20.1 Terms: Constants and Variables
This chapter begins a new stage in the textbook. In the remainder of the book we learn and use the final logical system, called FOL.
At this point, don't worry about what the name means.
FOL stands for first-order logic. The name is a bit unfortunate, because it is hard to even explain what the name means before you understand the system. Basically, the name means this: the quantifiers in the logic range over sets of objects in the domain, rather than sets of sets of objects.
See? Not very illuminating! At this point don't even worry about understanding the name. Just accept the name and learn the system.
Why not use a different name, then, you might ask? We use the name FOL because both the name and the system are ubiquitous, so you should be aware of how people refer to it. FOL is the logic of many important areas of mathematics, like analytic geometry and Zermelo-Fraenkel set theory.
Enough about the name. Let's see how FOL works.
In BOOL and PROP, the smallest units of the language were atomic sentences. The key development with FOL is that atomic sentences can now have parts.
One of those parts is terms. Terms refer to objects. They are a bit like nouns and noun phrases in English.
Terms come in two varieties: constants (names) and variables.
Constants are the same thing as names: we'll use the two words interchangeably.
Just like Pia is a name in English, we want a name in FOL that can refer to Pia. We will use
as our FOL name/constant for Pia.
Let's see if you can figure out the rest.
Terms are written in all lower case letters in FOL. That goes for both names and variables.
When we write out the names pia or quinn in FOL, we are using the long form.
There's another, equally good, way of writing names in FOL, the short form. Translating "Pia" into short form FOL is just the letter p.
FOL is a logical system we create to model and understand deductive reasoning. One way we can apply it is to model reasoning in mathematics. When we do that, we'll use names to refer to numbers.
For example, we'll use "one" as a name to refer to the number one, "two" as a name to refer to the number two, etc.
But it will be much easier to allow ourselves a conveniences that will be familiar: we will use Arabic numerals as shorthand names. So we'll use 1 as shorthand for "one", 2 as shorthand for "two", etc.
Names in FOL are also called constants because they always pick out the same thing: pia constantly refers to Pia.
For example, if we decide to use p as the short form for Pia, then we can't later use it for Peter. That is one reason why we have both short and long forms.
It is also essential that a name of FOL only picks out one object. If two people are name Pia, we can't use pia for both of them. You might, for example, use pia for one of them and p for the other.
But what if you have a third friend named Pia?
The solution is to create a convention for naming different people systematically.
We are already allowing Arabic numerals as names in FOL. So what we'll do is allow ourselves to mix numerals and lower-case letters together.
For example, if I have three friends named Pia, I could name them:
Since constants constantly refer to the same object, you can probably guess how variables are different.
What variables refer to can vary! Think of them like pronouns. In English, "she" doesn't always pick out the same person. Instead, context or other parts of the sentence will determine who "she" refers to in any particular case.
Variables of FOL will be a bit like that.
How it works is a bit more complicated, though, so we will focus on them later. For now, just note two things.
First, we will use the letters at the end of the alphabet for variables, like x, y, z, so that we can tell at a glance whether a term is a variable or constant.
Second, since variables are terms, they must always be written in lower case.