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Chapter 3: Problem 29

Determine whether each argument is valid or invalid. All \(A\) are \(B\), all \(B\) are \(C\), and all \(C\) are \(D\). Thus, all \(A\) are \(D\).

Short Answer

Expert verified
The given argument, 'all \(A\) are \(D\)', is Valid.

Step by step solution

01

Understanding the premises

The exercise starts with three premises: All \(A\) are \(B\), all \(B\) are \(C\), and all \(C\) are \(D\). There is an implication that \(A\), being a part of \(B\), is also a part of \(C\) through \(B\), and \(C\) is also part of \(D\). Therefore, it can be inferred that \(A\) is also part of \(D\)
02

Examining the argument

The argument directly stated in the exercise is 'all \(A\) are \(D\)'. This statement needs to be examined with the given premises to decide if the argument is valid or invalid.
03

Verifying the argument

If all \(A\) are \(B\), then anything that is true for \(B\) should also be true for \(A\). As per the premises, all \(B\) are \(C\), which now applies to \(A\) as well. The final premise says all \(C\) are \(D\), and since as per previous interpretation, \(A\) can be viewed as part of \(C\), \(A\) also falls under \(D\). So the argument '\(A\) are \(D\)' is valid.

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Most popular questions from this chapter

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

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 it is hot and humid, I complain. It is not hot or it is not humid. \(\therefore\) I am not complaining.

In Exercises 1-24, 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.

Use Euler diagrams to determine whether each argument is valid or invalid. All actors are artists. Sean Penn is an actor. Therefore, Sean Penn is an artist.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used Euler diagrams to determine that an argument is valid, but when I reverse one of the premises and the conclusion, this new argument is invalid.
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