Sorites Paradox

by | Oct 8, 2019


Sorites Paradox is a type of paradox dealing with how we categorize things. The classic example of Sorites Paradox concerns when something is a heap (in fact, the word sorites derives from the Greek word for heap).

Imagine a heap of sand. You carefully remove one grain. Is there still a heap? The obvious answer is: yes. Removing one grain doesn’t turn a heap into no heap. That principle can be applied again as you remove another grain, and then another… After each removal, there’s still a heap, according to the principle. But there were only finitely many grains to start with, so eventually you get down to a heap with just three grains, then a heap with just two grains, a heap with just one grain, and finally a heap with no grains at all. But that’s ridiculous. There must be something wrong with the principle. Sometimes, removing one grain does turn a heap into no heap. But that seems ridiculous too. Source.

A more formulaic way to think about this paradox is like this:

(1) Someone with zero hairs on his head is bald.

(2) If you are considered bald with zero hairs, you are still bald with one hair as it’s basically the same as having zero hairs. Thus, it can be said for any number n, if a person with n hairs is bald, then a person with (n + 1) hairs is bald.

(3) Thus, there is no specific number of hairs where someone switches from being bald to not bald. So, if 10 hairs is bald, so is 11. If 200 hairs is bald, so is 201, and then 202 is bald, as is 203. It goes all the way up. A person with 100,000 hairs on his head would still be bald based on the reasoning of the paradox. Source.

Similar examples use the concept of being rich – say having $1 million – and that having one dollar less is still rich. And a dollar less than that is still rich, and so on. Being 7 foot in height is tall. Being one inch shorter is till tall. So, if being 6’9″ is tall, so is one inch less at 6’8″ and so on.

The Sorites Paradox is a paradox because you accept the two underlying propositions to be true: a single grain of sand is not a heap and adding one grain of sand to something not a heap doesn’t transform it into a heap. This is an ancient paradox and there is no clear resolution. However, we know that there is something that is wrong with the reasoning.

A basic issue with the Sorites problem is that the paradox relies on defined categories. As discussed in a prior IFOD, there are different ways to categorize things.

The classic theory is the “Concept Theory” which provides that categories have a set of conditions or definitions and we place things in categories by referring to those conditions. Under the concept theory we would say that a person with little or no hair is bald and a person with a full head of hair is not bald. Slight gradations in characteristics aren’t handled well under the Concept Theory. Things fit the category or they don’t. The Sorites Paradox is problematic when using the Concept Theory to categorize.

A different way to categories is by the “Prototype Theory” which provides that something can be more or less of a category member and we decide that based on prototypes. Under the prototype theory a Labrador Retriever is more dog like than a Shih Tzu, and a robin in more bird like than an ostrich. Metallica is more like heavy metal than Van Halen. Brad Pitt fits our prototype of an actor better than Jack Black even though both are actors.

Under prototype theory we have a prototype for what a heap of sand is and a bunch of grains of sand are more or less heap-like depending on how many there are.

The Prototype Theory of categorization doesn’t solve the Sorites Paradox, but it does change the analysis. Each additional hair on someone’s head moves them further from the prototype of being bald. Each grain of sand removed from a heap of sand makes it less heap-like. Each dollar of wealth moves a non-rich person closer to the prototype of rich.

Read more about the Prototype Theory of categorization.


  1. I liked Ted’s comment

  2. I think a persons situation has a lot to due with the definition of a heap as well. A heap of sand may be different when it is in my shoe or in a gravel pit. Most of us have a different definition of some one being rich (having a heap of money). I contend that the real definition of rich is “Someone that has one more dollar than I have”


Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.


Subscribe To The IFOD

Get the Interesting Fact of the Day delivered twice a week. Plus, sign up today and get Chapter 2 of John's book The Uncertainty Solution to not only Think Better, but Live Better. Don't miss a single post!

You have Successfully Subscribed!

Share This
%d bloggers like this: