Sets and properties

Since our semantics will be formulated in set theory, sets will play a central role in it. One of these roles will be to represent properties. Some properties are ‘is red’, ‘is a mammal’, ‘is 6 feet tall’, ‘is the friend of Honghui’, and so on. These properties are either ‘had’ by certain things or not. The technical term is ‘exemplify’. So for example, every human has or exemplifies the property ‘being a mammal’, since every human is a mammal, but not every human is exactly 6 feet tall, so some have or exemplify the property ‘is 6 feet tall’, and some do not.

The simplest way to formally represent these properties is just to have a set of all those things that exemplify that property. So for example, one may form the set $H$ of all (and only) those things that are human (exemplify ‘is a human’), and thus the set $H$ will represent the property ‘is a human’. Similarly, one can represent the property ‘is a mammal’ by a set $M$ consisting of all (and only) the things that are mammals.

Now let’s look at some examples. Taylor Swift will be in the set $H$ and the set $M$, since she is a human and a mammal (exemplifies the property ‘is a human’ and ‘is a mammal’). However, if $S$ represents the property ‘is 6 feet tall’, she will not be in $S$, since she is not 6 feet tall (apparently, she’s 5’11”). The same goes for Jay-Z, since he is also a human and a mammal, and also not 6 feet tall (apparently, he is 6’2”). On the other hand, if we take Jay-Z’s Corvette C-1, it is neither in $H$, nor $M$, nor $S$, since it is a car that does not have any of these properties.

You can also look at which properties (understood as sets) are subsets of which other properties (understood as sets). For example, since every human is a mammal, the property ‘is a human’ is a subset of the property ‘is a mammal’, since every member of $H$ is a member of $M$. Indeed, $H$ is a proper subset of $M$, since there are many mammals that aren’t human. In other words, there are many animals in $M$ not in $H$.

There is a famous philosophical problem when it comes to identifying properties with sets. Namely, there are some properties that we would say are distinct, though they apply to exactly the same things, and hence they would be identified with the same set. For example, the property ‘is the the first rapper to be inducted into the Songwriters Hall of Fame’ and the property ‘is the first solo living rapper inducted in the Rock and Roll Hall of Fame’ seem to be different properties, yet if we identify properties with sets, they are the same set $\{\text{Jay-Z}\}$, so they are the same property. The wrong result! However, for our purposes (which is doing logic), it is okay to identify properties with sets.

We will return to the topic of properties and semantics in the next chapter.