Question

using the idea of the \bad row,\ fill in the entire truth table and test the validity of the following argument.
\\\

$$\begin{array}{c} p \ ightarrow q \\\\ q \ ightarrow r \\\\ \\hline \\sim r \ ightarrow \\sim p \\end{array}$$

\\

is the argument valid or invalid?
a. invalid
b. valid

Explanation:

Identify the argument components

Using the Truth Tables knowledge point
The argument consists of:

• Premise 1: \(p

ightarrow q\)

• Premise 2: \(q

ightarrow r\)

• Conclusion: \(\sim r

ightarrow\ \sim p\)

Construct the truth table columns

Using the Truth Tables knowledge point

$$
LATEXBLOCK0
$$

Analyze validity using the "Bad Row" method

Using the Argument Validity knowledge point
A "Bad Row" (counterexample) is a row where all premises are true (\(T\)) but the conclusion is false (\(F\)).

• Rows with all true premises: Row 1, Row 5, Row 7, Row 8.
• In all these rows, the conclusion \(\sim r

ightarrow\ \sim p\) is also true (\(T\)).

• Since there is no "Bad Row," the argument is valid.

Answer:

• (A) Invalid
• (B) Valid (Correct answer)