Monday, March 2, 2009

Eubulides' Paradoxes

I love paradoxes, and have tried to collect a considerable number of them, in philosophy as well as in other intellectual disciplines. I define paradox rather broadly as any real or apparent contradiction or just something that seems to defy common sense. In many cases the paradoxes are in a sense superficial since the apparent contradiction can be resolved.

In the ancient world most of the interesting paradoxes are either from from Zeno of Elea or Eubulides of Miletus. I'll talk about Zeno later, but I want to talk about Eubulides right now.

He has seven paradoxes attributed to him, but many of them are redundant. Ultimately he really has four unique paradoxes: the Masked Man paradox, the Horn Paradox, the Liar paradox and the Sorites paradox

The Masked Man paradox basically goes: if there is a masked man who is your father then someone might ask you, "Do you know your father?" To which you say "yes." Then they ask: "Do you know that masked man over there?" To which you answer, "No." But since the masked man is your father, you seem to be saying that you both know and don't know your father. This is only an apparent contradiction resolved by either seeing it as an equivocation on the word "know" The first "know" means "is acquainted with," and the second "know" means "can identify." Or the apparent contradiction is resolved by recognizing that the second answer, "No, I don't know that masked man," was simply made in error. Namely, you could say, "I didn't realize that I knew the masked man, because I didn't realize he was my father."

Next there is the horn paradox, which basically goes: If we assume that you have whatever you haven't lost, then from the observation that you've never lost horns, you therefore must have horns, but of course you don't have horns. This is also an only apparent paradox. It is just the result of a bad initial premise. No, we can't say that you have whatever you haven't lost. We can say, you have whatever how previously had and haven't lost."

The next paradox is the liar paradox, which has a few variations. For example, if I were to say, "Everything I say is a lie," or "This statement is false." Both variations are probably best explained as arising from some of the artificialities of language. It does appear to present some problems for logic, so a number of attempts have been made to resolve it. I think I'll return to it in more detail later.

Finally, there is the Sorites paradox ("sorites" means "heap"), another difficult paradox and I think Eubulides' most interesting paradox. The idea here is that if I have a heap of grains of sand, and I take away one grain of sand then it is still a heap. And if I take away another grain, it is still heap. And if I extend this logic, then at some point it will be too small to be a heap. But I can't make a heap into a non-heap by simply taking away one grain of sand. So, how did it become too small to be a heap?

Mostly this paradox is built on the vagueness of language, since "heap" and "non-heap" are not concretely defined terms. But when used in logical argument they are treated as a precise bivalent, black and white distinction.

Another way to think of it: If a million grains of a sand is a heap then certainly 999,999 is a heap, so is 999,998 and 999,997, etc. From this we infer that if x number of grains is a heap, then x-1 grains is a heap. If we iterate this reasoning again and again, we eventually can conclude that 1 grain of sand is a heap, as is 0 grains of sand. I would classify this as a vertical argument: an argument where each new premise is dependent on a previous conclusion. Vertical arguments grow weaker as one increases the steps since one increases the likelihood that there is a weak or fallible argument somewhere along the chain. Vertical arguments depend on an unbroken chain of infallible arguments to work (doable in math, but less so in philosophy). To conclude that 1 grain of sand is a heap requires a vertical argument of many thousands of steps, Any fallibility in the assumption that "if x number of grains is a heap, then x-1 grains is a heap" is amplified by the number of steps. The argument becomes rephrased "if x number of grains is a heap, then x-1 grains is a heap, and if x-1 grains is a heap then x-2 grains is a heap, and x-3 grains is a heap, and x-4 grains is a heap ... and x-999,999 grains of sand is a heap." We get an argument that grows weaker each step, just as correlatively the heap of grains of sand grows less likely to be called a "heap" the more grains of sand are removed, until we are left with a remaining bunch of grains that no one would call a heap and a vertical argument that has been stacked so high that it topples over.

1 comment:

Steven Walden said...

Could the answer be less about the numerical amount and more about the geometry of objects. You can have a heap of one if it 'looks' in an untidy mess - your coat is in a heap. Likewise you could theoretically do the same for 1 sand grain if you were close enough to know how sand grains normally look on the ground and how one in a heap would look. A heap is a description not of number but a subjective determination of its positioning - it looks odd. The number comes into play not because it is important but because there is more likelihood of grains of sand to 'appear odd and in a heap' with a higher number of sand grains to order. Steven Walden