Constants
Let’s start with the simplest case; constants. Constants in our language function like names. In particular, their meaning is just the thing they designate, or refer to.
Note that we are talking about two distinct ‘planes’ here. On the side of syntax, we have symbols without meaning. On the other side, we have things assigned to them. Giving a semantics to our language is then bridging a gap, assigning things to our constants. These things are not symbols, they are the things themselves. You may have heard of the phrase ‘domain of discourse’. The domain of discourse, or simply domain, is just the set of things we may talk about using a language.
Exercise 4.1. How does this notion of a domain relate to the notions of domain and codomain regarding functions?
In particular:
| Constants | Domain | |
|---|---|---|
| constant | thing |
You may remember the symbol from our discussion of set theory. It designates that a certain function outputs the right-hand side value given the value on the left-hand side. And indeed, giving meaning to our language is just specifying a function that, in part, assigns to every constant a thing (sometimes called an ‘object’) from our domain. Like this:
| Constants | Domain | |
|---|---|---|
| Robert J. Oppenheimer |
According to the above specification, the constant designates Oppenheimer in our language. Note that while is a symbol of the language, Oppenheimer, the thing (‘person’) it designates is, again, the real thing. We can continue assigning things to our constants to our own delight. For example, we can specify:
| Constants | Domain | |
|---|---|---|
| Robert J. Oppenheimer | ||
| Taylor Swift | ||
| the number | ||
| World War II | ||
| … | … |
As you can see, there is no limit to what a constant can designate. Thing and object is meant here in a very loose sense. It can be a person, a physical object, an event, an idea, whatever you want.
Let’s recap. We have constants, which are symbols of our language. We have objects, in a loose sense, which are members of the domain of discourse. And we have a function, which assigns to each constant a member of the domain of discourse, as illustrated in the table above. This function is usually called an interpretation function, since it interprets the uninterpreted symbols of a language. We can make this more precise as follows:
Definition 4.1. A domain (of discourse) is any set . An interpretation function for the constants of , denoted by , (relative to ) is a function . If is a constant of , then , and is what designates, denotes or refers to. Alternatively, we may say is the value of the symbol .
As you can see, interpretations are functions from the set of all constants to the domain. This means that to each constant, only one member of the domain corresponds. So unlike with real names like ‘Peter’, which may designate many different people, a constant of designates only one. On the other hand, the function need not be one-to-one. This means that some distinct constants may designate the same thing, just like ‘Miley Cyrus’ and ‘Hannah Montana’ designate the same person (in the hit TV show Hannah Montana). It also doesn’t need to be onto, so that some members of the domain may go nameless. Like the lack of injectivity, this is also natural, since many things do not have names in the real world. Just think of your left sock that fell behind the machine at the laundry.
Having constants or names that refer to things is the first step towards giving meaning to our expressions, but it is not enough to get us to truth. Names, by themselves, are neither true nor false, they just refer. The other ingredient we need is giving meaning to our predicates.
