# (p.531) Appendix C A New Version of the Paradox of the Heap

# (p.531) Appendix C A New Version of the Paradox of the Heap

In chapter 9, I presented the standard Hairiness/Baldness Sorites Paradox and showed, following a suggestion of Ryan Wasserman's, how it might be revised or reinterpreted so that it paralleled my Spectrum Arguments. In this appendix, I note how Wasserman's considerations might also apply to another well-known Sorites Paradox, the Paradox of the Heap.

The standard version of the Paradox of the Heap runs roughly as follows. Start with a large heap of sand. Surely, it is thought, one grain of sand will not be enough, by itself, to make a difference to whether or not there is a heap of sand. Call this the *Crucial Premise for the Standard Sorites Paradox of the Heap*, or *CPH*, for short. So, start with a pile of sand that is clearly a heap. Remove one grain. In accordance with *CPH*, it will still be a heap. Iterate. After each iteration, it will still be a heap, according to *CPH*. After enough iterations one will be left with a single grain of sand. This argument supposedly shows that a single grain of sand is a heap of sand. But that is obviously false. Hence the argument has clearly gone wrong, and the question is where.

Wasserman's revised version of the Paradox of the Heap proceeds differently. Wasserman contends that our criteria for determining whether or not a pile of sand is a heap are complex, and they vary depending on the alternatives being compared. Roughly, we think that two criteria must be met for a pile of sand to count as a heap. First, it must have a large number of grains of sand. Second, it must have a roughly mound-like shape. So, for example, a flat beach would not count as a heap of sand, no matter how many grains of sand it might contain. Neither would a large pile of sand roughly one inch deep, even if it had a few uneven undulations across its surface.

But then Wasserman suggests the following. Start with a relatively large pile of sand that is sufficiently mound-like in appearance as to clearly count as a heap. Suppose, for the sake of argument, that the sand heap is some twenty feet high. Wasserman suggests that given the fact that we have *two* criteria for
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assigning “heapness” to a sand pile—roughly the *number* of sand grains and the *shape* of the grains—we might all rightly agree that a *slight* worsening of a pile in terms of one criterion, say shape, could be more than made up for by a “sufficient” gain in the second criterion, say number.

So, Wasserman suggests, if we pushed down the top of a sand pile, lowering it by a few inches, but *doubled* the total number of grains in the pile, we might rightly agree that the second pile was even more of a heap than the first. The thought, of course, is that the second pile would be less like the first in terms of being a heap regarding its shape, but even more of a heap in terms of its number of grains of sand, and that in *this* case, since the change in shape was slight and the gain in numbers great, *all things considered*, the second would be even more of a heap than the first. Let us say that if one sand pile is even more of a heap than another, then it is “heapier.” So, in the example, the second pile is “heapier than” the first.

Wasserman then repeats the kind of transformation in question many times. Each time, he makes a *slight* alteration in the pile's heap-like *shape*, but he adds *many* times more grains of sand, so that everyone might agree that whether or not either of the piles was, in fact, a heap, the second would be heapier than the first. Of course, after many such iterations, one would be left with a flat beach-like surface, perhaps with some undulations, but less than one inch deep. As noted earlier, although everyone could agree that there were now *far* more grains of sand than one originally began with, the sand would *not* be a heap, and it would not be heapier than the original mound, which *was*, clearly, a heap.

One can see how this version of the Paradox of the Heap runs. There is a spectrum of piles of sand, ranging from the first to the last, such that the second is heapier than the first, the third is heapier than the second, and so on. If “heapier than” is a transitive relation, as one might have thought it *must* be, then the last pile must also be heapier than the first. But the last pile is like a beach. Correspondingly, we must either give up the view that the last pile is not a heap, and is not heapier than the first pile, which clearly *is* a heap; we must reject the view that “heapier than” is a transitive relation; or we must find at least one spot along the spectrum of alternatives where the spectrum's (*n* + 1)th member was not heapier than the *n*th.

The key to this argument, of course, is that we seem to apply *different* criteria in making different comparisons of sand piles. When we compare piles whose shapes are “sufficiently” similar, we give significant weight to the *number* of sand grains in determining which of two piles is heapier. But when the difference in shapes is large enough—and, more specifically, if one is clearly mounded, while the other is basically flat—then as long as there are *enough* grains of sand in the mounded pile, we will count that as a heap, and heapier than the flat-shaped pile, no matter *how* many more grains of sand there may be in the flat pile than in the mounded one.

(p.533) As should be clear, analogues of this argument could be presented for any Standard Sorites Paradox, as long as there were at least two dimensions underlying the Paradox's central notion, and we thought that they were related along the lines in question: specifically, if “sufficiently” small losses along one dimension could be made up for by “sufficiently” large gains along other dimensions, but “sufficiently” large losses along the first dimension could not be made up for by gains along the other dimensions, no matter how large those gains might be. As noted in chapter 9, such Revised Sorites Paradoxes will raise all the difficulties of my Spectrum Arguments and not be amenable to the same kind of solutions as Standard Sorites Paradoxes.