Clearer thinking about definitions

Do you believe that people waste too much time arguing over definitions?

I do too. But I also have a second problem: I’ve read some philosophy. And so, when I’m frustrated with pointless arguments over definitions, my frustration becomes compounded by the fact that nobody understands the thing that they’re arguing about, and the only way to solve the problem is by spending even more time arguing over useless stuff.

Case in point, in all the time you’ve ever spent arguing over definitions, have you ever once glanced at the relevant articles in either Wikipedia or the Stanford Encyclopedia of Philosophy? I’m guessing not, because I never thought to do such a thing myself for a long time.

So now that I’ve made everyone feel guilty, let’s talk about one of the things you’d learn from some basic research: intensional vs extensional definitions.

An intensional definition provides some rules about what a word refers to. These rules are called the intension of the word (not to be confused with “intention”). For example, the US president is the head of our executive branch.

An extensional definition provides the set of objects that the word refers to. This set is called the extension of the word. For example, we can extensionally define the current US President as Donald Trump.

Now consider the following statement: “The current US President could have been Hillary Clinton.” Clearly what is meant is that Hillary Clinton could have fallen under the intension of the current US president. But suppose we substituted the extension of the current US president: “Donald Trump could have been Hillary Clinton.” By ignoring the difference between extension and intension, we get nonsense.

Some philosophers argue that a way to deal with the extension/intension distinction is to use the language of possible worlds. The extension of a word is the set of things that the word applies to in the actual world. The intension of a word is the set of things that the word could apply to in all possible worlds. However, I do not think this is entirely satisfactory. Consider, for instance, the following intensional definitions of a triangle:

1. A triangle is a polygon with three sides.
2. A triangle is a polygon with three corners.

Since every polygon with three sides also has three corners, both of these kinds of triangles have the same extension in all possible worlds. Nonetheless, it would be incorrect to say that the two definitions are the same. One way to think about it is that the intension describes an algorithm to determine whether an object is a triangle or not. The two definitions of a triangle correspond to two distinct algorithms, and these algorithms happen to always output the same conclusion, but that does not mean that the algorithms were one and the same.

Question: Is the extensional or intensional definition more important?

We often like to believe that the intensional definition is the important one. The intension definition is, after all, what you find in dictionaries. It’s the definition that you try to provide whenever someone asks for a definition.

But there are many reasons to believe that the extensional definition is also important. For example, though I provided two intensional definitions of triangles, most people hardly care which is the “correct” definition. Both definitions specify the correct extension, and that’s pretty much all that matters.

Or consider the definition of “religion”. When a discussion group is asked to define religion, people will think about it and come to different conclusions. But even amidst disagreement, there are at least a few points of agreement. We mostly agree that the extension of religion includes Christianity, Judaism, Islam, Buddhism, and Hinduism; it excludes baseball, vegetarianism, and university. And should someone happen to disagree on any of these aspects of the extension, then that might cause us to object to their preferred definition.

What this suggests is that the extension of religion is actually more fundamental than the intension. Or at least, some aspects of the extension are fundamental. We don’t necessarily agree on whether, say, the Sunday Assembly is in the extension of religion. Of course, definitions don’t need to be complete and unambiguous, but it’s often nice to be able to classify new things that are brought to our attention. So usually, the approach is to come up with a reasonable intensional definition, such that it matches most or all of our preconceptions about the extension. If the intensional definition doesn’t exactly match the extensional definition, we can choose to either modify the intension, modify the extension, or else be satisfied with multiple definitions.

Why is the extension so fundamental? As I’ve pointed out before, all words must ultimately be defined by pointing to examples. Defining by pointing is called an ostensive definition, and it’s basically a variety of extensional definition.

Extensional definitions also appear to be a fundamental aspect of our psychology. I am especially fond of pointing out the existence of prototype theory. The idea is that we think of most concepts by imagining one or more prototypical examples, and then we classify new examples by judging their similarity to the prototypical examples. In other words, for many words, we start with an idea of its extension, and classify new objects according to their similarity to the extension.

Of course, different words behave differently. I avoid claiming that all words operate according to prototype theory. I’m fairly sure, for instance, that our understanding of the US president is based on an intensional definition rather than an extensional one, if only because we don’t like to think about the current extension.

Go forth, and argue about definitions in a more enlightened way. Or don’t, if you prefer.


  1. sennkestra says

    Well, as someone with a linguistics background, I would usually at some point just throw my hands in the air and yell “all ‘definitions’ are inherently imperfect and inaccurate and political tools so just pick something that has decent utility already! Preferrable mine :)”

    As far as specific models of definitions / approaches to semantics, though, I wouldn’t describe prototype theory to be an example of an extensional approach to definitions – in linguistic semantics, at least, prototype theory is actually considered to be a separate approach with contrasts with and rejects some of the aspects of both extensional and intentional semantic approaches, which are both very grounded in set theory. (Extensional and intentional definitions are popular in things like logical semantics, which attempt to construct more formal or algorithmic models of meaning, whereas prototype theory is more linked to cognitive science and attempts to model how human brains actually tend to handle things like definition and categorization).

    The distinction – which is sort of addressed in the linked wiki article – is that both extensional and intensional approaches to semantics typically have a more binary approach of “is this a member of the set, or is it not” (either by comparing it to a list of set members, or checking it against a list of properties that determine the boundaries of a set), with little differentiation between members of a set, whereas prototype theory does not draw clear boundaries between what is and is not part of a set (and uses set language/theory less overall), and allows for hierarchies of how “x-ish” something can be). So in many ways, prototype theory actually involves rejecting both extensional as well as intensional definitions. I’d argue that extensional and intensional approaches both have more in common with each other (as part of set theory approaches) than either does with prototype theory.

    That said, prototype theory is also very much a descriptive approach to semantics, and when people talk about “definitions” in the context of identity debates, that’s all about prescriptive definitions – no one in those kinds of debates actually care about how a linguist would define those words (which would usually be like, “there are 10+ possible definitions of this word depending on the dialect/sociolect of the speaker and context of the utterance, here are some of them and when they occur”). The goal of “definition” debates is typically to settle on one (artificially constructed) single definition to ensure consistency. So as much as I love prototype theory for descriptive approaches, both extensional and intensional approaches tend to be better for prescriptive things like making dictionaries and glossaries, where brevity is valued even at the expense of accuracy.

    So in terms of the question “which is more important?”:

    From a linguistic/descriptive approach, i would say both are bad, that’s not how human brains work, lets play with more complex models like prototype theory.

    In the context of definitional debates, though, I would argue that both can be useful, but for human identity labels where potential set members can number in the millions, intensional approaches are perhaps more inescapable – while one could fairly easily create a workable model of sexuality that’s completely intensional, it’s impossible to make one that’s purely extensional (which would literally involve writing out the names of every human being and assigning them to a set, and which would always be incomplete because new people are constantly being born). That said, I think it’s common to have a mix of both – for example, “queer” might sometimes* be defined extensionally as the set of lesbians + gay men + bi people + trans people, but in order to process that definition you still need intensional sub-definitions of each of those subcategories, like “men who are attracted to other men” or “men who are sexually attracted to other men” etc. (*Not my personal definition)

    That said, for practical lexicography – that is, actually writing things for dictionaries and glossaries, as opposed to more abstract linguistics or philosophy or cognitive science – “definitions” also include additional components like near synonyms, potential translations, supplemental examples or exclusions/contrasts, common contexts, etc. that don’t rely on set theory at all – I think this “definition of a definition” can be interesting from that regard (SIL is a linguistic organization but one that often has a more practical/applied focus):

  2. says

    Thanks for this!

    My own strategy for limiting time invested in quarrelling about definitions is to try to adapt to whatever terminology my interlocutor prefers, *provided* that the vocabulary doesn’t make it literally impossible for me to say what I want to say. “Defining” a phenomenon or category out of existence by arranging it so that there isn’t a word for it is distinctly not-cricket. It’s a rule that quickly distinguishes between people with strong vocabulary preferences — often for professional or technical reasons — from people who are just trying to control a debate by controlling the terms. (That said, I’m happy to bolt made-up terms onto an existing vocabulary, as long as you’re willing to discuss “bleen” and “farble” along with “privilege-as-you-define-it” and “gender-as-you-define-it” or whatever.)

Leave a Reply

Your email address will not be published. Required fields are marked *