Subsection 2.1.2 The Building Blocks of Statements
The first question we have to answer is: What are the fundamental objects that we can talk about?
In arithmetic, there are numbers, such as 0, 127, -4, and 7.865.
In Boolean logic, there are truth values. More specifically, there are two truth values, True (which we’ll often write just as T) and False (written as F).
You’ve probably noticed one difference already between arithmetic and Boolean logic: there is an infinite number of numbers but only two truth values. We’ll see later that this matters; it is one reason why Boolean logic is as useful as it is. But, lest you think that logics can only deal with very small sets of objects, we’ll just mention here that, later in this course, we’ll describe a more powerful logical system that lets us talk about infinite universes (including numbers).
Next, we need variables (or names for things that may take on any of the fundamental values):
In algebra, there are variables. You are used to giving them names like x and y.
In Boolean logic, there are also variables. We will give them names like p and q.
In algebra, we use variables to stand for quantities that we want to reason about. For example, we might let x stand for Mary’s age and y stand for John’s age.
In Boolean logic, we do the same thing except that now we want to reason about claims that may be true or false. So, for example, we might let p stand for the claim, “It is raining,” and q stand for the claim, “The sidewalks are wet.”
Recall that a statement (sometimes also called a proposition) is something to which we can attach a truth value. So we now have the beginning of a formal definition of a Boolean logic statement (also sometimes called a well-formed formula or wff, pronounced “woof”):
True (usually abbreviated T) is a statement. It is always true.
False (usually abbreviated F) is a statement. It is always false.
Individual claims (which we may write in English or we may denote with variables like p) are statements. They may be either true or false.
In the next section, we’ll see how to combine simple statements to make more expressive ones.