The Sorites Paradox

The Sorites paradox is a hairy chestnut inherited from the ancient Greeks. There are a lot of variants, but the paradox of the man with a full head of hair will do.

Consider a man with a full head of hair. If we were to remove one hair from his head then he would still have a full head of hair. But if that is true then we can keep removing hairs from his head until none are left and he will still have a full head of hair.

Clearly there is something wrong here. The fault is not in the logic; rather it is in the premise. Apparently somewhere along the way we switch from a full head of hair to a not so full head of hair. There are various ways to “fix” the problem. Thus we can rule that any one with fewer than N hairs on their head does not have a full head of hair, where N is some arbitrary number. Or we can define a metric so we can say that Fred has a 70% fullness of head hair whereas John has a 92% fullness. Then again, we can introduce the notion the notion of vague attributes that have no definite boundaries and try to make that fly. The striking thing about all of these fixes is that they all smell like hacks.

It seems to me that there is a fundamental error in the premise, to wit, things like big and small, tall and short, baldness and full head of hair are not attributes. What do I mean by that? One way to put it is that attributes are intrinsic. For example, a red car is red regardless of whether other cars are blue or black or whatever. However, when we say that something it is big there always is a context. An elephant is larger than a mouse but smaller than a planet. There is more to it than that; it also depends upon the observer. Thus, from the perspective of a microbe, elephants and mice are enormous; their differences in size are inconsequential.

So what have we said here? We have seen that terms like “big” are context dependent and go with the observer. What does this have to do with the sorites paradox? Very simply, the removing of hairs are hypothetical and our observations are extrapolated. (Some of us have extravagant ideas about “simple”.) Put it this way: We think that removing one hair from this head of hair will not affect its fullness – in our opinion. It doesn’t follow that we will think the same thing about some other head of hair, nor even the same thing about this head of hair after it has lost some hair. For that matter in the future we might change our mind about whether this head of hair is full.

If this seems obscure, consider a landscape with a hill and valley. Where is the boundary between them? The answer is that there is no specific boundary – everyone has their own say as to which is hill and which is valley. Even if everyone agrees that a particular patch of ground is part of the hill, that doesn’t settle the matter – someone might come along and say that it is part of the valley.

In short, whether one has a full head of hair or not depends on who is looking, when they are looking, and upon the phase of the moon.