DRMacIver's Notebook
Defining definition
Defining definition
Another abandoned draft, this one dredged up from an old Google doc. It was meant to be part of a longer book-length project but I lost steam, because that’s what happens when I undertake longer book-length projects.
When we use a word, we might know exactly what we mean by it, but unless other people know what we mean by it our use of it isn’t very useful. If I wrote a long essay about the strengths and weaknesses of incorporating scamprooftThese are of course all completely made up words. into our florbnoggle practices, this would not be very helpful to a reader that doesn’t know what scamprooft is and hasn’t ever practised florbnoggle in their life.Of course, such a reader probably won’t benefit from reading such an essay in most circumstances, but they might if you can adequately communicate them that florbnoggling is basically the same as bungipopping, a practice they are quite familiar with, except you zeft when in bungipopping you would smork. We can solve this problem by giving people definitions of concepts we are using that they might not be familiar with, in order to help them understand what we mean by them.
A definition, informally, is an explanation of what some concept (usually labelled with a word or a phrase) means. That is, it tells you how to apply the concept.“Definition” is used in a variety of other related ways. For example, you can “define the problem”, you can “define the terms of an agreement” etc. When I talk about definitions in this chapter, I don’t mean those senses, I mean specifically defining the meaning of some term.
The key thing that a definition lets you do is understand when some item fits the concept. This provides you with conditions (things you can check) that allow you to tell whether something is an instance of that concept or not. That is, a definition is supposed to communicate to you some sort of decision procedure that you can use to check whether the concept applies in a given instance.
Most definitions we use in practice are in some sense incomplete - there will be some things that the definition gives you a clear yes to, some that it gives you a clear no to, and a large class of states (often, but not always, somewhere between the two) where it doesn’t give you a clear answer. Sometimes that’s a limitation of the definition and a better definition would decide between them, sometimes it’s just that the precise boundary is just genuinely fuzzy.
Additionally, some things are just extremely hard to define without introducing some sort of circularity or non-linguistic explanation. We explain the colour “red” by pointing to red things until you get the pattern.
Defining things perfectly is hard to impossible, and often even undesirable, but some amount of definition is necessary if we’re to be able to communicate at all. This chapter is about how to use definitions well, and to do that we’re going to have to dig in to what a definition actually does, and how to take advantage of definition’s strengths while compensating for its weaknesses.
Necessary and sufficient conditions
The ideal of a definition is the sort of definition you would see in mathematics, which takes some concept and provides strict necessary and sufficient rules for when it applies. For example: a square is a shape with four straight sides, all of equal lengths, and each corner having the same angle.
What it means to say that these are necessary conditions is that if something doesn’t satisfy all of them it is not a square. For example, an equilateral triangle is a shape with three straight sides, all of equal lengths, and each corner having the same angle. You can tell that it’s not a square because it fails the condition of having four straight sides even though it satisfies all the others.
What it means to say that these are sufficient conditions is that if something satisfies all of them it’s definitely a square. There’s no possibility of saying “Oh whoops I forget that there’s a secret extra requirement”, these conditions lay out what it means to be a square and anything that satisfies them is one. We will often write sufficient conditions as an “if” statement. e.g. “A shape is a square if…”.
To look it another way: A necessary condition is something you can check in order to show that an item of interest isn’t the thing. A sufficient condition tells you something you can check to show that an item of interest is the thing. We will often write necessary conditions as an “only if” statement. e.g. “A shape is a square only if…”.
Having a set of conditions which is both necessary and sufficient is very convenient because it gives you a precise
It’s possible to have sets of conditions that are necessary but not sufficient.
For example, you weaken a set of necessary conditions by removing parts and they’ll still be necessary, so e.g. “a shape with four sides of equal lengths” is a condition that is necessary for being a square (every square has four sides with equal lengths) but not sufficient because e.g. a rhombus (a square which has been skewed so that all sides have equal lengths but some corners are pointier than others) satisfies these but is not a square.
Similarly you can strengthen a set of sufficient conditions and still have a set of sufficient conditions. e.g. Anything satisfying “a shape with four sides of length one and all angles equal” is a square, but a square with sides of length two does not satisfy these conditions so they are not necessary.
Generally with non-mathematical objects we’re not so lucky as to have simple necessary and sufficient conditions. Those that are tend to refer to human-constructed concepts that are built as having necessary and sufficient conditions deliberately. e.g. legal concepts tend to come with a very explicit decision procedure for determining that they apply. You’re “officially married in the country of X”Normally one would just say “married” here, but this blurs multiple related but distinct concepts together, and also elides historical marriages from before state records begin. if you’re officially recorded as married in X or by some country whose marriages are recognised by X (and if your marriage is of a type that is recognised by as permissible by X due to e.g. constraints on gender).It’s almost certainly more complicated than this. I am not a lawyer.
For things that are more about describing things that actually exist in the physical world, what you tend to find is that it’s easy to provide sufficient conditions and it’s easy to provide necessary conditions, but there’s a large gap in the middle. e.g. the sorites paradox (how many grains of sand count as a heap?) can be thought of as being about this gap: It’s easy to provide a sufficient condition (e.g. if you’ve got between one and two million grains in a pile, that counts as a heap), it’s easy to provide a necessary condition (e.g. you need more than one grain to count as a heap), but somewhere in the middle the question gets hard to resolve. You’ll have this problem with most continuous quantities - how tall do you have to be before you count as tall? How old do you have to be before you count as elderly? There isn’t a clear boundary, but it’s easy to create necessary conditions and easy to create sufficient conditions, it’s just not easy to create conditions that are both.
Running into problems like this is often what stops people coming up with good definitions for things: They are aiming for a mathematical style set of necessary-and-sufficient conditions, this doesn’t work, and then they give up on the whole thing as indefinable.
In fact, definitions are perfectly possible for real-world concepts. We have to give up on them being as nice as mathematical definitions, but we can still use them to provide people with helpful handles on what we mean, and it starts by recognising that we’re often going to want to provide separate necessary conditions and sufficient conditions, and we’ll almost always have ambiguous edge cases that any definition we provide won’t cover. This doesn’t stop them being useful.
For example, a necessary condition for being a definition is that it is an attempt to explain what a term means using language. This is not a sufficient condition (e.g. talking about the history of the use of a term satisfies this condition but is not really a definition).
A sufficient condition for something being a definition is that it has the form “A (term) is a (thing) which…”. This gives you something very like the mathematical type of definition, and you can sure that something that looks like this is, or at least can function as, a definition.
Much of the interesting work of defining things lies between the regions set by these two conditions.
Ostensive Definition
One useful way of defining things is by ostensive definition, which means roughly “definition by showing”. Classically, you point out a bunch of examples and define your term as meaning “this sort of thing”. When you see it in writing it will often look like “By this term, I mean things like…”
Ostensive definition as defined this way is bad and I don’t like it, but the thing that it is trying to point to is good, so I’m going to shamelessly redefine the term (in a way that includes the original usage) as meaning a definition which uses examples in some essential way (i.e. not just as illustrations of some point).
The limitation of definition by just pointing at examples is this: You can think of an example of a concept as being an extremely specific sufficient condition, and you need at least some sort of necessary condition preventing you from generalising that condition too far. “You see this thing? It’s a cat.” gives you a sufficient definition for being a cat, but it doesn’t tell you what else is a cat. It would be perfectly reasonable from this one example to assume that cat meant, for example, animal, and start calling dogs cats as well.This isn’t pedantry! It’s an actual problem in language acquisition, both first language and foreign languages. At the bare minimum, defining things ostensively requires you to point at both examples of the concept and non-examples of the concept.
In theory, you can define anything you like ostensively as long as you’re able to interact with the recipient. You give them a bunch of examples, and then they point at more examples and you say whether they are or aren’t an instance of the thing.
In practice, most of the time, this doesn’t work very well. Both because you’re not necessarily getting to interact with the people you’re defining things for (e.g. if you’re writing it down), and also it’s very slow because it can take a very long time for them to figure out the right decision procedure for the concept.
The problem is that people are not necessarily generalising in the way that you want them to. If you show them a bunch of examples and they come up with a rule for distinguishing them, there can be many rules that work.This means that they will consistently have errors in their understanding of the concept.
Worse, often you won’t even be able to tell what those errors are, because either they’re in important but rare edge cases, or the back-and-forth testing has blind spots because they’re looking for examples that confirm their theoryThere’s a classic experiment called the Wason reflection task which asks you which cards you need to turn over to see if a rule is satisfied. People very regularly fail to turn over the cards that they would need to determine that the rule isn’t satisfied. For example, given the rule “if a card shows an even number on one face, then its opposite face is blue”, you need to turn over red cards and even cards, but instead people turn over blue cards, which can provide no information. People don’t tend to show this error in subjects with which they’re familiar, but they do show it with subjects with which they’re not, which is exactly the situation you find yourself in when learning a new concept., and you don’t understand what the examples you need to show their theory is wrong are because you don’t know what their theory is.
You can solve, or at least mitigate this, by giving each of your examples a generalisation - in the case of a positive example (an example where the concept holds), a sufficient condition that that example satisfies that makes it an instance of the concept, and in the case of a negative example (an example where the concept doesn’t hold) a necessary condition it fails to satisfy.
For example, I’ve given you a couple of examples of definitions, but I haven’t given you examples of things that aren’t definitions.
Here’s an easy example of something that isn’t a definition: A cat. You can tell it’s not a definition because it’s not a use of language, it’s a cat. A necessary condition for being a definition is that it involves some degree of use of language (even if it’s just to say “This is an X” when pointing at an X).“Is a cat an instance of this concept?” is a surprisingly good starting point in general, because it forces you to define the boundaries of what sort of thing this concept applies to.
Here’s a more interesting example of something that is not a definition:
Priests are identifiable by their consistent attire and groomed appearance. When not occupied with their official duties, they can be spotted around town engaged in leisurely activities. They reside in modest accommodations among the public. Priests dedicate time to advancing their knowledge and serving the community. They make themselves available to offer guidance. Though often solitary, priests derive purpose from their vocation. During infrequent breaks, they partake in simple enjoyments.
This is a description, not a definition. It tells you what priests are likeAt least stereotypically. Very few of these things are universally true of priests of course., not what a priest is. This may be useful information about priests and how to spot one, but it doesn’t tell you anything about what makes someone a priest, or really even how to tell if they are a priest.
Using examples like this is an important part of doing ostensive definition well: If you only pick examples that are far from the boundary between being instances of the concept and not being instances, you don’t help people really learn how to make the decision in the cases where it’s difficult.
Another thing that can be helpful is to provide examples where it’s genuinely ambiguous as to whether it’s an instance of the concept, or at least it’s not a very good instance. For example, here’s an alternative paragraph about priests:
Priests are people who wear special clothes and live in churches. They spend a lot of time reading old books and talking to people about being nice. Priests tell stories to children and sometimes help people when they are sad.
Is this a definition? I guess? It tells you just enough about what a priest is and what they’re for that I suppose I’d count it as a definition. But it’s not one I’d be particularly happy with giving.
This sort of ostensive definition is often useful to provide even if you’re providing a more direct sort of definition, because it greatly helps the reader understand how the definition works to have examples and illustrations of how they fit into the definition.
Working definitions
There’s one small defect with this view of definitions in terms of necessary and sufficient conditions: It’s wrong. Or, at least, it’s very limiting - it fails to cover many definitions that exist in practice and serve a useful functional process.
A nice example of this is from Bernard Suits’s book “The Grasshopper”, which is about philosophy of games. He offers two definitions of a game. The first is:
To play a game is to attempt to achieve a specific state of affairs [prelusory goal], using only means permitted by rules [lusory means], where the rules prohibit use of more efficient in favour of less efficient means [constitutive rules], and where the rules are accepted just because they make possible such activity [lusory attitude].
He then offers the following “portable” definition:
a voluntary attempt to overcome unnecessary obstacles
These do not define the same concept - satisfying either of them is neither a necessary nor a sufficient condition for satisfying the other.
For example, suppose you’re going somewhere and you decide to take the long way around just to add some variety. You discover there’s a road blocked for maintenance. You could backtrack, but you’re feeling stubborn, so you instead try to find your way around the road block.
This is a voluntary (you’re doing it for no other reason that you feel like it) attempt to overcome an unnecessary (you could always backtrack) obstacle. It’s not a game in the full sense that Suits defines it, because it lacks constitutive rules. The thing that is stopping you from backtracking is not that it would be a rules violation, it’s that you’re feeling stubborn.
You could argue that there are constitutive rules that you’re making up on the fly, but I think that’s cheating and requires diluting the meaning of “rules” too much.I think a better argument would be that Suits is putting too much emphasis on “rules” and that the things that constitute a game may not be rules in any sort of formal sense, and any sort of taking on a restriction to achieve an activity that is constituted by that restriction is at least somewhat gamelike. Under this view, this example is a game but not a “Suitsian game” to use Nguyen’s term for it. This is particularly true because I think what happens when you violate these putative rules is different than what happens if you violate the rules of a game.
An example from C Thi Nguyen’s account of games is mountain climbing: There you have the prelusory goal of getting to the top of the mountain, and a set of constitutive rules about how to do it (you have to climb the mountain). You could achieve the goal of getting to the top of the mountain by travelling there in a helicopter. You would still have achieved the goal, but in violating the rules you are not doing mountain climbing. In contrast, in our detour example it’s not that you’ve failed to do something well-defined. There’s no game of “take the long way around” that you’re now not playing. It’s just not what you want to do.
In the other direction, suppose someone threatens to kill you unless you beat them at chess. Your playing of this game of chess is in no way voluntary, but you’re still playing a game in the full definition.
So, in the strictly idealised mathematical sense of definitions as providing necessary and sufficient conditions, these aren’t the same definition, they’re barely even related. And yet, it sure seems like Suits’s portable definition adds something to your understanding of the meaning of games on top of the full definition.
Part of what it adds is convenience - the full definition of games is somewhat unwieldy, and introduces many other terms as part of its definition. It’s strictly a better definition in terms of giving you the full concept in detail and making it easier to analyse something in terms of Suits’s conception of a game, but it’s hard to get much of an intuitive sense of.
In contrast “a voluntary attempt to overcome unnecessary obstacles” is pretty intuitively comprehensible and it is easy to see its gamelike structure.
Afterword
And that’s where it randomly cuts off. Chalk this one up to another victim of “things I wish other people understood but that I don’t find worth the effort of explaining”.