Problem

An advertisement for a tennis magazine state, “If I’m not playing tennis, I’m watching tennis. And if I’m not watching tennis, I’m reading about tennis.” We can assume that the speaker cannot do more than one of these activities at a time. What is the speaker doing?

Solution

To solve this, we will translate the speakers sentences into our formal language and create a truth table. Let \(A\) be the statement “I’m playing tennis,” let \(B\) be the statement “I’m watching tennis,” and let \(C\) be the statement “I’m reading about tennis.” Since the speaker can only be doing one activity at a time, there are only the following four truth assignments to consider \begin{array}{|c|c|} A & B & C & (\lnot A) & (\lnot B) & ((\lnot A)\to B) & ((\lnot B)\to C)
\hline T & F & F & F & T & T & F
F & T & F & T & F & T & T
F & F & T & T & T & F & T
F & F & F & T & T & F & F \end{array} From the speakers statement, we want a truth assignment where both \(((\lnot A)\to B)\) and \(((\lnot B)\to C)\) are true. From our observation of the truth table, we see that there is only one such truth assignment where both statements are true, that is the one where \(B\) is true. Thus, the speaker is watching tennis.