Answer
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.
Work Step by Step
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.
