Example 0.2.2.

Consider the statement:

If Bob gets a 90 on the final, then Bob will pass the class.

This is definitely an implication: \(P\) is the statement “Bob gets a 90 on the final,” and \(Q\) is the statement “Bob will pass the class.”

Suppose I made that statement to Bob. In what circumstances would it be fair to call me a liar? What if Bob really did get a 90 on the final, and he did pass the class? Then I have not lied; my statement is true. However, if Bob did get a 90 on the final and did not pass the class, then I lied, making the statement false. The tricky case is this: what if Bob did not get a 90 on the final? Maybe he passes the class, maybe he doesn't. Did I lie in either case? I think not. In these last two cases, \(P\) was false, and the statement \(P \imp Q\) was true. In the first case, \(Q\) was true, and so was \(P \imp Q\text{.}\) So \(P \imp Q\) is true when either \(P\) is false or \(Q\) is true.