bald, then a man with n+1 hairs is bald' is not supertrue—there is always some precisification where it fails. Supervaluationism also handles 'penumbral connections'—predicates like 'red' and 'orange' must remain consistent across precisifications, so something cannot be both red and orange. This framework shows how vagueness can be accommodated without sacrificing logical structure, offering a middle path between epistemicism and many-valued logic.