## Thinking Mathematically (6th Edition)

Use letters to represent simple statements in the argument. p: A doctor tells the truth. q: A doctor destroys the base on which the placebo rests. r: A doctor jeopardizes a relationship build on trust. Express the premises and the conclusion symbolically. $p\to q$ If the doctor tells the truth, he destroys the base on which the placebo rests. $\tilde{\ }p\to r$ If he doesn’t tell the truth, he jeopardizes a relationship build on trust. $\frac{r}{\therefore q}$ $\frac{\text{The ALA says that secondary cigarette smoke is a health threat}\text{.}}{\text{It }\!\!'\!\!\text{ s wrong to smoke in public}\text{.}}$ 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 last column of the truth table is true for every possible case. Thus, the provided argument is valid. Hence, the valid argument is “the doctor either destroys the base on which the placebo rests or jeopardizes a relationship build on trust.”