When A Paradox Is Not A Paradox

“This blog is false.” ~ Theory Parker

In my free time, which is almost never these days, I like to think about paradoxes and the flaws that lead to their apparent existence. Paradoxes – statements that seems contradictory but may be or seem true – come in several types; they may be paradoxes of logic, of self-reference, of statistics, of probability, of vagueness, of mathematics, of geometry and even of physics. For the sake of brevity, I am going to focus on three paradoxes today that relate more to philosophy and demonstrate why they are in fact not paradoxes. I will start with one of the most famous of all, one of Zeno of Ela’s paradoxes, Achilles and the tortoise.

Achilles and the Tortoise

This “paradox of motion” basically states that in his race with a tortoise, the much faster Achilles must travel at least half the distance covered by his slower opponent an infinite number of times in order to even get close to the tortoise. Since the tortoise always moves ahead some distance, Achilles has to travel to the tortoise’s starting point of that distance before reaching the tortoise. So, as long as the tortoise is moving, Achilles can never catch up and pass the tortoise. Or, as Aristotle himself relates it in his work Physics, “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

Solutions and rebuttals to this paradox have been offered by Aristotle, Archimedes, Thomas Aquinas, Bertrand Russell and philosopher Nick Huggett, with only the physicist Peter Lynds and mathematician Hermann Weyl coming close but still missing the mark. I’ve always found this puzzling as the solution to this paradox – thus not making it a paradox – is incredibly simple: The paradox is only a true paradox if we’re talking about Achilles and the tortoise as two-dimensional beings on a two-dimensional plane. But this is not the case in reality. In reality we live in three dimensions of space and one of time, to say nothing of the fact that the Earth is rotating while hurdling through space during which our solar system orbits the galactic center which is itself winding its way away from other galaxies as space is expanding. So, when Aristotle says something like, “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started,” it cannot be true that the faster runner ever occupies the same point in space (much less time) that the slower runner has started from, no matter the given starting point for the slower runner. Another way to put it would be to say that if we point to the starting point of the slower runner and tell the faster runner to go there, the faster runner may go to the same geographical location but the geographical location has changes its position in space-time by the time the faster runner gets there. And if the faster runner is fortunate to have the Earth rotate towards them as they run, they are in fact that much quicker (though this increase in speed is not detectable by human senses).

Zeno’s paradox here, like his other paradoxes of motion, winds up failing on account that he perhaps didn’t know about the four dimensions we live in. Furthermore, Zeno appears to make the mistake that time is static; that there is no flow to time. While time may occur in discreet packets at the quantum level (i.e. Plank time), there appears to be no interruption in the flow of time at the macro-level of the physical world. If Zeno has bothered to listen to his contemporary Heraclitus who said, “Everything is in flux,” Zeno would have readily seen the flaw in his paradox and never come up with his tale of Achilles and the tortoise.

“This Sentence is False.”

“This sentence is false,” is probably the most well-known version of the Liar Paradox and is categorized as a self-referential paradox. Quite simply, the apparent paradox is this: If "This sentence is false," is true then the sentence is false, but if "This sentence is false," is false, then the sentence is true.

I deny this is a paradox on two accounts. The first is that despite appearances, the sentence actually has no subject with which to refer to. The subject in question, “This sentence…” if taken by itself as a phrase, can have no truth value in the same way a phrase like ‘this book’

does, seeing how ‘this sentence’ is an abstract concept and ‘this book’ refers to an object in reality. That is to say, abstractions cannot have truth values, or perhaps it is more precise to say they have no truth values in reality, outside of individual minds. While a unicorn may exist as an abstract within our minds, unicorns don’t actually exist in reality. As such, I am of the position that abstractions have no truth or false values. If something has no truth value universally to all people everywhere, it simply has no truth value at all; the word, phrase, or sentence in question is ultimately devoid of truth or falsity.

Furthermore, I object that abstractions can reference themselves and maintain any truth value simply because referring to itself as being true or false cannot be verified. If a unicorn says, “This unicorn is true,” we don’t know what about the unicorn the unicorn is saying is true much less know whether or not the unicorn is lying, especially when unicorns don’t even exist in reality. 

“This sentence is false,” is merely a nonsensical utterance derived from the contrivances of language.

Ship of Theseus Paradox

A little more challenging is this paradox brought to us by the Greek historian, Plutarch. He writes in his Theseus, "The ship wherein Theseus and the youth of Athens returned from

Crete had thirty oars … they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same." What Plutarch is asking is if parts are taken away from the ship and replaced which new parts though of the same design, is it still the same ship? Furthermore, would it still be the same ship if all the pieces were eventually replaced? Further furthermore, if all the parts were eventually replaced with new ones and the old pieces were reassembled just as they had been on the ‘old’ ship, which ship is actually Theseus’ ship?!

Aristotle thought he’d solved this paradox by invoking four criteria (he called them ‘causes’) for the ‘identity’ of the ship. They include the Formal Cause (the ship’s design), the Efficient Cause (how the ship was built), the Material Cause (what the ship is made of), and the Final Cause (the purpose for which the ship is made). Aristotle was mostly concerned with a thing’s formal cause, so as far as he considered the problem, there was no paradox because despite the replacement of parts – no matter how extensive – resulted in the same design. This also satisfied Aristotle’s second most import cause, the Final Cause, since the design still allowed for sailing. So, if we were only concerned with the idea of Theseus’ ship, the ‘form’ of Theseus’ ship, then both ships are Theseus’ ship (which puts aside the question of ownership). But since Aristotle added criteria that to him indicated that only the ship that was still worthy of sailing was Theseus’ ship. Of course, if both ships are sailable, Aristotle’s argument sinks.

The solution that most philosophers ignore because it is just too obvious, is to say that Theseus’ ship is whichever one he owns and is at the helm of. If Theseus had paid for one single ship, there might be a question as to which ship he actually owns, but by adding that Theseus must be at the helm of the ship as well quite quickly resolves the apparent paradox. While the pieces of the ‘old’ ship may have been taken away and reassembled, it is not Theseus’ ship. It was Theseus’ ship but not anymore, just like there was a paradox but not anymore.

Have a paradox you need solved? Just send them to pi3.onefour@gmail.com and I’ll get back to just as soon as I return from time-travelling to the past to kill my grandfather.