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Chapter 3: Problem 24

Use Euler diagrams to determine whether each argument is valid or invalid. All multiples of 6 are multiples of 3 . Eight is not a multiple of \(6 .\) Therefore, 8 is not a multiple of 3 .

Short Answer

Expert verified
The argument given is invalid. Just because eight is not a multiple of 6 doesn't directly imply that it is not a multiple of 3. The Euler Diagram representation supports this, 8 is outside both the multiples of 6 and 3 circles.

Step by step solution

01

Understanding the Problem

Start by understanding what is given. In the problem, there are two premises given. The first premise is that all multiples of 6 are multiples of 3 and the second premise is that eight is not a multiple of 6. The conclusion that has been arrived at based on these premises is that eight is not a multiple of 3. The task here is to determine whether this argument is valid or invalid using Euler diagrams.
02

Draw the Euler Diagrams

Euler Diagrams represent sets and subsets graphically. So, create two circles - one for multiples of 6 and one for multiples of 3. The multiples of 6 circle will be completely inside the multiples of 3 circle to indicate that all multiples of 6 are multiples of 3. Place an '8' outside both circles to represent that it's neither a multiple of 6 nor a multiple of 3.
03

Analyze the Diagram

Consider the placement of the '8'. 8 is outside of the multiples of 6, supporting the second premise that eight is not a multiple of 6. However, it is also outside of the multiples of 3 circle whilst the conclusion claims that eight is not a multiple of 3, which is not necessarily true according to our diagram. This suggests a flaw in the logic of the argument.

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Most popular questions from this chapter

Use Euler diagrams to determine whether each argument is valid or invalid. All thefts are immoral acts. Some thefts are justifiable. Therefore, some immoral acts are justifiable.

Use Euler diagrams to determine whether each argument is valid or invalid. All writers appreciate language. All poets are writers. Therefore, all poets appreciate language.

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. You may use a truth table or, if applicable, compare the argument's symbolic form to a standard valid or invalid form. (You can ignore differences in past, present, and future tense.) If The Graduate and Midnight Cowboy are shown, then the performance is sold out. Midnight Cowboy was shown and the performance was not sold out. \(\therefore\) The Graduate was not shown.

Use a truth table to determine whether the symbolic form of the argument is valid or invalid. $$ \begin{aligned} &p \rightarrow q \\ &\underline{q \rightarrow r} \\ &\therefore \sim p \rightarrow \sim r \end{aligned} $$

If you are given an argument in words that contains two premises and a conclusion, describe how to determine if the argument is valid or invalid.
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