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Chapter 3: Problem 41

What are equivalent statements?

Short Answer

Expert verified
Equivalent statements are those that have the same truth value in all possible interpretations. If two statements are equivalent, they are either both true or both false at the same time in all scenarios.

Step by step solution

01

Define Equivalent Statements

In logic, equivalent statements are statements that, regardless of the truth value assigned to its variables, have the same truth value. In simple terms, if you have two statements, they are equivalent if both are true or both are false at the same time in all possible scenarios.
02

Explain 'Truth Value'

Truth values are the values that a statement can have which, in classical logic, are exclusively 'true' or 'false'. For example, the statement 'It is raining' may be true or false depending on whether it is actually raining or not.
03

Provide Examples

Let's consider two simple statements: 'If it is raining, then the ground is wet' and 'If the ground is not wet, then it is not raining'. These are logically equivalent statements, because they are both true or both false in the same circumstances.

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Most popular questions from this chapter

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

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I made Euler diagrams for the premises of an argument and one of my possible diagr芒ms did not illustraate the conclusion, so the argument is invalid.

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 some journalists learn about the invasion, the newspapers will print the news. If the newspapers print the news, the invasion will not be a secret. No journalists learned about the invasion. \(\therefore\) The invasion was a secret.

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

Write an original argument in words that has a true conclusion, yet is invalid.
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