I recently wrote a post about ADHD, where I made mention of the ‘vagueness’ of the symptom criteria. So I thought, just for fun, today I would write a post about vagueness and show you that it’s actually a thing.

In the article “ADHD – probing the controversy” where I summarize the main criticisms of characterizing ADHD as a disorder, I mentioned that vagueness was one bone of contention for skeptics.

I also mentioned that just because it was ‘vague’ didn’t mean it was wrong.

Now I doubt anyone who read that post gave any extra thought to what I meant when I said that the symptoms were vague. But it turns I had something fairly specific in mind.

Get ready to have your minds bent.

The Sorites paradox

So you have a pile of sand… just pretend ok.

Suppose the pile is exactly 100,000 grains of sand.

Suppose you take one grain of sand off the pile.

It’s now 99,999 grains of sand.

Is it still a pile?

Yes?

I think most would agree that it is.

Suppose we take another grain of sand.

It’s now 99,998 grains of sand.

Is it still a pile of sand?

Yes?

It seems fair to say that if it was a ‘pile of sand’ at 100,000 grains, then it’s a pile at 99,999 and 99,998 also right?

Would you agree then that one grain of sand is not enough to make the difference between a ‘pile of sand’ and something that is no longer a pile of sand?

If I remove one grain at a time, each time I remove one grain, it’s just not enough to say that it is not a pile of sand.

If 15,543 grains makes a pile, then surely 15,542 is also a pile right?

So every time we take one grain and ask the question, is it still a pile, we kind of have to say yes right?

So what we can say is this:

If 10k grains = ‘a pile of sand’

Then 9,999 grains = ‘a pile of sand’

If 9,999 grains = ‘a pile of sand’

Then 9,998 grains = ‘a pile of sand’

… And so on and so forth …

Until eventually:

1 grain = ‘a pile of sand’

If 1 grain = ‘a pile of sand’

Then 0 grains = ‘a pile of sand’

Therefore 0 grains of sand equals a pile of sand.

If there’s no specific point that a pile of sand stops being a pile, then it seems to follow that zero grains of sand still constitutes a pile.

Do you see the problem?

The idea is that, we all think of a ‘pile of sand’, and we all know what we mean. We also get the idea that two ‘piles of sand’ beside each other, even if they’re not exactly the same size are still both piles. So it’s fair to say that ‘1 grain of sand is not the difference between a pile and a non-pile’. But if you start with a pile, and remove one at a time, then it never stops being a pile… even when there’s none left!

Think about it…

Keep thinking…

Blowing your mind yet?

Turns out this is a very famous logical problem known as the Sorites paradox, which takes its name from the Greek word soros for ‘heap’.

It’s considered a paradox because the logic of the argument appears completely valid (in fact is very simple and based on the simplest most well established principles of formal logic), and yet the conclusion seems obviously false.

Even in a completely conversational, non-rigorous sense it’s quite perplexing isn’t it? We ‘know’ what a pile of sand is, we know that two piles can be differing amounts, we ‘know’ that if there is no sand, then there is no pile… but we can’t quite, exactly, specifically say when it stops being a pile. If we can’t exactly say when it stops being a pile then, in a sense, we can’t really say that it ever does stop being a pile.

Another way to think about it, is that there are things that we talk about that have what we would call ‘borderline cases’. There’s a point at which we’re less comfortable with the label.

You can clearly see the difference between green and yellow here, but the bit in between is icky. When does yellow stop being yellow and start being green?

It is a problem of vagueness and has fascinated philosophers for millennia. The term ‘pile’ or ‘heap’ is what’s known as a ‘vague predicate’, which is a fancy way of saying it’s just a word that describes something without being very specific.

A pile is a pile is a pile. It’s not made up of a specific number and it’s not supposed to. We don’t want it to have a number, we just want a word to describe a pile of sand without having to count.

Even if you don’t know exactly when it stops being a pile of sand, you might be able to say when it’s definitely not a pile anymore. Like maybe 5 grains can’t make a pile because it’s just too small. Even if we tried to mark a specific point where it stopped being a pile, people’s opinion will differ. But someone else likely thinks 5 is a perfectly acceptable amount to make a pile. So not only can we not specifically pin point the pile/no pile threshold, but even if we could, we would have differing opinions on where it was.

Everything is vague…

It’s not just piles of sand either. A great many things are vague. In fact just about every physical object can be thought of as vague. Even the concept of vagueness is kind of vague.

If I chip the corner of a table, is it still a table? what about a chip in the opposite corner as well… still a table?

You, yes you, are made up of ~100 trillion cells.

If 100, 000, 000, 000, 000 cells makes a human a human, then certainly 99, 999, 999, 999, 999 cells does to… Can you see where I’m going?

If the seriousness, and the depth of the problem has sunk in at all, then by now you’re probably hoping I will rescue you from this existential wormhole with some clever logical or philosophical solution.

Unfortunately I can’t.

While there are certain attempts and answers that have been given, they tend to rely on specific assumptions. They generally all have important and valid counter arguments and flaws, and none of them are considered the consensus in philosophy.

For example Williamson and Sorensen argued that in fact there is an exact point that it stops being a pile, it’s just that we don’t know where it is, and never can!

But as I said, sometimes we’re using words because they don’t have a boundary, that’s the reason we’re using them. Like saying to someone ‘I’ll just be a sec’… You don’t mean exactly a second, in fact you’re being deliberately vague.

Another guy named Peter Unger has argued that actually, the argument is completely valid, and sound, and actually vague objects just don’t exist!

But like many other interesting philosophical paradoxes, the answers to this one are still being debated and there is no universal agreement on any one (so far as I know anyway).

Welcome to the frustration and, paradoxically, the joy of philosophy (pun intended).

There’s something funny about philosophy and problems of logic like this. They’re so frustrating and problematic, it’s kind of creepy. But alas, despite the problem, the world keeps turning. We do not fall into a black hole of absurdity all of a sudden because the Sorites paradox hasn’t been satisfyingly solved. But there are some situations where it can be a very real problem.

A real world example

Put this back into the context of my previous article, the problem is that traits like ‘hyperactivity’, ‘body mass index’, ‘height’, ‘arm length’, are all vague.

What I mean by vague, is that they are measured on a continuous scale.

What is a continuous scale? It’s one that has an infinite number of possible values. Take height for example. There is a ‘shortest person’ and there is a ‘tallest person’, but there is an infinite number of potential heights in-between. No one is exactly, precisely the same height as someone else.

Of course when we measure height we usually say someone is ‘X’ cm tall, and that is an exact number, but we do that for a couple of reasons. Firstly because our ability to measure height is limited by the accuracy of our measuring devices, so we tend to round off. Also we do it because it’s practical. For everyday purposes it’s not very informative to say that someone is 176.9287432984732974243cm tall. It’s much easier and much more useful to say 177cm.

But in theory, we could measure people with infinite specificity. So human height is measured on a ‘continuous’ scale because human height varies ‘continuously’ from one person to the next. If it helps, the alternative type of measurement is a ‘discreet’ measurement. This is when there are only specific categories that some value can take. Like hair colour (technically). You can have ‘brown’, or ‘blonde’ hair, or some other fairly well-defined colour. I’ll go into this in a dedicated post in future, but for now just know that there are ‘continuous’ variables, and ‘discrete’ variables.

What this means is that continuous measurements are vague, and hence the vagueness paradox is relevant to them.

That’s why I mentioned that the vagueness in the ADHD behaviour criteria doesn’t necessarily make it wrong. The same with BMI. There is a threshold for obesity. The threshold is somewhat arbitrary, but that doesn’t matter. It’s there for practical reasons.

Doctors cannot sit around all day speculating over the Sorites paradox and the problem of vagueness. They need to help people. So while it’s vague, and the thresholds are somewhat arbitrary, they are there because they need to be.

ADHD is the same, in theory.

Whether it’s a disorder or not is irrelevant to the vagueness of the criteria. Diagnosticians at some point have to draw the line. It doesn’t mean that it is exactly correct, but just because the behaviours are vague doesn’t mean it’s incorrect.

So that’s what I meant when I said that the behaviours described in characterizing ADHD were vague.

Further reading

https://en.wikipedia.org/wiki/Sorites_paradox – Yes I know it’s Wikipedia, but it does provide a nice overview in quite accessible language and references many of the original sources

https://plato.stanford.edu/entries/sorites-paradox/ – This is a much more technical and in-depth article, which just happens to have been written by my old philosophy lecturer, who taught me the Sorites paradox originally! A brilliant and talented philosophy and logic lecturer.