Part I: Lesson

A helpful method of understanding categorematic terms is the use of the concepts of extension and intension. To review once again, a categorematic term is a term that refers to a category. A category is just a collection of particular things that share some relevant common property or properties. This common property is referred to by the categorematic term itself. For instance, the term “chairs” picks out the category “chairs”, which is just a collection of all of the things in existence that have the property of being a chair. All of those chairs are members of the category “chairs”. It is worth noting that we can label a category in the singular or the plural. To be specific, “chair” and “chairs” are the same category: they have all things in reality that are chairs as members. Now we can understand the following definitions:

• Extension of a categorematic term: The totality of all the particular members of the category that the term refers to.
• Intension of a categorematic term: The meaning/definition of the term; The description of the common property/properties that all members of the extension of the term share.

It’s worth noting that the intension of a term can also be thought of as its definition. We can use the two terms interchangeably. Let’s use an example to more easily understand these ideas. Let’s say we have the term “triangle.” The intension, or definition, of the term “triangle” is “a three-sided polygon”. The extension of the term “triangle”, then, is the totality of the things in reality that have the common properties of being both:

1. three-sided

and

2. a polygon (straight sides, closed, and planar)

If there is something that is both (1) three-sided and (2) a polygon, then it is in the extension of “triangle.” In other words, such a thing is a member of the category “triangle”, since it has both of the common properties that define such a category. It is important to note that “define” comes from the latin “de” + “finire,” or “to bound/limit completely”. So, what defines the limit of a category? The definition/intension of the term that refers to it. And what things are in such a category? Only those things that have the common property/properties that a term’s intension defines of them. Here’s a picture to help visualize this:

As we can see, when we use the term “triangle”, we refer to a category that contains the totality of things we call “triangle” (this totality is the extension of the term “triangle”). What objects are in this extension is defined by the intension of “triangle” given. These are the common properties held by all things inside the category but are not held by anything outside the category. In other words, everything inside the circle, or everything in the extension of the term, (1) has three sides and (2) is a polygon. Everything outside of the circle either does not have three sides, is not a polygon, or is neither a polygon nor three-sided.

It is important to note that these two concepts are, in a way, inversely related. As the extension of a term increases, the term’s definition must cover a wider range of things: its intension must become less specific. Similarly, as the extension of a term decreases, the term’s definition covers a narrower range of things: its intension becomes more specific. As a general rule, the greater number of things to which a term refers, the less specific in meaning it must be.  

For instance, consider an obtuse triangle. “Obtuse triangle” is a more specific term than “triangle” (it not only is a three-sided polygon, but it also has an angle larger than 90°). Thus, the intension has become more specific than that of a simple triangle. In other words, things must have more properties to be in the category “obtuse triangle”. However, because the meaning of the term has become more specific, the collection of things that the term refers to in the world decreases. This general relation holds for almost all terms. We now have:

Notice that the category has become much smaller, because that which is limiting it (the intension) is more specific. There are now triangles outside of the category, because they are not obtuse triangles. 

Intension is very important because it plays a crucial role in our definitions. When we define something, we want to choose a definition that is as specific as possible to fit the intuitive extension we associate with a term. Why is this the case? Take the triangle example again. If we were to define “triangle” with a definition that is broader than the generally-accepted intension of triangle, then we would have non-triangles in the category of “triangle.” For instance, if we defined “triangle” as “a geometric shape,” then squares would be members of the category “triangle,” because squares have the property of being geometric shapes! Our bad intension would yield something like the diagram below:

This diagram shows that our definition/intension is way too broad, because now squares, stars, and trapezoids are in the category “triangle”. Therefore, we know that we need to make our definition/intension more specific. Since our goal in argumentation is to avoid as much vagueness and ambiguity as possible, we want to make sure that our definitions include only the objects we are talking about. Of course, we also do not want our definition to be too specific either. We would like all and only triangles inside the category “triangle”. 

[Activity 1] With a partner, play a game that demonstrates the ideas of extension and intension. Have your partner define an intension. Now you must point to the extension of the intension defined. For instance, your partner might define the intension, “things which have four legs.” Now, you might point to all of the chairs and desks (if they all have four legs). Once you do this with your partner a few times, switch roles: you define an intension and your partner will point to all the things that fall within the relevant extension.

[Activity 2] Now have your partner define an extension by pointing to certain things in the room. Based on what your partner chooses, come up with an intension to describe what was chosen. For instance, if your partner points only to chairs, then your job is easy! You would simply label the intension as “chair.” Now, if your partner points to pencils, paper, rulers, etc., then you might define the intension as “school supplies.” Try to define an intension that is as specific as possible. For instance, in the previous example, say something like “school supplies” rather than “things.” Once you do this with your partner a few times, switch roles: you define an extension and your partner will define an intension that fits that extension.