The Paradox of the Heap

The paradox of the heap argues in this way:

A large pile of sand is composed of grains of sand. But taking away a grain of sand from a pile of sand cannot make a pile of sand stop being a pile of sand. Therefore if you continually take away grains of sand from the pile until only one grain of sand remains, that grain must still be a pile of sand.

A similar argument can be made with any vague word that can differ by an apparently continuous number of degrees. Thus for example it is applied to whether a man has a beard (he should not be able to change from having a beard to not having a beard by the removal of a single hair), to colors (an imperceptible variation of color should not be able to change a thing from being red to not being red), and so on.

The conclusion, that a single grain of sand is a pile of sand, or that a shaven man has a beard, or that the color blue is red, is obviously false. In order to block the deduction, it seems necessary to say that it fails at a particular point. But this means that at some point, a pile of sand will indeed stop being a pile of sand when you take away a single grain. But this seems absurd.

Suppose you don’t know the meaning of “red,” and someone attempts to explain. They presumably do so by pointing to examples of red things. But this does not provide you with a rigid definition of redness that you could use to determine whether some arbitrary color is an example of red or not. Rather, the probability that you will call something red will vary continuously as the color of things becomes more remote from the examples from which you learned the name, being very high for the canonical examples and becoming very low as you approach other colors such as blue.

This explains why setting a boundary where an imperceptible change of color would change something from being red to being not red seems inappropriate. Red doesn’t have a rigid definition in the first place, and assigning such a boundary would mean assigning such a definition. But this would be modifying the meaning of the word. Consequently, if the meaning is accepted in an unmodified form, the deduction cannot logically be blocked, just as in the previous post, if the meaning of “true” is accepted in an unmodified form, one cannot block the deduction that all statements are both true and false.

Someone might conclude from this that I am accepting the conclusions of the paradoxical arguments, and therefore that I am saying that all statements are both true and false, and that a single grain of sand is a pile, and so on.

I am not. Concluding that this is my position is simply making the exact same mistake that is made in the original paradoxes. And that mistake is to assume a perfection in human language which does not exist. “True,” “pile,” and so on, are words that possess meaning in an imperfect way. Ultimately all human words are imperfect in this way, because all human language is vague. The fact that logic cannot block the paradoxical conclusions without modifying the meanings of our words happens not because those conclusions are true, but because the meanings are imperfect, while logic presupposes a perfection of meaning which is simply not there.

In a number of other places I have talked about how various motivations can lead us astray. But there are some areas where the very desire for truth can lead us away from truth, and the discussion of such logical paradoxes, and of the vagueness of human thought and language, is one of those areas. In particular, the desire for truth can lead us to wish to believe that truth is more attainable than it actually is. In this case it would happen by wishing to believe that human language is more perfect than it is, as for example that “red” really does have a meaning that would cause something in an a definitive way to stop being red at some point with an imperceptible change, or in the case of the Liar, to assert that the word “true” really does have something like a level subscript attached to its meaning, or that it has some other definition which can block the paradoxical deductions.

These things are not true. Nor are the paradoxical conclusions.

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