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Food for Thought: Ship of Theseus and the Heap

Updated on July 14, 2014


Ship of Theseus

A classic paradox, Ship of Theseus poses a problem of perception and threshold. At what point in a process can we say that the process is successful or complete – and does that point exist in a single moment, or do all of the moments make up this process?

As Ship of Theseus is a thought exercise, there is no correct solution. The problem is merely intended to rile up a philosopher or prod at your brain.

Throughout history, many have attempted to come up with reasonable logic to understand the paradox, but have all too often come to the same problem. Additionally, several philosophers have stumbled upon relevant derivations of the paradox.

  • George Washington’s axe had had its head replaced twice and its handle three times, but never simultaneously. Is it still George Washington’s axe?

  • John Locke happened upon this paradox while mending a pair of socks. If a sock has a hole, and that hole is covered with a patch, it is reasonable to conclude that it is the same sock. However, after a second, third, fourth, … hole has been patched, at what point is the sock no longer itself?

The Paradox

The original instance (that we know of) considering this logically fallacy comes from Plutarch’s Theseus

"The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place, 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."

The paradox posed by Plutarch is such:

Consider a ship made of wooden planks. As they naturally decay, old planks are replaced with new, stronger planks. This process continues until each plank has been replaced. Now that each plank has been replaced, is it still the same ship?

It can be argued that the planks required to completely renovate the ship could have built a second ship, thus each new plank changes the ship. In contrast, if each plank is replaced individually (not simultaneously), and we assume that replacing one plank does not change the ship as an entity, then the ship is still itself after each repair; thus, it is itself after all repairs. Now, assume that we decide definitively that one plank does not alter the ship, but all planks do. How many planks must be replaced before the ship is no longer itself? Some are inclined to answer that just over 50% (a majority) of the planks must be replaced before the ship transforms into a new ship. However, consider that if just over 50% of the planks are removed entirely from random points so that the ship’s shape is retained, the result is a ship missing planks – but a ship, nonetheless, which is the same ship it was before losing the planks.


Sorites paradox

The “heap” paradox – so translates the word sorites from ancient Greek – requires similar reasoning and arbitration. The original paradox is attributed to the philosopher Eubulides, famous for his brain-ticklers.


The Paradox

Consider, first, that you have a heap of sand. If you carefully remove one grain, is it still a heap of sand (yes, it is.)? if you remove a second grain, do you still have a heap (yes, you do.)? Following this logic, you will eventually have a single grain of sand which must be a heap. The fallacy of logic lies in our vague description of the sand – a heap. Without defining the amount of sand in a heap, it is impossible to define the heap itself. That is why a heap less one grain of sand is still a heap. Because it is still a heap, the same logic is applied.

We can also consider the opposite: if we have one grain of sand – is it a heap? No, it isn’t. Thus, we add a grain of sand, but it still is not a heap. Since it is “not a heap,” we add a grain of sand. It is still “not a heap,” and will therefore never be a heap.

Another version of this paradox is the simultaneous result of the previous heap paradoxes. We have a “heap” and a grain of sand – “not a heap.” We take one grain from the “heap” and place it onto or next to “not a heap,” the single grain. Clearly, the heap is still a “heap” and the two grains are still “not a heap.” Following the previous logic, both “heap” and “not a heap” will remain respectively, and subjectively, unchanged. However, at a certain point, there will be exactly the same number of sand grains in the “heap” and in “not a heap.” They must therefore be either both “heap” or both “not a heap,” but which one? At the exact average between the original grains in each pile, there is no inclination toward either designation. They cannot each be both “not a heap” and “heap,” but one grain earlier, they were uniquely “heap” and “not a heap.”


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