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Vagueness is ubiquitous (is someone with 1,000,001 hairs bald?), yet traditional logic assumes precise truth-conditions. Sorites arguments show how innocent-seeming principles lead to paradox. Supervaluationism, degree semantics, and epistemicism offer competing solutions with different implications for meaning and logic.
You already understand the Sorites paradox: take a heap of sand, remove one grain — still a heap. Repeat a thousand times. At no point does a single grain make the difference between heap and non-heap, yet we end with one grain and the argument forces us to call it a heap. The paradox arises because "heap" is vague: there is no sharp boundary between heaps and non-heaps, and cases in the middle — a hundred grains, perhaps — are borderline cases where it is genuinely unclear whether the term applies. The challenge is to explain what's happening in borderline cases without accepting the paradox.
The simplest response is epistemicism, defended by Timothy Williamson. Vague predicates actually have perfectly sharp extensions — there really is a precise number of hairs below which someone is bald — but we cannot know where the boundary falls because our concepts are not precise enough to detect it. On this view, classical logic is fully preserved: every statement is either true or false; it's just that we can't always know which. The counterintuitive implication is that removing a single hair sometimes makes someone bald, we just can't tell when. Epistemicism saves logic at the cost of making meaning radically inaccessible.
Supervaluationism takes a different route. A vague predicate like "tall" can be "precisified" — made arbitrarily sharp — in many different ways. Someone 5'10" might count as tall on some precisifications and not tall on others. Supervaluationism says a sentence is supertrue if it is true on all precisifications, superfalse if false on all, and indeterminate (neither true nor false) if it varies. Classical tautologies like "John is tall or not tall" remain supertrue — true on every precisification — even when "John is tall" is indeterminate. This preserves classical logical laws while allowing truth-value gaps at borderline cases. The cost: some instances of classical inference fail; you can have a disjunction that is supertrue even though neither disjunct is true.
Degree semantics (developed by Kamp, Fine, and others) assigns sentences involving vague predicates degrees of truth between 0 and 1, rather than just true or false. "John is tall" might have a degree of 0.7 if he's 5'11". Logical connectives then operate on degrees: "not" inverts, "and" takes the minimum, "or" takes the maximum. Borderline cases are cases with intermediate degree, not missing truth values. The challenge for degree semantics is explaining what these degrees represent — are they objective features of the world, facts about our dispositions, or something else? — and handling higher-order vagueness: the boundary between "clearly tall" and "borderline tall" is itself vague, threatening an infinite regress of borderlines. Each theory reveals something important: vagueness is not merely a linguistic imprecision to be cleaned up, but a deep feature of how language engages with a continuous world.
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