Chapter 3.4, Problem 54E

In a complicated argument with many variables, it is not practical to use truth tables because of their size. We can, however, use valid argument forms to reason without using truth tables. For example, consider the following argument:

$\begin{array}{l}\underset{\_}{\begin{array}{l}p\\ p\to q\\ p\wedge q\to r\\ \sim s\to \sim r\end{array}}\\ \therefore s\end{array}$

We assume that all the premises are true, and we reason like this to prove that the argument is valid:

1. We assumed that p and $p\to q$ are both true: therefore, by the law of detachment, we have that q is true.

2. Now both p and q are true, so $p\wedge q$ is also true.

3. By the law of detachment again, $p\wedge q$ is true and $p\wedge q\to r$ true forces r to be true.

4. Because the statement $\sim s\to \sim r$ is equivalent to its contrapositive, we then know that $r\to s$ is true.

5. Knowing that r and $r\to s$ are both true, we conclude that s is also true, we conclude that s is also true.

Therefore, by assuming that all the premises are true, we were able to reason that the conclusion s also must be true. This means that the argument is valid. In Exercises 53 and 54, reason similarly to prove that each argument is valid.

$\begin{array}{l}\underset{\_}{\begin{array}{l}p\to \sim q\\ p\to r\\ \sim s\to q\\ p\end{array}}\\ \therefore r\wedge s\end{array}$