Zeno’s Paradox of the Arrow

A reconstruction of the argument

(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.

Aristotle’s solution

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.

Perhaps instants and intervals are being confused

“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.

A final reconstruction

In this version there is no confusion between instants and intervals. Rather, there is a fallacy that logic students will recognize as the “quantifier switch” fallacy. The universal quantifier, “at every instant,” ranges over instants of time; the existential quantifier, “there is a place,” ranges over locations at which the arrow might be found. The order in which these quantifiers occur makes a difference! (To find out more about the order of quantifiers, click here.) Observe what happens when their order 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 throughout 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.

We will use the following abbreviations:

L(p, i) The arrow is located at place p at instant i
R The arrow is at rest throughout the interval between t0 and t1

The argument then looks like this:

1c. If there is a p such that for every i, L(p, i), then R.
$"i L(p, i) ® R

2c. For every i, there is a p such that: L(p, i).
" $p 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)
$"i L(p, i)

The reason they are not equivalent is that the order of the quantifiers is different. (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.

The Arrow and Atomism

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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Go to previous lecture on the Zeno’s Paradox of the Race Course, part 2

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Copyright © 2003, S. Marc Cohen