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lecture on the Zeno's Paradox of the Race Course, part
2.
(following Aristotle, Physics 239b5-7 = RAGP 10):
1. When the arrow is in a place just its own size, it’s at rest.
2. At every moment of its flight, the arrow is in a place just its own size.
3. Therefore, at every moment of its flight, the arrow is at rest.
Weakness in Aristotle’s solution: it seems to deny the possibility of motion or rest "at an instant". But instantaneous velocity is a useful and important concept in physics:
The velocity of x at instant t can be defined as the limit of the sequence of x’s average velocities for increasingly small intervals of time containing t.
In this case, we can reply that if Zeno’s argument exclusively concerns (durationless) instants of time, the first premise is false: "x is in a place just the size of x at instant i" entails neither that x is resting at i nor that x is moving at i.
"When?" can mean either "at what instant?" (as in "When did the concert begin?") or "during what interval?" (as in "When did you read War and Peace?").
1a. At every instant at which the arrow is in a place just its own size, it’s at rest. (false)
2a. At every instant during its flight, the arrow is in a place just its own size. (true)
1b. During every interval throughout which the arrow stays in a place just its own size, it’s at rest. (true)
2b. During every interval of time within its flight, the arrow occupies a place just its own size. (false)
Both versions of Zeno’s premises above yield an unsound argument: in each there is a false premise: the first premise is false in the "instant" version (1a); the second is false in the "interval" version (2b). And the two true premises, (1b) and (2a), yield no conclusion.
In this version there is no confusion between instants and intervals. Rather, there is a "quantifier switch" fallacy. One quantifier ("at every instant") ranges over instants of time; another quantifier ("there is a place") ranges over locations at which the arrow might be found. Observe what happens when the order of these quantifiers gets illegitimately switched:
1c. If there is a place just the size of the arrow at which it is located at every instant between t0 and t1, the arrow is at rest during the interval between t0 and t1.
2c. At every instant between t0 and t1, there is a place just the size of the arrow at which it is located.
Abbreviating "the arrow is located at place p at instant i" by L(p, i), we get:
1c. IF there is a p such that for every i, L(p, i), THEN the arrow is at rest.
2c. For every i, there is a p such that: L(p, i).
But (2c) is not equivalent to, and does not entail, the antecedent of (1c):
There is a p such that for every i, L(p, i)
(2c) says that the arrow always has some location or other ("at every instant i it is located at some place p") -- and that is trivially true as long as the arrow exists! But the antecedent of (1c) says there is some location such that the arrow is always located there ("there is some place p at which it is located at every instant i") -- and that will only be true provided the arrow does not move!
So one cannot infer from (1c) and (2c) that the arrow is at rest.
Although the argument does not succeed in showing that motion is impossible, it does raise a special difficulty for proponents of an atomic conception of space. For an application of the Arrow Paradox to atomism, click here.
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lecture on the Zeno's Paradox of the Race Course, part
2.
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