An Introduction to the Sorites Paradox

Meet Charles. Charles has graciously volunteered himself for a demonstration. Now, Charles is a middle-aged man and, unfortunately, has started to bald. Now, he isn't quite bald yet; he still has well-groomed mane. But there is a bald spot that for the past few weeks has been growing. Let n be the number of individual hairs on Charles' head. Suffice for our purposes, n is a pretty big number. For those that work better without variables, feel free to plug in 100,000.

(1) Charles has n hairs.
(2) Someone with n hairs is not bald.
(3) Therefore Charles, with n hairs is not bald.

Now comes the experiment. We sit Charles down on a comfortable chair and give him a big, juicy rib eye steak for his troubles. We then take a pair of tweezers and pluck out one of Charles' hairs. It seems obvious that we did not do much to change Charles' being bald or not. Sure, we may have put him one hair closer to complete baldness. But we did not make him bald by removing a single hair.

(4) Charles has n-1 hairs.
(5) Someone with n-1 hairs is not bald.
(6) Therefore charles, with n-1 hairs, is not bald.

What if we kept plucking out a hair of Charles, one by one, and asking ourselves at that juncture if he were bald? Surely, at some point he must become bald. After all, a man with no hair on his head is most certainly bald. But where is that point? A man with only a single hair would still, presumably, be bald. So, too, would a man with two hairs - as evidenced think Homer Simpson. Furthermore, the following principle seems to hold:

(7) If someone with n hairs is bald, then someone with n+1 hairs is bald.

This is grounded in the implausibility that a single hair makes the difference between baldness and non-baldness. Imagine two men standing next to each other, one bald and one not bald. Would you expect there to be only a single hair to separate the two? Is that even possible? Even more troublesome is that it appears we can also reason in the opposite direction.

(8) If someone with n hairs is not bald, then someone with n-1 is not bald.

This is justified in exactly the same way as (7). Losing a single hair cannot move anyone into a state of baldness. Thus, fully presented, the argument runs:

(1) If someone with n hairs is not bald, then someone with n-1 hairs is not bald.
(2) Someone with 100,000 hairs is not bald.
(3) Someone with 99,999 hairs is not bald. (From 1 and 2)
(3) Someone with 99,998 hairs is not bald. (From 1 and 3)
...
(100,001) Someone with 0 hairs is not bald. (From 1 and 100,000)
(100,002) Someone with 0 hairs is bald. (From common sense!)
(100,003) Contradiction!

As noted above, the argument could be run in reverse, with the intuition at (100,002) reading, "Someone with 100,000 hairs is not bald."

This type of puzzle is known as the sorites paradox. It is not restricted, of course, to baldness. Parallel arguments can be made for things like heaps of sand (or garbage). There are a variety of replies to the paradox. We will first, however, apply the paradox specifically to material composition.