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Chapter 3: Problem 53

Translate each argument into symbolic form. Then determine whether the argument is valid or invalid. It is the case that \(x<5\) or \(x>8\), but \(x \geq 5\), so \(x>8\).

Short Answer

Expert verified
The argument It is the case that \(x<5\) or \(x>8\), but \(x \geq 5\), so \(x>8\) translated into quantifiable logic statements is It is the case that \(P\) or \(Q\), but \(R\), so \(Q\). The argument is valid, as the conclusion \(Q\) or \(x>8\) necessarily follows from the premises.

Step by step solution

01

Translate the argument into symbolic form

First, translate the arithmetic inequalities to quantifiable logic statements. Let's use 'P' to represent the statement \(x<5\) , 'Q' to represent the statement \(x>8\), and 'R' to represent the statement \(x \geq 5\). So the initial argument translates to: It is the case that \(P\) or \(Q\), but \(R\), so \(Q\).
02

Evaluate the validity of the argument

In logic, argument is considered valid if the conclusion necessarily follows from the premises. Here, given that either \(P\) or \(Q\) is true, and \(R\) is true, because \(x \geq 5\), we can already reject \(P\) as false, leaving us with \(Q\) or \(x > 8\), as the necessarily valid conclusion.

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