Crisp and fuzzy categorisation

By Gordon Rugg

Categorisation occurs pretty much everywhere in human life. Most of the time, most of the categorisation appears so obvious that we don’t pay particular attention to it. Every once in a while, though, a case crops up which suddenly calls our assumptions about categorisation into question, and raises uncomfortable questions about whether there’s something fundamentally wrong in how we think about the world.

In this article, I’ll look at one important aspect of categorisation, namely the difference between crisp sets and fuzzy sets. It looks, and is, simple, but it has powerful and far-reaching implications for making sense of the world.

I’ll start with the example of whether or not you own a motorbike. At first glance, this looks like a straightforward question which divides people neatly into two groups, namely those who own motorbikes, and those who don’t. We can represent this visually as two boxes, with a crisp dividing line between them, like this.

However, when you’re dealing with real life, you encounter a surprising number of cases where the answer is unclear. Suppose, for instance, that someone has jointly bought a motorbike with their friend. Does that person count as being the owner of a motorbike, when they’re actually the joint owner? Or what about someone who has bought a motorbike on hire purchase, and has not yet finished the payments?

This was a problem that confronted classical logic right from the start. Classical logic as practised by leading Ancient Greeks philosophers typically involved dividing things into the categories of “A” and “not-A” and then proceeding from this starting point. The difficulty with this, as other Ancient Greek philosophers pointed out from the outset, is that a lot of categories don’t co-operate with this approach. A classical example is the paradox of the beard. At what point does someone move from the category of “has a beard” to the category of “doesn’t have a beard”? Just how many hairs need to be present for the category of “has a beard” to kick in?

An obvious response is to use categories that form some sort of scale, such as one ranging from “utterly hairless” via “wispy” to “magnificently bearded”. We can show that visually like this.

How does the motorbike example map onto these two approaches? One way is to combine them into three groups, namely “definitely owns a motorbike” and “sort-of owns a motorbike” and “definitely does not own a motorbike”. We can represent this visually like this. (I’ve included blue lines around the central category to show where it begins and ends.)

As an added refinement, we could make the width of the boxes correspond with how many people fit into each category, like this.

So far, so good. We now have a simple, powerful way of handling categorisation, which can be represented via clear, simple diagrams. As an added bonus, there’s a branch of mathematical logic, namely fuzzy logic, which provides a rigorous way of handling fuzzy categories. Since its invention by Zadeh in the 1960s, fuzzy logic has spread into a huge spread of applications; for example, it’s used in engineering applications ranging from washing machines to car engines.

At this point, you may be wondering suspiciously whether there’s a catch. You may well already know the catch, and know that it’s a big one.

The catch

So far, I’ve used gentle, non-threatening examples of categorisation, such as motorbike ownership and beard classification. The catch is that a lot of categorisation involves scary, threatening, morally charged issues that are usually linked with strong emotions, value systems, identity, and vested interests.

Here’s an illustration of how a lot of value systems work.

They divide the world into two crisp sets, where one set consists of goodness and light, and the other set consists of evil and darkness.

So how does this type of value system handle the “sort-of” problem? Typically, it will go through each type of “sort-of” and decide whether to include it in the “goodness and light” category or in the “evil and darkness” category.

This would lead to all sorts of problems if everyone within a particular ideological group made those judgments for themselves, so usually the judgments are made by a chosen individual or set of individuals, on behalf of the group. These judgments are then treated as laws, whether in the purely legal sense, or in the religious sense (e.g. dietary laws) or in a political or some other sense (as in the stated principles of a political group). I’ve blogged about the basic issues here, and about religious categorisations of gender and clean/unclean here.

This arrangement gives a lot of power to whoever decides where to position the ideological red line between light and darkness.

How would that “whoever” respond to a suggestion such as the one below, where there is a greyscale category between “definitely good” and “definitely evil”?

The usual response is to condemn it vehemently as moral relativism which puts everything on a greyscale, like the one below, where everything is tainted by the grey of evil and is therefore impure (and consequently likely to suffer the punishments prescribed by the value system). Yes, that’s a misrepresentation of what the diagram above is actually saying, but when emotions run high, facts tend to be early casualties.


So, where does this leave us?

First conclusion: The concepts of crisp sets and fuzzy sets are extremely useful, and allow us to make clean sense of messy problems. They’re easy to represent visually, which helps clarify what people mean when dealing with a specific problem.

Second conclusion: The combination of crisp, fuzzy and crisp is a particularly useful way of handling problems which initially look like messy greyscales.

Third conclusion: Categorisation is often associated with emotionally charged value judgments, so any plan to change an existing categorisation or to introduce a new one should check for potential value system associations.

Although these concepts look simple, they have far reaching implications, both for communication of what we actually mean within a category system, and for how we structure the categorisation systems at the heart of our world.

Notes and links

There’s a good background article about fuzzy logic here:

You’re welcome to use Hyde & Rugg copyleft images for any non-commercial purpose, including lectures, provided that you state that they’re copyleft Hyde & Rugg.

There’s more about the theory behind this article in my latest book:

Blind Spot, by Gordon Rugg with Joseph D’Agnese

You might also find our website useful:

Overviews of the articles on this blog:




7 thoughts on “Crisp and fuzzy categorisation

  1. A very nice post thank you, and a topic I am fascinated by.

    I have a number of things which are unclear to me, but I am hoping you might point me to in some way resolving the following…

    Issue 1: is not categorisation (mainly) an issue of grouping objects and categories, categories which reprsent groups of objects. Are not therefore categories eventually grounded in real world existents. And existents have characteristics and behaviours. They either fall in the category or they dont. Your example of the motorcycle for instance, if we define ownership the people either own it or they dont. Someone buying of hire purchase does not ‘almost’ own the motorcycle although once they do own it one can look back and say they almost did. If theye didnt pay the final payment then they will never own it and one doesnt know in advance when they ‘nearly’ own it or not. They do not own it.

    My concern is therefore that categorisations cannot be fuzzy, i really do not see how they can be. Categories are bounded and limited by their definitions. No definition, no category. So while I can see that categorisation is necessary to people, it is not the categories that are fuzzy and confused, just the definitions of people.

    Issue 2: The issue of fuzzy logic is surely not and issue of categories it is an issue of logic (definition of logic I will not even try). What I mean by this is that logic uses definitions and categories to deduce/induce relationships and therefore propositions based on relationships. Isn’t fuzzy logica just logic that tries to make inductions/deductions that really aren’t definite. Isn’t fuzzy logic just ‘best guesses’ based upon imperfect categorsations.

    It will be clear that I am an amateur when it comes to this stuff. I know my limits but you are stretching me a bit here. Probably why the post fsacinates me.

    Any help appreciated.


    • Excellent points; there’s a lot about this on the Wikipedia page I linked to in the notes. I’ve made similar points in my article about gender and categorisation, where I take the same line as you in saying that some gender categories (e.g. chromosomal XXY) are separate crisp sets with a clear definition. However, a practical risk with relying on crisp definitions is that in some applications this could end up with explosive fan-out into unwieldy numbers of definitions, so there can be pragmatic advantages in using a fuzzy approach. Legal guidelines on criminal sentencing are essentially based on this approach, within the overall crisp set of “found guilty”.

      For your second point, concepts such as “old” and “young” lend themselves well to a fuzzy approach. The age of a person may be precisely known, so it’s not just an issue of imperfect knowledge; the cut-off points for different categorisations of age are highly debatable, so a fuzzy approach provides a practical way of side-stepping a debate which might be unresolvable, and allowing modelers, engineers etc to produce something that works.

      I’m not an expert in these areas, so I may have made some errors in my descriptions. The thing that I’ve found most interesting about the concepts in this post was the reactions I’ve seen to the crisp/fuzzy/crisp visualisation when it’s explicitly distinguished from the single greyscale visualisation. Surprisingly often, people have said that it gave them a powerful new way of handling problems that had previously looked intractabley messy.

      I hope this helps.

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