Zeno's Paradoxes--"True Implies False"

Zeno's Paradoxes: True Implies False

by Greg Moler
allthatcounts.com

Summary: The conclusions of the paradoxes of Zeno are based on faulty logic. The solution has nothing to do with infinite series or any other mathematical construct.

Introduction

           Consider the following two equations:

x + 2 = 5
x + 5 = 10
           I now announce the above set of equations constitutes a "paradox". Would you believe that? No, the equations are simply contrary. Yet it is the nature of paradoxes that they mask such logical errors with clever wordings.
           Consider this one from Bertrand Russell:

"The barber shaves everyone in town who
doesn't shave himself. Who shaves the barber?"
The barber statement represents an impossible set of conditions. Russell has in effect created two equations (like the ones above) and masked their contraries by clever wording. Russell could have replaced the above with this simpler "paradox":

"This statement is false."

The Rules of Logic

           An implication consists of a premise and a conclusion. The premise and the conclusion may be each true or false. The rules of logic state:

1) True implies False -- the implication is false
2) False implies True -- the implication is true
           Sometimes it is not obvious which of the premise or conclusion is true/false. Consider the well-known proof that that square root of 2 is irrational:

1. Let sqrt(2) = R/S, where R and S are integers, not both
even (if they were, the even numbers could be divided out).

2. Then 2 = R²/S², and
R² = 2*S²; therefore R is even.

3. If R is even, then R = 2 * Q, where Q is an integer.

4. Then the equation (2) becomes
4 * Q² = 2 * S², or
S² = 2 * Q².

5. from (2) and (4), both R and S are even, contrary to (1).
           Now, I don't recall anyone referring to the above proof as a "paradox". The assumption is the premise (1) is false. Therefore, the argument is of the form "false implies true" so that the implication (i.e., the proof) is logically valid.
           Now let us turn to two of Zeno's Paradoxes.

The "Half-Way" Paradox

           A man cannot get anywhere. The reason is that he must first go half way, then when he gets half way, he must go half way of the remaining distance, then half way of the remaining distance, ad infinitum. He thus is forever getting half way to his goal, never reaching it.
           Solution: the paradox's logic requires our friend to get "half way", but the conclusion is that he cannot get anywhere. Thus, the paradox requires as true that which it is trying to prove false. The implication is of the form "true implies false", and is false.
           Note that in this case it is obvious that the premise (he must first go half way) is true and conclusion is false. Common sense here.

The Arrow Paradox

Consider an arrow in flight. At each moment in flight, it occupies exactly one position and no other, and therefore is at rest at that moment. It is therefore at rest at each moment in flight. Thus, the arrow cannot move.
Solution: the premise is that the arrows moves (is in flight), and the conclusion is that it cannot move. True implies false again.