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 192: 58

Answer

First, we will use a letter to represent each simple statement of the argument: $p:$ An argument is valid. $q:$ An argument produce truth. \[r:\] An argument is sound. Now, the above statement can be written in form of premises and conclusion symbolically as: $\sim p\to \sim q$ If an argument is invalid, it does not produce truth. $\left( p\wedge \sim r \right)\to \sim q$ A valid unsound argument also does not produce truth $\sim p\vee \left( p\wedge \sim r \right)$ Arguments are invalid, or they are valid but unsound. $\therefore \ \sim q$ No arguments produce truth. Rewriting the conditional statements in symbolic form: $\left[ \left( \sim p\to \sim q \right)\wedge \left\{ \left( p\wedge \sim r \right)\to \sim q \right\}\wedge \left\{ \sim p\vee \left( p\wedge \sim r \right) \right\} \right]\to \sim q$. Now, with all the information above, we will construct a truth table for the conditional statement,

Work Step by Step

The entries in the final column of the truth table are all true, so the conditional statement is a tautology. Thus, the given argument is valid but sound. The argument is valid but sound.
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