Question

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

\\
\

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

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

Explanation:

Identify the column headers and premises

Using the Truth Tables knowledge point
The argument is:
Premise 1: \(p
ightarrow q\)
Premise 2: \(q
ightarrow r\)
Conclusion: \(\sim r
ightarrow \sim p\)

The three blank column headers in the truth table correspond to the premises and the conclusion:
Column 4: \(p
ightarrow q\)
Column 5: \(q
ightarrow r\)
Column 6: \(\sim r
ightarrow \sim p\)

Evaluate the truth values for each row

Using the Truth Tables knowledge point
Row 1: \(p=T, q=T, r=T\)

• \(p

ightarrow q = T
ightarrow T = T\)

• \(q

ightarrow r = T
ightarrow T = T\)

• \(\sim r

ightarrow \sim p = F
ightarrow F = T\)

Row 2: \(p=T, q=T, r=F\)

• \(p

ightarrow q = T
ightarrow T = T\)

• \(q

ightarrow r = T
ightarrow F = F\)

• \(\sim r

ightarrow \sim p = T
ightarrow F = F\)

Row 3: \(p=T, q=F, r=T\)

• \(p

ightarrow q = T
ightarrow F = F\)

• \(q

ightarrow r = F
ightarrow T = T\)

• \(\sim r

ightarrow \sim p = F
ightarrow F = T\)

Row 4: \(p=T, q=F, r=F\)

• \(p

ightarrow q = T
ightarrow F = F\)

• \(q

ightarrow r = F
ightarrow F = T\)

• \(\sim r

ightarrow \sim p = T
ightarrow F = F\)

Row 5: \(p=F, q=T, r=T\)

• \(p

ightarrow q = F
ightarrow T = T\)

• \(q

ightarrow r = T
ightarrow T = T\)

• \(\sim r

ightarrow \sim p = F
ightarrow T = T\)

Row 6: \(p=F, q=T, r=F\)

• \(p

ightarrow q = F
ightarrow T = T\)

• \(q

ightarrow r = T
ightarrow F = F\)

• \(\sim r

ightarrow \sim p = T
ightarrow T = T\)

Row 7: \(p=F, q=F, r=T\)

• \(p

ightarrow q = F
ightarrow F = T\)

• \(q

ightarrow r = F
ightarrow T = T\)

• \(\sim r

ightarrow \sim p = F
ightarrow T = T\)

Row 8: \(p=F, q=F, r=F\)

• \(p

ightarrow q = F
ightarrow F = T\)

• \(q

ightarrow r = F
ightarrow F = T\)

• \(\sim r

ightarrow \sim p = T
ightarrow T = T\)

Analyze argument validity using the "Bad Row" method

Using the Argument Validity knowledge point
A "Bad Row" (counterexample) is a row where all premises are true, but the conclusion is false.
The premises are \(p
ightarrow q\) and \(q
ightarrow r\).
Rows where both premises are true:

• Row 1: Conclusion is \(T\)
• Row 5: Conclusion is \(T\)
• Row 7: Conclusion is \(T\)
• Row 8: Conclusion is \(T\)

Since there is no row where all premises are true and the conclusion is false, there is no "Bad Row." Therefore, the argument is valid.

Answer:

Question 1

The completed truth table is:

\(p\)

\(q\)

\(r\)

\(p

ightarrow q\) | \(q
ightarrow r\) | \(\sim r
ightarrow \sim p\) |

T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	T
T	F	F	F	T	F
F	T	T	T	T	T
F	T	F	T	F	T
F	F	T	T	T	T
F	F	F	T	T	T

Question 2

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