Subsection 2.1.6 Formal Definitions of the Logical Operators
Suppose we know that the statement “Jim is tall” is true but that the statement “Joe is short” is false; what is the truth value of “Jim is tall or Joe is short”? You say, “It’s true,” and you are correct. However, had we asked what is the truth value of “Jim is tall and Joe is short” (under the same circumstances), you should say, “It’s false”. Obviously, then, the action of the operator and versus the action of or is critical for determining truth values.
In answering the questions that we just asked, you used our everyday notions of what and and or mean. We have those notions because they are useful. They help us to reason about the world.
We are embarking on a study of formal logic. Our goal is to describe a system so precisely that there can be no disagreement about what we mean or about whether an argument that we make is valid. So we need to provide precise definitions of every operator that we’ll use. But, of course, we still want operators that let us do useful things (as opposed to formally justified, yet silly things).
So, in the next several sections, we will define precisely what we mean by each of the operators that we will use. We hope that you’ll agree that we’ve chosen to define a set of operators that correspond well to our intuitions about sound reasoning and, thus, that we’ll be able to use them in useful ways.