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The Sorites Puzzle of the Heap
The Sorites problem was one of a number of paradoxes created by the 4th century BCE Megarian philosopher Eubulides, who was a pupil of Euclid.
The Greek word
Would you describe a single grain of wheat as a heap? Not at all. Another variation is to start with a genuinely large heap, claim that the following two premises are true, then remove grains of sand.
A million grains of sand is a heap of sand After removing enough grains, we get to the borderline cases of the paradox. The second premise shows that one grain is absolutely not a heap, because removing one grain leaves nothing, let alone a heap. Sorites problems are also called "little by little" because small changes may be indiscernible in large objects but they become obvious when applied long enough and the object becomes small. A characteristic of all Sorites puzzles is the breakdown of truth conditions at some point along the soritical chain of steps from one end to the other. This is often considered a logical paradox, but it seems to be created by our ambiguous language..
Sorites paradoxes appear to resemble proofs by mathematical induction. If F, and given any _{n+1}n where F is true, then it is true for all _{n}n.
The Stoics are said to have backed away from the strong conditional A ⇒ B to a weaker material implication where A → B is true just in the case that either A is false or B is true, or not (¬A ∨ B) . But this did hot help them.
Viewed from the point of the infinite series of mathematical induction, the problem can be found in the fact that for some n is small), while for other values of n, F is true.
_{n}
∀
But there is no particular point n along the chain where the failure is obvious, since each step seems too small to make the difference. Put another way, there is no transitivity of truth back and forth
Some philosophers regard this failure at some point midway between One semi-formal way out might be say that either/or soritical terms need a third option or even a "dialectical" acceptance of "both." This is similar but not identical to the failure of bi-valence in statements about the future that are neither true nor false. We are often somewhere in the middle between extremes, neither rich nor poor, but middle class, neither hot nor cold, but Goldilocks "just right." Accepting "both" might be statements like, "He's bald but he's not that bald."
Another workaround for sorites paradoxes might be to notice that neither/nor can be said of the truth value for situations in the vagueness gap. For example, somewhere between small and large, we might say it's neither small nor large. Then if we say that small = "not large," we can say that in the gap we have neither small nor not small is true. Since it is always true that everything is either small or not small, without knowing which, some metatheorists imagine a " The fact that large objects appear not to change when small, indiscernible changes are made is also called a vagueness problem. A classic example is Peter Unger's observation that a few water molecules at the edge of a cloud may be removed with no obvious change in the cloud.
See also David Wiggins's version of Tibbles the Cat as really 1,001 cats by selectively excluding one of Tibbles' 1,000 hairs.
Unger's conclusion was that the water molecules may compose many clouds by selectively excluding or including just a few molecules. This is known as the Problem of the Many, but Unger's first response was to say that the ambiguity meant that there are no clouds at all, a position known as mereological nihilism that is now endorsed by Peter van Inwagen.
Liar Paradox
Eubulides also created a variation on Sorites with the number of hairs on a bald man's head and the much more famous Liar's Paradox
A man says that he is lying. Is what he says true or false? A modern self-referential variation is Russell's Paradox
The statement in this box is false
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