Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 3 - Logic - 3.7 Arguments and Truth Tables - Exercise Set 3.7 - Page 194: 88

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.
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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.
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