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: 87

Answer

Use letters to represent both simple statements in the argument. p: You only spoke when spoken to, and I only speak when spoken to. q: Nobody would ever say anything. Express the premises and the conclusion symbolically. \[p\to q\] If you only spoke when spoken to, and I only speak when spoken to, then nobody would ever say anything. \[\frac{\tilde{\ }q}{\therefore \tilde{\ }p}\]\[\frac{\text{Everyone would say something}}{\text{You wouldn }\!\!'\!\!\text{ t speak when spoken to, and I always speak}}\] Thus, the symbolic form of the provided argument is \[\begin{align} & p\to q \\ & \frac{\tilde{\ }q}{\therefore \tilde{\ }p} \\ \end{align}\] 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\[\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p\]. Construct a truth table for the conditional statement \[\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p\] as shown below:
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Work Step by Step

It can be seen from the truth table that the symbolic form \[\left[ \left( p\to q \right)\wedge \tilde{\ }q \right]\to \,\tilde{\ }p\] is always true. Thus, the valid argument is people sometimes speak without being spoken to. Hence, the valid argument is,people sometimes speak without being spoken to.
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