This was written in 2009. (Revised 2022.) 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!

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.

Colourwise, any two adjacent squares look the same.

These facts may seem innocuous but they in fact generate a paradox since they allow us to “deduce” that the

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.

The reasoning is brutally simple, but it must surely contain a flaw, since the bottom line (“The last square looks red”) 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, it remains unclear where exactly the reasoning goes wrong.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.

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Ergo, the last square looks red.What

Some people think that pointing this out “resolves” the paradox, but it’s not that simple, of course, because it does not tell us where the reasoning above goes wrong, which is the crucial thing. After all, the two statements that we began with remain true:

The first square looks red.

Colourwise, any two adjacent squares look the same.

But, from these two statements, we seem able to “prove” that the last square looks red, as demonstrated above. This reasoning needs to be addressed explicitly, which is what has proven difficult to do.Colourwise, any two adjacent squares look the same.

The ancient Greeks themselves discussed the paradox in terms of a heap of sand, or sometimes wheat.

Consider a heap of sand—comprised of a billion grains, say, which is roughly the amount pictured below. If you remove just

will still be a heap of sand. After all, a

Repeated reasoning then “proves” that, no matter how many grains of sand you remove, you will still have a heap of sand, even if you pare it down to just one grain! The reasoning, as before, is this:

A billion grains is a heap.

The conclusion of this reasoning (“One grain is a heap”) is again absurd and virtually everyone agrees that it cannot be swallowed. So, as with the coloured squares, it seems that we must “reject the reasoning”—the only question being how. Following the Greeks, this is often called the paradox of the heap. It is one of the oldest, simplest, and yet hardest paradoxes around. Removing one grain from a heap still leaves you with a heap.

Ergo, 999,999,999 grains is a heap. Removing one grain from a heap still leaves you with a heap.

Ergo, 999,999,998 grains is a heap⋮

Ergo, one grain is a heap.Amazingly, there is no consensus to this day on exactly where the reasoning above goes wrong, although various ideas have been proposed. Let’s examine a couple of them here, which we may discuss in terms of the heap of sand. (Similar things may be said of the coloured squares.)

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Some people trace the problem to the term ‘heap,’ which they accuse of being

This way of “rejecting the reasoning,” favoured by the American logician and philosopher W. V. Quine (pictured above), among others, assumes that such vagueness or imprecision is a

For example, in the majority of contexts in which the term ‘heap’ is used, a single grain really makes no difference to what counts as a heap. If you are building a sandcastle, say, and need a heap of sand, it would make no difference if your “heap” contained a grain more, or less, of sand. One grain would make no difference.

If you require a heap of sand for building a road, then your heap would need to be larger, of course, but, even so, one grain of sand would make no difference. In cases like these, a “heap” of sand is just whatever (bell-shaped) accumulation of sand befits the purpose at hand, the precise number of grains being immaterial.

And even if there was some special context where the precise number of grains

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Moved by these concerns, some people have suggested a completely different way of “rejecting the reasoning.”

Granting that words like ‘heap’ and ‘red’ are meant to be vague, they have attempted instead to formulate a more adequate

On this view, championed by the American philosopher Kenton Machina (pictured above), among many others, the fewer the number of grains in a given accumulation, the

By these lights, the mistake in the reasoning was to fail to notice this gradual dwindling (or leakage) of truth. Consider, for example, our opening syllogism:

A billion grains is a heap

On the classical view, since both premises are true, the conclusion must be true as well, which begets the slippery slope. On the current view, however, truth comes in degrees. The first premise possesses the maximum degree of truth, let us suppose. (“A billion grains is Removing one grain from a heap still leaves you with a heap

Ergo, 999,999,999 grains is a heapowing to the slight dwindling of heaphood caused by the removal of one grain. So the conclusion, when drawn, ends up being slightly less true than the first premise.

So the syllogism is not watertight, where truth is concerned, and, by the time we are down to a single grain, the conclusion will have been completely drained of truth. (“A single grain is definitely

As promising as this account may seem, it faces its own hurdles. Many of these are technical ones, relating to the idea that truth comes in degrees. Thus, what determines the degree of truth of a given statement? What determines how much truth is transmitted across a logical inference from premises to conclusion? And so on. These difficulties are vexed and we can’t examine them here, but here’s a simple example of one such hurdle, just to give the idea.

Suppose that a billion grains is

A billion grains is definitely a heap;

which would appear to defeat the point of introducing degrees of heaphood in the first place. Removing one grain from something that is definitely a heap still leaves you with something that is definitely a heap.

Ergo, 999,999,999 grains is definitely a heap;⋮

Ergo, one grain is definitely a heapTo be fair, all of these matters are still up in the air and consensus on them remains elusive. We’ve only scratched the surface thus far, but you can find out more about the above two and various other attempts to resolve this ancient paradox from this book (shown on the right) or from this Stanford encyclopedia entry.

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For the moment, let’s leave the paradox unresolved and return briefly to the point made in the previous section. Recall our observation that a line of reasoning may have a flaw so subtle that we’d never suspect its existence

For example, opponents of abortion often point out that if it is impermissible to abort a foetus one hundred days after conception, when the foetus is recognizably human, then it must be impermissible ninety-nine days after conception too.

After all,

Repeated reasoning then “proves” that the foetus cannot permissibly be aborted even

We cannot blame them for seeing no flaw in this logic, since even those who know that there is a flaw cannot agree on what it is.

This sort of case should worry us, 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

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

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 paradox where the same danger proves to arise.

Achilles & the tortoise

The surprise exam

Newcomb’s problem

Newcomb’s problem (sassy version)

Seeing and being

Logic test!

Philosophers say the strangest things

Favourite puzzles

Books on consciousness

Philosophy videos

Phinteresting

Philosopher biographies

Philosopher birthdays

Draft

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