This was written in 2009. It explains the notion of a logical paradox in a way that comes naturally to me. The conclusion drawn at the end is probably the main thing for me (“We are blinder than we can imagine in matters of straightforward logical reasoning”) but it is not always emphasized by others!
7. The heap
Consider the myriad coloured squares below and these two obvious facts about them:
The first square looks red. Colourwise, any two adjacent squares look the same.
These two facts generate a paradox since they allow us to “deduce” that the last square must look red, a proposition that is patently false, since the last square is clearly blue.
All the same, the deduction is as follows:
The first square looks red. But any two adjacent squares look the same. Ergo, the second square looks red. But any two adjacent squares look the same. Ergo, the third square looks red. But any two adjacent squares look the same. Ergo, the fourth square looks red. But
. . .
Ergo, the last square looks red.
This reasoning is extremely simple, but it must also be completely flawed, since the bottom line cannot possibly be swallowed. This was known to the ancient Greeks, who knew this paradox well, but, to this day, despite no shortage of suggestions, no one has given a satisfactory account of the “flaw” in the reasoning!
The ancient Greeks themselves discussed the paradox in terms of a heap of sand, or sometimes wheat.
A lone grain of sand, they observed, is obviously not a heap of sand. But, equally obviously, adding one grain of sand to something that is not a heap of sand, will not suddenly turn it into a heap of sand.
It follows that two grains of sand is not a heap of sand either.
Repeated reasoning then “proves” that you can never have a heap of sand, no matter how many grains you add! Following the Greeks, this is often called the paradox of the heap. It is one of the oldest and hardest paradoxes around.
It also illustrates our point well since, with our coloured squares, it was only the outrageous conclusion (“The last square looks red”) that made us question the reasoning. If the conclusion had seemed agreeable, we would almost certainly have bought the reasoning! After all, the reasoning seems impeccable even when we know it must be flawed. What more if there was no indication?
Indeed, people unfamiliar with this paradox employ this sort of reasoning all the time, perfectly convinced of their logic!
For example, in the oft-heated abortion debate, “pro-lifers” often argue that if it is impermissible to abort a foetus one hundred days after conception (when everyone agrees that the foetus is recognizably human), then it must be impermissible 99 days after conception too.
After all, one day can’t make a difference – the foetus doesn’t suddenly turn human overnight!
Repeated reasoning then “proves” that the foetus cannot permissibly be aborted even one day after conception, notwithstanding its then just being a clump of cells.
We cannot blame them for seeing no flaw in this logic, since even those who know there is a flaw cannot say what it is.
This case should worry us though, for how many others like it exist that we have no inkling of? In the present case, we are lucky enough to have realized that the given style of reasoning is flawed, even if we can’t quite pin down the flaw, and so word can at least go around not to trust this style of reasoning. (Logicians call it the slippery-slope fallacy.)
But might there be other cases where it hasn’t yet dawned upon us that a certain style of reasoning is flawed, even though we employ it all the time, perfectly convinced of its validity? The answer is almost certainly yes, so the only question is how widespread the phenomenon is.
Unfortunately, by its very nature, we have no real way of knowing. But there is reason to suspect that the phenomenon is more widespread than we might think. Let’s consider another famous case where the same danger proves to arise.