For some reason, the sorites paradox seems quite a bit like the supposed paradox of Achilles and the turtle with a head start: every time Achilles reaches where the turtle had been, the turtle moves a little bit forward, and so by that line of reasoning, Achilles will never be able to reach the turtle. Yet, when we watch Achilles chase the turtle in real life, he catches it and passes it with ease. By shifting the level of perspective from the molecular to the macro level, so to speak, we move beyond the paradox into a practical solution. If we try to define "heap" by specifying the exact number of grains of sand it takes to differentiate between "x grains of sand" and "a heap of sand," aren't we merely perpetuating the same fallacy, albeit in a different way, by saying that every time Achilles reaches where the turtle had been, the turtle has moved on from there? If not, how are the two situations qualitatively different? Thanks.

In my opinion, the reasoning that generates the paradox of Achilles and the tortoise isn't nearly as compelling as the reasoning that generates the sorites paradox. The Achilles reasoning overlooks the simple fact that Achilles and the tortoise are travelling at different speeds: if you graph the motion of each of them, with one axis for distance and the other axis for elapsed time, the two curves will eventually cross and then diverge as Achilles pulls farther and farther ahead of the tortoise. All of this is compatible with the fact that, for any point along the path that's within the tortoise's head start, the tortoise will have moved on by the time Achilles reaches that point: that's just what it means for the tortoise to have a head start. It's not that the Achilles reasoning is good at the micro level but bad at the macro level. It's just bad.

By contrast, the only thing overlooked by the sorites reasoning is the principle that a small quantitative change (e.g., the loss of one grain of sand) can change the category to which something objectively belongs (e.g., a change from being a quantity of grains large enough to make a heap of sand to being a quantity of grains too small to make a heap of sand). This principle is forced on us by the facts that (a) zero grains is objectively too small a quantity to make a heap of sand, (b) some number of grains (e.g., 1 billion) is objectively large enough to make a heap of sand, and (c) classical logic holds without exception. Although the principle forced on us is true, many people find the principle hard to accept, but I think they may be confusing the principle with the stronger (and false) claim that we can always identify or specify where these objective cutoffs occur. It may be that we never can.

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