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Chapter 3: Problem 40

Explain how to use Euler diagrams to determine whether or not an argument is valid.

Short Answer

Expert verified
Euler diagrams can be used to test the validity of arguments by visually representing the premises and checking whether the conclusion logically follows. The diagrams plot categories (sets) as circles where overlaps and inclusions represent relationships between categories. The argument is proven to be valid if it follows the structure of the given premises.

Step by step solution

01

Understand the context

The first thing you need to do is to examine the structure of the argument, noting particularly the premises and the conclusion. Identify the categories (sets) that are involved and what is being claimed about them. Examples of some statements could be: 'All A are B', 'No A are B', 'Some A are B', 'Some A are not B'.
02

Drawing the Euler Diagram for the Premises

Begin by plotting the categories in the syllogism using circles. The area within the circle represents the category it labels; where circles overlap, this represents categories that have elements in common. Draw the diagram according to the premises. For 'All A are B', A is completely inside B. For 'No A are B', A and B circles don't overlap. For 'Some A are B', A and B circles overlap partially. For 'Some A are not B', A and B circles overlap but part of A is outside B. There could be several ways to represent the premises, and all of them should be considered
03

Check the Conclusion with the Diagram

Once you've drawn the diagram based on the premises, check whether the conclusion must be true based on that diagram(s). If you've drawn the diagram correctly and the conclusion derived from the premises stands true in the visual representation, the argument was sound and thus valid. But if the conclusion isn't true in any of the diagrams from step 2, the argument is invalid

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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 \(6 .\) Therefore, 8 is not a multiple of 3 .

Use Euler diagrams to determine whether each argument is valid or invalid. All insects have six legs. No spiders are insects. Therefore, no spiders have six legs.

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 a metrorail system is not in operation, there are traffic delays. Over the past year there have been no traffic delays. \(\therefore\) Over the past year a metrorail system has been in operation.

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 all people obey the law, then no jails are needed. Some people do not obey the law. \(\therefore\) Some jails are needed.

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.
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