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Chapter 3: Problem 4

Use Euler diagrams to determine whether each argument is valid or invalid. All cowboys live on ranches. All cowherders live on ranches. Therefore, all cowboys are cowherders.

Short Answer

Expert verified
The argument is invalid because the fact that cowboys and cowherders both live on ranches does not imply that all cowboys are cowherders.

Step by step solution

01

Draw Individual Circles for Each Group

First, draw three separate circles - one each for 'cowboys', 'cowherders', and 'ranches'.
02

Mark the Overlapping Areas According to the Premises

Then, according to the first premise 'All cowboys live on ranches', overlap the 'cowboys' circle with the 'ranches' circle. Following the second premise 'All cowherders live on ranches', overlap the 'cowherders' circle with the 'ranches' circle. This shows that all cowboys and all cowherders are residing on ranches, but it does not necessarily mean they are part of the same group. Two different groups can reside at the same place (ranches) without being the same.
03

Analyze the Diagram

Now, the diagram does not show any overlap between cowboys and cowherders, because there's no premise stating that there's a connection between cowboys and cowherders. Therefore, from this diagram, it cannot be inferred that 'All cowboys are cowherders', despite both being on ranches.

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Most popular questions from this chapter

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 3 . Therefore, 8 is not a multiple of 6 .

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.) We criminalize drugs or we damage the future of young people. We will not damage the future of young people. \(\therefore\) We criminalize drugs.

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.) There must be a dam or there is flooding. This year there is flooding. \(\therefore\) This year there is no dam.

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

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