Heaps of Paradox
Using a philosophical puzzle to look more closely at how human concepts work
Readers who have been following for a while may remember a previous post about the Liar Paradox - the topic of my doctoral thesis. There is another philosophical paradox that sometimes gets discussed alongside the Liar Paradox - the Sorites Paradox - named after the Ancient Greek word for a ‘heap’. Like the Liar Paradox, thinking it through reveals useful information about the relationship between language (and how we use it), logic and reality. In each case, the paradox reveals problems with common (philosophical) assumptions.
The classic presentation of the paradox begins by imagining there is a heap of sand sitting on the ground somewhere. Then imagine someone carefully removing one grain of sand from it. We then ask if it is still a heap of sand afterwards. Obviously, removing one grain of sand will make no difference. However, if that person (who clearly would have to be as pedantic as a philosopher) was to keep on removing individual grains of sand from the heap, at some point there would no longer be a heap of sand.
The paradox arises because while removing one grain of sand never makes a difference, removing one grain of sand lots and lots of times does. To put it in slightly more formal terms, it seems obviously true that "If N grains of sand make a heap, then (N-1) grains is still a heap" for any number N. But also it is true for some big enough number K that "If N grains of sand make a heap, then (N-K) grains is not a heap". However, on any rigorous logic, both can’t be true.
While the classic form of the paradox is about sand, the same problem arises with many common concepts. We could set it up by asking how many metres we have to take off the top of a mountain before it becomes a hill. Or at what point shifting the shade of colour of an apple will turn it from red to orange. Or how much we need to shrink the back of a chair before it becomes a stool. All of these concepts are described by philosophers as vague concepts as they don't have precise definitions.
One natural response to the Sorites Paradox is to wonder why anyone cares. It is obvious that many of our concepts are not precisely defined and therefore trying to force precise definitions on those concepts is not going to work. However, looking at the Paradox, and the types of solutions proposed, helps us better understand how human concepts and language work.
Possible solutions - and why they don't work
Logically, the natural way to resolve a paradox like this is to question the assumptions that lead to it - and the obvious target is the statement that "If N grains of sand make a heap, then (N-1) grains is still a heap."
One option is to argue that removing one grain of sand does make a difference - it makes it ever so slightly less of a heap. If each grain of sand makes it less of a heap, then there can be no surprise that we eventually end up with no heap left. On this, fuzzier, way of thinking, whether something is a heap of sand is not a straightforward yes/no binary, but instead there are shades of grey. If we wanted to quantify it, we would be dealing with a way of thinking that would argue that in a particular situation it is 85%, or 42%, a heap.
While this resolves the strict paradox, it doesn't resolve very much in a useful way. At some point, as we remove grains of sand, there will come a point where we agree that there is no longer a heap. But where do we draw that line? To quantify it, is it at 60% or 40% or 20%? We have just shifted the problem, rather than solving it.
A related response to the paradox is to argue that there is no sharp line between there being a heap of sand and there no longer being a heap. Instead, there are a range of situations in the middle where it isn't clear and we wouldn’t decide either way. There is significant intuitive appeal behind this, but it doesn't resolve the paradox. We now have two puzzling boundaries instead of one. At what point does it go from definitely being a heap to it being unclear? And then from being unclear, to definitely not a heap? Again, we have just shifted the problems rather than resolving them.
A further approach is to bite the bullet and argue that "If N grains of sand make a heap, then (N-1) grains is still a heap" isn't always true as there is, in fact, a precise cut off. This is the approach that geologist and geographers, for example, sometimes take for mountains. For example, in the United Kingdom, a summit of at least 2,000 feet (610 metres) is officially defined to be a mountain and anything smaller is a hill.
A big problem with this approach is that people commonly disagree about where the precise line should be and there is rarely an accepted authority who would decide. Who definitively decides when a collection of grains of sand becomes a heap?
A more interesting approach is to argue that each individual person has a precise line in mind when they use the concept, but different people have different individual, albeit substantially overlapping, definitions.1 Colours are a useful example of this idea. Two people will often agree on one thing being red, and another being orange, but strongly disagree on the colour of something that falls in between.
While this provides a neat resolution, it doesn't comfortably match how we use concepts. In all the sorts of examples we have looked at, there are a range of cases in the middle where our responses will typically be: "I'm not sure" or "It's somewhere in the middle" or "It depends." For example, there are geographic features that if I am asked whether it is a hill or a mountain, my response would be non-committal: "It could be either." As humans, we will often resist drawing a line when we are pushed to provide a precise definition, which suggests we don’t actually have precise definitions in mind.
There are other approaches to the paradox that argue that we need to take into account context, or group consensus, in our definitions. And there have been a number of complex proposals build on various mathematical approaches. However, at any pragmatic level, these all tend to just shuffle the problem around rather than solve it.2 Instead it is more useful to think about the assumptions underlying the arguments that produce the paradox.
Think differently about concepts
The problems arising from the Sorites Paradox all bite when we use a particular idea of what a concept is and how it should work. Philosophers, mathematicians and scientists are commonly in the business of formulating precise definitions of concepts, as these are assumed to be the gold standard. On this way of thinking a genuine concept is one that is precisely defined and fully determined, so that for every relevant situation it will be clear and decidable whether the concept holds or not. This matters: for example, we don't want vets to be confused about whether the animal in front of them is a cat or a dog.
However, the default assumption is often that this is the only way that concepts can work. If a concept isn’t precisely defined and fully determined, then it isn’t a genuine concept. And if this is the case, then the Sorites Paradox creates problems as there are many useful concepts that can’t really be precisely defined.
My suggestion is that, similar to the situation with the Liar Paradox, this just shows there is a gap between how language works and how we think it should work. We should not assume human concepts are precisely defined and fully determined. Instead, like our statements and broader language, our concepts are more like sketches of reality, not precise and accurate descriptions.
A sketch of a house picks out a few features that helps us know some useful things about it. Similarly, a concept like a heap or a mountain picks out a few features of reality that help us know some useful things about it. But a sketch of a house is rarely fully determined in the sense above. It sometimes isn’t clear whether it is a picture of the house in front of us. In the same way, a concept like a heap or a mountain is sometimes indeterminate as it isn't clear whether it applies to what is in front of us.
Like sketches, many of our concepts are abstractions of reality that are partial (i.e. not fully determined) and pick out some relevant features of reality, but don't provide a fully precise definition. So a mountain signifies something like a particularly large, tall and prominent landform that isn't easy to walk all the way up. And a heap of sand means there is enough sand in the one spot to be usable or to get in the way - you can't ignore a heap of sand in the way you can ignore grains of sand. These partial abstractions tell us a lot of useful and relevant information, but not everything.
For the purposes of philosophy and science, we prefer definitions that are context neutral and universally applicable. So we want a scientific definition of a heap of sand to be the same everywhere, and so we tend to start counting grains of sand. Similarly, we apply precise measurements of height to mountains and wavelengths of light to colours. However, our human concepts are often partial abstractions - sketches - that work very differently.
For one, many of our concepts are anchored to, or defined by, functional considerations. For example, a chair is something designed for you to sit on and has a backrest for your back. A stool, by contrast, doesn't have a functional backrest. There are plenty of designs which have something of a backrest but it is unclear whether it plays the right function, or it depends on who is using it. Something might be a chair for a little child, but only a stool for me. The application of concepts like these vary by outcome, function and context.
Other concepts are relative, or contrastive, in how we use them. Deciding whether someone is tall is entirely a matter of contrast with the norm in some context. A tall gymnast will look very small compared to a small NBA player. The important fact here is tall denotes a noticeable height difference to the norm, where the norm depends entirely on context and the relevant reference group - and where the extent of difference is also not precisely defined.
These examples show ways in which our concepts only pick out a few useful features of reality and can’t be translated into any kind of universal, precise definition. They often tell us the information that is useful in a particular context and don't try to do anything more. They are useful sketches and abstractions, not precise definitions.
To go back to the Sorites Paradox, the problems arise when we decide that the number of grains of sand are the crucial factor for any definition of a heap. However, what we would consider a heap varies by context. If I found a heap of sand on my kitchen floor, it would look very different from what I'd consider a heap of sand on the construction site of a new stadium. And other factors, like shape, can change my decision even if the number of grains don’t change. If I accidentally kick the heap of sand in my kitchen and spread it everywhere, I have the same number of grains of sand in the room but no longer have a heap.
This means that the concept of a heap picks out something about the function of the sand and its importance in the broader system and context, not just something particular about its size. Likewise, many of our concepts are useful, partial sketches that pick out some aspects of reality, but are not precise and universal definitions as they are silent about many other aspects of reality.
Timothy Williamson is well known for advancing this position. See https://plato.stanford.edu/entries/sorites-paradox/#EpisTheo
Both Wikipedia and the Stanford Encyclopedia of Philosophy have good, and relatively accessible, summaries of the various solutions that have been proposed. Detailed critiques are beyond the scope of this post.
Years ago I briefly studied vagueness in the forensic science context, mostly focused on the writing of reports. It now strikes me that the main ideas of your text are similar to recent developments in the interpretation of forensic evidence: a “new” paradigm is underway, and it adopts the Bayesian approach to probability theory. The subjective (personal) assignment of a probability value to describe one’s degree of belief is also dependent on context, available information, data, and experience. The frequentist approach to probability tends to “focus on the number of grains of sand”, and leads forensic scientists to ask the wrong questions and examine the wrong data.
Very interesting. It strikes me that the Sorites Paradox might also describe a key challenge in public policy. Often (almost always) public policy efforts are designed to achieve more or less of something. What remains unclear is what a 'good' let alone what a 'right' amount of that something is. Success is defined by difference against the status quo rather than difference with a defined ultimate objective. The end result is that, while some sand remains, it is considered by policy as a heap, because there is no other way of deciding that it is not.