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The philosophical paradox of indistinguishable colors, known as the sorites paradox, focusing on the assumption that there could be a phenomenal continuum of only three color patches. The text argues that if a four-patch long continuum exists, then a three-patch long continuum also exists, leading to the surprising result that longer phenomenal continuums imply the existence of shorter ones.
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One especially interesting version of the sorites paradox is the one involving a long series of color shades, each indistinguishable from the preceding one, but which is such that the first is clearly distinguishable from the last.
That version of the paradox went like this:
This argument seems like an especially hard to resist version of sorites reasoning, since the sorites premise of this argument — for any objects x, y, if x looks the same as y and x is red, then y is red — seems especially attractive. How could two objects be completely indistinguishable in respect of color, and yet one be red and the other not?
This version of the sorites argument, unlike the others, also relies on an as- sumption: the assumption that there could be a phenomenal continuum of color patches such that the first is distinguishable in color from the last, and yet none is distinguishable from the adjacent patch in the series. However, this assump- tion clearly seems to be true — if we had a million color patches, couldn’t we set up such a case?
We can also imagine sitting in front of a movie screen whose color is changing so slowly from red to orange that you can never notice a change from one moment to the next, and yet the initial color is clearly distinct from the final color. (Or imagine waking up and watching your room get lighter and lighter, very gradually.)
So it seems that there could be a million-patch long phenomenal continuum. But could there be a phenomenal continuum of only 3 color patches? It might seem not — it would be hard to come up with just 3 patches such that patch 1
and patch 2 are indistinguishable, and patch 2 and patch 3 are indistinguishable, though patch 1 and patch 3 are distinguishable.
The surprising result is that these two natural views are in conflict: if a 1 million patch long phenomenal continuum is possible, so is a 3 patch long phenomenal continuum. A sort of proof of this result can be presented as follows:
Imagine that there is a 4 patch long continuum. Then:
1 & 2 are indistinguishable 2 & 3 are indistinguishable 3 & 4 are indistinguishable 1 & 4 are distinguishable
Now consider 1 and 3. Are these distinguishable, or not?
Suppose that they are. Then, since 1 & 2 and 2 & 3 are indistinguishable, 1,2, and 3 compose a 3-member phenomenal continuum.
Suppose that they are not distinguishable. Then, since 3 & 4 are indistinguish- able and 1 & 4 are distinguishable, 1, 3, and 4 compose a 3-member phenomenal continuum.
So, if there is a 4-member phenomenal continuum, there is a 3-member phe- nomenal continuum.
Can you see how to adapt this argument to show that if there is a 5-member phenomenal continuum, there is also a 4-member phenomenal continuum?
Can you see how to adapt this argument to show that if there is a n-member phe- nomenal continuum such that n > 3, there is also an n − 1 member phenomenal continuum?