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Chapter 3: Problem 11

Use Euler diagrams to determine whether each argument is valid or invalid. All professors are wise people. Some professors are actors. Therefore, some wise people are actors.

Short Answer

Expert verified
The argument 'All professors are wise people, some professors are actors, therefore some wise people are actors' is valid.

Step by step solution

01

Define the Categories

Identify the different categories from the statements. In this case, the categories are: 'professors', 'wise people', and 'actors'.
02

Draw First Two Premises

Create an Euler diagram depicting the first two premises. Draw a circle to represent the category 'professors' entirely within another circle that represents 'wise people', indicating 'all professors are wise people'. Regarding the statement 'some professors are actors', create an overlapping area between 'professors' and 'actors', signifying the intersection between the two.
03

Analyze the Conclusion

Subsequently, check if the logical conclusion 'some wise people are actors' is true in the previously drawn diagram. It is observed that there is indeed an overlap between 'wise people' and 'actors'. This area represents wise people who are also actors. This is due to the overlap between 'professors' (all of whom are wise people) and 'actors'.
04

Conclusion of Validity

From the analysis of the drawn Euler diagram and given the premises, it is safe to conclude that 'some wise people are actors', therefore, the argument is valid.

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Most popular questions from this chapter

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.) He is intelligent or an overachiever. He is not intelligent. \(\therefore\) He is an overachiever.

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

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 Tim and Janet play, then the team wins. Tim played and the team did not win. \(\therefore\) Janet did not play.

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 I'm at the beach, then I swim in the ocean. If I swim in the ocean, then I feel refreshed. \(\therefore\) If I'm not at the beach, then I don't feel refreshed.

Determine whether each argument is valid or invalid. No \(A\) are \(B\), some \(A\) are \(C\), and all \(C\) are \(D\). Thus, some \(D\) are \(B\)
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