The Paradox of the Heap

Consider the following. Before you stands a heap of sand. You have just returned home from the beach, and while there, you grabbed a bunch of sand, placed it in a jar, took it home, and dumped it onto the floor. These individual grains of sand are collectively a heap. We can both agree to this as “heap” is a rather vague description for a mound or pile of a particular substance. Now, there are probably hundreds upon hundreds of grains of sand in this heap. Let’s suppose there are exactly 10,000 grains. If we take exactly one grain of sand out of the heap, we would then have 9,999 grains of sand, and we would both still agree that 9,999 grains of sand is a heap. Likewise, if we remove another grain of sand, we are left with 9,998 grains, and it would still be a heap. Removing one grain of sand is so trivial a difference that it does not change whether the remaining grains are a heap.

You might already see a potential problem with this argument. If we accept the premise that removing one grain of sand does not change whether the remaining grains are a heap, we can then continue removing grains of sand until we reach a fantastic conclusion:

• 9,997 grains are still a heap.
• 9,996 grains are still a heap.
• 9,995 grains are still a heap.
• ...
• 3 grains are still a heap.
• 2 grains are still a heap.
• 1 grain is still a heap?

This is a famous paradox known as the sorites paradox, and it is an example of what logicians call a deductive argument, in which the premises of the argument by definition rule out all the possible rival conclusions except for one. A classic example of such an argument is:

• All humans are mortal.
  
• Mr. Stroh is human.
• Therefore, Mr. Stroh is mortal.

The key term here is “all”. By definition, the first premise rules out every conceivable conclusion that a human is not mortal, so when the second premise asserts that Mr. Stroh is a human, we must conclude that Mr. Stroh is mortal. There is no other choice. Since there is only one possible conclusion, this argument is deductive. Deductive arguments can be extremely powerful. They are the most common type of argument used in mathematics—mathematical operations are often expressly defined, ruling out different things. 1+1 is defined as 2. We wouldn’t even consider 3 or 4 to be a rival conclusion to 2, since 2 is the only possible conclusion by definition.

But even deductive arguments can fail, as the sorites paradox demonstrates. The premise that "removing one grain of sand does not change whether the remaining grains are a heap" rules out every conceivable conclusion where removing a single grain of sand turns a heap into a non-heap. All such conclusions are eliminated if we accept this premise. It must follow, then, that if we continue to remove grains of sand individually in the heap, we must conclude that one grain of sand is a heap. There is no other choice.

The opposite (roughly speaking) of a deductive argument is an inductive argument, which is a collection of premises put forward to support a conclusion among multiple possible rival conclusions. For example:

• A car crashed on the side of the road the other day, killing the driver.
  
• There were no skid marks, indicating that the brakes were not being applied at the time of the crash.
• An investigation into the steering wheel and braking system indicated that they were functional at the time of the crash.
  
• The crash occurred in the evening, and driver had a long day and did not get any sleep the night before.
  
• Therefore, the car crashed because the driver fell asleep at the wheel.

Unlike a deductive argument, the conclusion that the driver fell asleep at the wheel is not the only possible conclusion that could explain the crash—for example, the driver could have been trying to commit suicide—but the existence of rival conclusions per se does not make this a bad argument. In fact, in my opinion, the argument is quite sound.

There is no academic consensus as to how the sorites paradox should be resolved. Some scholars propose that we should simply accept as a definition that an arbitrary number of grains of sand is no longer a heap. Some other scholars propose more complicated methods of using group consensus. The source of the problem’s complexity is the vagueness of the term “heap”. If there were a universally accepted definition of what is and what is not a heap, then we would not have this problem. (Perhaps we could universally accept that one grain is not a heap, but what about 10 grains or 25 grains or 50 grains? At what point is the cut-off, and why is a one grain difference so significant in that case?) But instead, we may end up having to resort to arbitrary definitions of “heap" that varies by individual.

The sorites paradox reminds us that even deductive arguments are not infallible. We might logically rule out all possible rival conclusions to a problem, but the conclusion we are left with just does not make any sense. We as humans tend to highlight our ability to reason and apply logic as a trait that sets us apart from other animals, but ironically, even deductive logic fails occasionally to solve problems to our satisfaction.