## Thinking Mathematically (6th Edition)

Use letters to represent simple statements in the argument. p: Secondary cigarette smoke is a health threat. q: It’s wrong to smoke in public. r: The ALA says that secondary cigarette smoke is a health threat. Express the premises and the conclusion symbolically. $p\to q$: If secondary cigarette smoke is a health threat, then it’s wrong to smoke in public. $\tilde{\ }p\to \tilde{\ }r$: If secondary cigarette smoke is not a health threat, then it’s not wrong to smoke in public. $\frac{r}{\therefore q}$:$\frac{\text{The ALA says that secondary cigarette smoke is a health threat}\text{.}}{\text{It }\!\!'\!\!\text{ s wrong to smoke in public}\text{.}}$ Thus, the symbolic form of the provided argument is \begin{align} & p\to q \\ & \tilde{\ }p\to \tilde{\ }r \\ & \frac{r}{\therefore q} \\ \end{align} Write a symbolic statement of the form $\left[ \left( \text{premise 1} \right)\wedge \left( \text{premise 2} \right) \right]\to \text{conclusion}$. The symbolic statement is $\left[ \left( p\to q \right)\wedge \left( \tilde{\ }p\to \tilde{\ }r \right)\wedge r \right]\to q$. Construct a truth table for the conditional statement $\left[ \left( p\to q \right)\wedge \left( \tilde{\ }p\to \tilde{\ }r \right)\wedge r \right]\to q$as shown below.
It can be seen from the truth table that the symbolic form $\left[ \left( p\to q \right)\wedge \left( \tilde{\ }p\to \tilde{\ }r \right)\wedge r \right]\to q$ is always true. Thus, the provided argument is valid.