You're Underestimating the Significance of the Problem

About the Author

Bryan Frances is the world’s only intellectual wisdom coach. He’s a former professor of philosophy & logic, doing research & teaching at universities in the US, UK, Europe, Asia, and the Middle East. He teaches you how to become the wisest thinker in the room—which is different from being the most knowledgeable or having the highest IQ. Contact for a free session.

After many years of doing philosophy, I have come to the conclusion that most people underestimate the significance of certain traditional philosophical problems. This is true even of professional philosophers.

One such problem is the sorites paradox. Many philosophers think it’s a problem only if you embrace classical logic, or have false views about ambiguity, context dependence, theories of truth, or semantic incompleteness. I disagree. It’s not difficult to see that the fundamental problem can be generated while sidestepping any claims about ambiguity, context dependence, or semantic incompleteness. In addition, I don’t think it has anything to do with truth. Moreover, in order to see the difficulty of the problem, we don’t have to fuss with any of the more complex aspects of vagueness, such as the nature of borderline cases.

I won’t present the rigorous arguments for those claims here. Instead, I’ll present the more user-friendly versions.

1. Alleged Commonsense about Vagueness

Most sentences we use contain vague words. The word ‘yellow’ is vague because there are many shades of color on the border between yellow and orange (as well as yellow and green). You can’t say they’re yellow but you can’t say they’re not yellow either. Even if you have the shade of color right in front of you, your eyes are working just fine, and you know all about its physical properties (such as the wavelengths of light that it reflects), you still wouldn’t be able to classify it as ‘yellow’ or ‘not yellow’.

The source of the problem is not that you need to know more about color. Instead, the problem is that the term ‘yellow’ doesn’t have an exact definition that allows us to figure out, for any particular shade of color, whether it counts as yellow or not. We can figure it out for many color shades, but not all of them. No one ever gave the word ‘yellow’ a perfectly exact meaning.

Or consider the word ‘chair’. There are weird pieces of furniture that are somewhat similar to normal chairs, but they are really different as well. Even if you know everything about chairs, and you know everything about the weird piece of furniture in front of you, you may still not be able to classify it as a chair or not. The term ‘chair’ doesn’t have an exact definition that allows us to figure out, for any particular piece of furniture, whether it counts as a chair. Most of the time it’s easy to tell whether a given object is a chair, but not always.

Sometimes it’s a whole phrase, instead of a particular word, that’s vague. Consider the phrase ‘tall for a 15 year old girl in Hong Kong in 2025’. Suppose you know that Mia is a 15 year old girl living in Hong Kong in 2025, she is 170 cm tall, and the average height for a 15 year old girl in Hong Kong in 2025 is 160 cm. Even with all that information, you can’t tell whether she is “tall” for that group, since ‘tall’ is too imprecise. Your failure to be able to classify Mia as tall or not tall isn’t due to some ignorance on your part. You might know the height of literally every single girl in Hong Kong that year. Instead, your ignorance is due to ‘tall’ not having an exact meaning. If Mia was 162 cm, she definitely wouldn’t count as tall; if she were 190 cm, she would definitely count as tall. But there are lots of intermediate heights that don’t count as either ‘tall’ or ‘not tall’, due to the incompleteness of the meaning of ‘tall’.

Someone could come along and attach highly precise meanings to ‘yellow’, ‘chair’, and ‘tall for a 15 year old girl in Hong Kong in 2025’. For instance, we might say that an object counts as “yellow” in this technical sense = 95% of the light it reflects is 570-590 nanometers in length. That’s fine. We do things like that on certain occasions, like when economists and lawmakers offer relatively precise definitions of ‘poverty’ for the purposes of legislation, record keeping, and economic forecasting. But when those words are used in ordinary, everyday circumstances, they are vague in the ways I described.

Everything I just wrote strikes most people as being pretty obvious. It’s hardly an earth-shaking announcement to say that lots of words and phrases are vague. Even so, there’s a powerful argument that seems to show that there’s a fatal mistake in what I wrote above.

2. Sharpism

Suppose Joe says to his friend, ‘The restaurant I was telling you about is a short walk from here’. That’s the sentence he uttered out loud. Joe and his friend are in Manhattan, just saw a play at a theater, and now want to get a bite to eat. If the restaurant is a mile away, then his sentence is false. That’s because a mile is definitely not a “short walk” when you’re in Manhattan. If the restaurant is 50 feet away, then his sentence is true. If the restaurant is 50 feet and one inch away, then his sentence is true. It’s still true if the restaurant is 50 feet and two inches away.

According to the view I’ll call Sharpism, Joe’s sentence stops being true at a sharp cutoff: it’s just plain true when the restaurant is a certain distance away, but it’s not just plain true if the restaurant is even just one nanometer farther away—even though Joe has never made any fancy linguistic definitions and doesn’t even know what a nanometer is. He is an ordinary speaker of English. Sharpism says that Joe’s sentence had a perfectly precise meaning, totally unbeknownst to him and despite the fact that no one has ever been odd enough to attach a fantastically precise meaning to his words.

In general, sharpism says that our sentences from ordinary, everyday life have supernaturally precise meanings even though no one ever gave them such meanings. (For those who read the literature, epistemicism is only one of the varieties of sharpism.)

You might be thinking to yourself ‘Why would anyone accept a view as crazy as sharpism? Does anyone actually believe it?’ The incredible truth is that there is an extremely strong argument for sharpism, one that can be illustrated with a story.

3. Jo at the Farm, Looking for a Pumpkin

A few days before Halloween, Jo and her niece are walking on Farmer Fred’s farm, where there is a pumpkin patch. Her niece wants to pick out a pumpkin to take home and carve for Halloween. Jo says to her, after they get out of the car, ‘There is a pumpkin by the tree’. Jo can see a single large tree about 100 feet away. She wasn’t able to see any pumpkins by that tree, since her view was blocked by a tool shed between her and the tree, but she remembers from last year that there were pumpkins near that huge tree she and her niece see towering behind the shed.

Suppose that you, Jo, uttered that sentence to your niece at noon. Ordinary common sense says that you spoke truly: just as you said, there was a living pumpkin roughly a foot away from the very tree you indicated. Call the situation you were in at that time S₁; so your sentence ‘There is a pumpkin by the tree’ is true when evaluated with respect to S₁. S₁ is the whole situation you were in at noon on that day: it’s a snapshot of what the ordinary world was like at a certain instant in time.

Now I want you to imagine the story being ever so slightly different: imagine that the top of the pumpkin had had one fewer molecule on it. Otherwise, everything is exactly the same: your body, your mind, your entire history, the tree, the rest of the universe, the English language, and so on. Let S₂ be that alternative imaginary but perfectly realistic situation, with the single microscopic difference from S₁. In both S₁ and S₂ you do the very same things in the very same ways, down to the last detail: bringing your niece to the farm, seeing the tree, and saying ‘There is a pumpkin by the tree’. Everything is exactly the same in your own mind and behavior. Our initial question is this: in S₂, was your sentence ‘There is a pumpkin by the tree’ true?

Obviously, it’s very reasonable to think that the answer is ‘Yes, of course’. The only difference between S₁ and S₂ is the absence of a single molecule. So, given that your pumpkin sentence was true in S₁, it must have been true in S₂.

Now imagine situation S₃, which is exactly like S₁ except that the top of the pumpkin has two fewer molecules on it. S₃ is exactly like S₂, which is exactly like S₁, with the exception of a molecule or two on the top of the pumpkin. Our new question is this: in S₃, was your sentence ‘There is a pumpkin by the tree’ true? And of course, just as before with S₂, it’s very reasonable to think that the answer is ‘yes’.

Repeat the process over and over: each time consider a pumpkin with one fewer molecule in it, starting from the top of the pumpkin and working down. For each imaginary S situation consider the question ‘Was Jo’s pumpkin sentence true in that situation?’ We are evaluating your pumpkin sentence with respect to many zillions of situations, the Ss.

One could do this literally a billion times and the first and the billionth situations, S₁ and S_billion, would still look identical to the naked eye: a difference of a billion molecules from the top of a pumpkin would be literally undetectable without scientific instruments, as molecules are so small and so numerous in a pumpkin.

But if we add up enough of these microscopic differences, all that will be left of the pumpkin is … nothing. The tree will still be there, and there will be grass around it, but there will be no pumpkin material whatsoever. So S_big, which is a situation when there’s no more pumpkin material, isn’t a situation with a pumpkin by the tree. Your pumpkin sentence evaluated with respect to that situation, S_big, is false. When your niece runs over to the tree in S_big she complains to you that there’s no pumpkin by the tree—and of course she’s right.

At what point in the sequence of situations, the Ss, did your sentence ‘There is a pumpkin by the tree’ there stop being true? We started with a nice, healthy pumpkin and then considered a sequence of possible situations, each nearly identical to the one before in the sequence. Eventually, we had a situation with no pumpkin by the tree. So, when did Jo’s sentence ‘There is a pumpkin by the tree’ go from just plain true to something else?

The sharpist (the person who accepts sharpism) says this: I don’t know when the pumpkin sentence goes from just plain true to something else, but I do insist that there’s got to be two consecutive situations in which that’s exactly what happens.

However, when faced with the question ‘When did Jo’s pumpkin sentence go from just true to something else?’ most people will insist that there is something wrong with the question. The question is demanding a particular cutoff: for some number N, in situation S_n the pumpkin sentence is true, but in situation S_N+1, which differs from S_N by exactly one molecule, the pumpkin sentence isn’t just plain true. But of course, most of us are inclined to say, there is no such sharp cutoff! Instead, we insist, the pumpkin goes away in a gradual manner, not all of a sudden. Imagine seeing a sequence of photographs of the pumpkinish thing, starting with S₁ and proceeding through S₂, S₃, and the rest in rapid succession. It would look like a pumpkin being very slowly destroyed from the top down, particle by particle. There would be no one point where you could say with any confidence ‘Right there! That’s the very point when the pumpkin sentence goes from being true to something else. An instant before it was true, but with that one particle gone it’s no longer just plain true’.

Yes, that is the entirely reasonable thing to say, at least before we delve into logic and philosophy. No doubt about it. However, a compelling line of reasoning seems to prove that it’s wrong. And after 2000 years of investigation by philosophers and logicians, there is no stable consensus regarding what’s wrong with the reasoning, if there’s anything wrong with it at all.

4. The Sharpist’s Argument

We have a single sentence, Jo’s pumpkin sentence, evaluated in each of many very similar situations S₁, S₂, S₃, etc. The first situation involves a perfectly ordinary, full-grown, and healthy living pumpkin by the tree. Each situation differs from the immediately previous situation by a comically miniscule difference. We end up with the following puzzling table:

Situation Truth Status of the Pumpkin Sentence

• S₁ True

• S₂ True

• S₃ True

• … …

• S_n ???

• … …

• S_{big – 2} False

• S_{big – 1} False

• S_big False

Perhaps the entries start out, at the top of the right column, with ‘true’ and then, after many rows, change to ‘false’ for the next row. Some people might disagree, thinking that there are statuses other than ‘true’ and ‘false’, such as one or more of these:

• true and false

• neither true nor false

• neither true nor not true

• both true and not true

• meaningless

• indeterminate

• indeterminately indeterminate

• indeterminately indeterminately indeterminately ….

• incoherent

• 93.07768% true

• sort of true

• [no status at all]

• [weird statuses]

For my purposes, it doesn’t matter what goes in the right column, other than this one simple and apparently obvious fact: not all the entries in the right column are the same. I don’t care what all the fancy truth statuses might be, like in the bullet list above. I don’t know, and I don’t care. But I do know that not all the statuses are the same!

If that’s true, then the string of ‘true’ entries starting from the top has to stop at some point. I don’t care what goes in the next row. Maybe it’s ‘false’ or ‘less than 100% true’ or ‘neither true nor false’ or one of the other ideas listed above. In any case, for some number N, there are situations S_N and S_N+1 such that the pumpkin sentence doesn’t have the very same truth status in those two situations.

In order to see why there must be two consecutive rows with different entries in the right column, consider a different table in which the entries in the right column are a bit different:

Situation Status of the Pumpkin Sentence

• S₁ 1

• S₂ 1

• S₃ 1

• … …

• S_n ???

• … …

• S_{big – 2} 56

• S_{big – 1} 56

• S_big 56

You don’t know what the numbers mean and you don’t know what limitations there are on the numbers that can appear in the right column (e.g., whether there can be fractions like 7/8 in addition to whole numbers like 1 and 56). However, even when saddled with this ignorance a quick glance at the table is all you need to feel perfectly confident in concluding that the sequence of ‘1’s that starts the top of the right column has to end at some point, and thus there will be a pair of consecutive rows X and X + 1 such that row X is the last one of the initial sequence of ‘1’s and in the next row X + 1 there is something other than the simple ‘1’. That’s a sharp cutoff. You have no idea whether at the first cutoff the numbers go from ‘1’ to ‘56’ or ‘1’ to ‘2’ or ‘1’ to ‘-34.45’ or ‘1’ to ‘1.0000000000001’ or ‘1’ to ‘Beethoven’ or ‘1’ to ‘I love you’. For all you know, after the initial string of ‘1’s there is a blank in the next row. But you do know, immediately and with very little thought, that there simply must be a last row, counting down from the top, that has a simple ‘1’ in it and the next row will have something else (or a blank).

So, for some number N, there are situations S_N and S_N+1 such that Jo’s pumpkin sentence doesn’t have the very same truth status in those two situations.

Remember that any two “consecutive” situations differ only in one molecule from the pumpkin thing. In particular, you are identical in the two situations: same body, same mind, same thoughts, same behavior, same everything. Perfectly identical. And the English language is identical too, with all the same words with all the same meanings. So, it makes sense to think that your sentence ‘There is a pumpkin by the tree’ has the very same meaning in each situation.

Since the sentences in S_N and S_N+1 have the same meaning, like I just suggested, then a perfectly ordinary use of ‘There is a pumpkin by the tree’ has one truth status with respect to one situation, but a perfectly ordinary token of that sentence with the very same meaning doesn’t have that status with respect to a situation that differs from the first in just one molecule. That is, there is at least one sharp cutoff.

5. Why the Sharp Cutoff Seems Impossible

The primary problem with sharp cutoffs is the lack of any plausible story of the acquisition of such cutoffs: we can’t even come close to seeing how an ordinary vague sentence could get such a cutoff.

We can see how ‘The pumpkin by the tree weighs at least 8.888 kg’ is true when evaluated with respect to one situation A and then not true when evaluated with respect to situation B even though the only difference between A and B is miniscule: a loss of one gram of mass. But in this particular case the sentence contains words that allow us to see where the sensitive meaning comes from. The same holds for sentences such as ‘I have more than 43,398.23 USD in my bank accounts at this moment’.

This is a valuable lesson: at least some sharp cutoffs for vague sentences aren’t puzzling—when we can see, at least in outline, how the sentence acquired such a sensitive meaning.

The problem with sharpism is that we don’t see anything in ‘There is a pumpkin by the tree’ that could give it a comically sensitive meaning, one that makes it change truth statuses when evaluated with respect to situations X and Y even though the only difference between X and Y is incredibly miniscule. Philosophers and linguists think that the meanings of our words are fixed by a whole bunch of factors, such as a person’s intentions, assumptions, linguistic behavior, and mental states, along with facts about her words that don’t come from her as an individual. But no one has ever come up with any idea of how those factors could make ‘There is a pumpkin by the tree’ have a meaning that changes its truth status with the subtraction of a single molecule. That would require a miracle.

We can run the sharpism argument on ‘I wish I had a healthy snack to eat’, ‘Please bring me a glass of cold water’, or ‘Stand over there while I go use the restroom’ to show that there are sharp cutoffs corresponding to wishes, requests, and demands instead of fact-stating sentences such as the pumpkin one. You say ‘Please bring me a glass of cold water’, and in the first situation they bring you a full glass of water at a temperature of 37 degrees Fahrenheit. Your request is fulfilled. In the second situation the water is at 37.000001 degrees; in the third situation it’s at 37.000002 degrees. And so on.

Another example. In one situation my daughter and I are in a public place and I say to her ‘Stand over there while I go use the restroom’ while I point to an area right outside the restroom. She does so, and my demand has the status of being complied with. In an alternative situation everything is exactly the same but she stands one picometer farther to the left (one picometer = one trillionth of a meter). In the next situation she is two picometers to the left. And so on. Are we really supposed to believe that my sentence ‘Stand over there while I go use the restroom’ had a meaning such that it changed its “complied with” status in the space of a picometer?

*****

Summing up, there is an excellent argument that Jo’s pumpkin sentence has a fantastically precise meaning, but there’s also an excellent argument that it can’t have such a meaning. That’s the paradox.

There’s a lot more to say about what the basic problem is, but I think I’ve said enough to give you a rough idea of the problem.