Here's an argument I came across recently:

I don't think any of us would be willing to accept this conclusion. However, the structure of the argument is fine: it's the same form of argument that mathematicians use routinely (in proofs by induction). And both premises seem to be reasonable - at least, at face value. So, what gives?

Perhaps one of the premises is false. My first instinct when confronted with this argument was to deny the first premise: I would define 'heap' as 'a collection of object(s) arranged in no particular order'. This would mean that a single grain of sand would be a 'heap', albeit a very small one. However, this doesn't seem very satisfactory: if someone asked me for a heap of sand, and I gave them a single grain of sand, then they'd probably just throw the grain of sand back in my face . Even if I'm willing to accept that one grain of sand is a 'heap', most others won't.

So, perhaps the second premise is false? Perhaps there's some magic number of grains of sand (say, 10,000 grains) at which a non-heap becomes a heap. Unfortunately, I'm not happy with this solution either: it seems very arbitrary, because a pile of 9,999 grains of sand is not meaningfully different from a pile of 10,001 grains of sand. Heck, if I piled up 9,999 grains of sand, and then piled up 10,001 grains of sand, and asked which of these two piles had the most grains, I don't think any of us could give the correct answer with any confidence (I certainly couldn't!). Or, going back to my earlier attempt to solve this, I could just modify my definition of 'heap' to 'a collection of multiple objects arranged in no particular order'. At this point, one grain of sand is not a heap, but two grains of sand would be, provided that they weren't arranged in any particular order. This seems slightly more reasonable; however, if someone asked me for a heap of sand, and I gave them two grains of sand, then I suspect the outcome would be pretty much the same (the only difference is that I'd have twice as much sand thrown back in my face as before ).

So, I decided to look up a proper definition of 'heap'. The Oxford English Dictionary defines the 'heap' as "An untidy collection of objects placed haphazardly on top of each other". So, we'd have three separate conditions for a bunch of sand to be a 'heap':

So, if I have four grains of sand lying on the floor next to each other, they don't form a 'heap', because they aren't on top of one another. However, if I put a fifth grain of sand on top of those four, then I've created a 'heap' (thus refuting the second premise). The heap I've made is still very small; however, it has all of the required structural features to be a 'heap'.

So, are you happy with that solution - or, are you going to throw those five grains of sand back in my face ?

Quote:Premise 1: A single grain of sand is not a heap.

Premise 2: Adding one grain of sand to a non-heap does not turn it into a heap.

Conclusion: There is no such thing as a heap of sand.

I don't think any of us would be willing to accept this conclusion. However, the structure of the argument is fine: it's the same form of argument that mathematicians use routinely (in proofs by induction). And both premises seem to be reasonable - at least, at face value. So, what gives?

Perhaps one of the premises is false. My first instinct when confronted with this argument was to deny the first premise: I would define 'heap' as 'a collection of object(s) arranged in no particular order'. This would mean that a single grain of sand would be a 'heap', albeit a very small one. However, this doesn't seem very satisfactory: if someone asked me for a heap of sand, and I gave them a single grain of sand, then they'd probably just throw the grain of sand back in my face . Even if I'm willing to accept that one grain of sand is a 'heap', most others won't.

So, perhaps the second premise is false? Perhaps there's some magic number of grains of sand (say, 10,000 grains) at which a non-heap becomes a heap. Unfortunately, I'm not happy with this solution either: it seems very arbitrary, because a pile of 9,999 grains of sand is not meaningfully different from a pile of 10,001 grains of sand. Heck, if I piled up 9,999 grains of sand, and then piled up 10,001 grains of sand, and asked which of these two piles had the most grains, I don't think any of us could give the correct answer with any confidence (I certainly couldn't!). Or, going back to my earlier attempt to solve this, I could just modify my definition of 'heap' to 'a collection of multiple objects arranged in no particular order'. At this point, one grain of sand is not a heap, but two grains of sand would be, provided that they weren't arranged in any particular order. This seems slightly more reasonable; however, if someone asked me for a heap of sand, and I gave them two grains of sand, then I suspect the outcome would be pretty much the same (the only difference is that I'd have twice as much sand thrown back in my face as before ).

So, I decided to look up a proper definition of 'heap'. The Oxford English Dictionary defines the 'heap' as "An untidy collection of objects placed haphazardly on top of each other". So, we'd have three separate conditions for a bunch of sand to be a 'heap':

- There must be multiple grains of sand. One grain of sand is not a 'heap'.

- The sand must be in an untidy arrangement. This block of sand is not a 'heap', because it's too orderly.

- The grains of sand must be on top of one another. The grains of sand on the underside of sandpaper are not a 'heap', because they are not on top of one another (and also because sandpaper no longer has real sand on its underside )

So, if I have four grains of sand lying on the floor next to each other, they don't form a 'heap', because they aren't on top of one another. However, if I put a fifth grain of sand on top of those four, then I've created a 'heap' (thus refuting the second premise). The heap I've made is still very small; however, it has all of the required structural features to be a 'heap'.

So, are you happy with that solution - or, are you going to throw those five grains of sand back in my face ?

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Moonface (in 'Woman runs 49 red lights in ex's car')' Wrote: If only she had ran another 20 lights.

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