Question

suppose the conditional ( p \to q ) is true and the conditional ( q \to r ) is false. must the conditional ( p \to r ) also be false? explain. choose the correct answer below

a. no, ( p \to r ) is not false. if ( p \to q ) is true and ( q \to r ) is false, then the hypothesis of ( p \to r ) is false. this means ( p \to r ) is true.

b. no, ( p \to r ) is not necessarily false. if ( q \to r ) is false, then ( q ) is true and ( r ) is false. this means that either ( p ) is true and ( r ) is false, so that ( p \to r ) is false, or ( p ) and ( r ) are both false, so that ( p \to r ) is true. therefore, ( p \to r ) could be either true or false.

c. yes, ( p \to r ) is false. if ( q \to r ) is false, then ( q ) is true and ( r ) is false. this means that either ( p ) is true and ( r ) is false or ( p ) and ( r ) are both false. in either case, ( p \to r ) is false.

d. no, ( p \to r ) is not necessarily false. if ( p \to q ) is true and ( q \to r ) is false, then the conclusion of ( p \to r ) is false. this means ( p \to r ) could be either true or false.

e. no, ( p \to r ) is not false. the two conditionals ( p \to q ) and ( q \to r ) can be combined to form the single conditional ( p \to r ). this means ( p \to r ) is true.

f. no, ( p \to r ) is not false. if ( q \to r ) is false, then ( q ) is true and ( r ) is false. this means that either ( p ) is true and ( r ) is false or ( p ) and ( r ) are both false. in either case, ( p \to r ) is true.

Explanation:

To solve this, we analyze the truth values of conditional statements. A conditional \( p \to q \) is false only when \( p \) is true and \( q \) is false (by the definition of a conditional: \( p \to q \) is \(
eg p \lor q \), so it's false when \( p \land
eg q \)). Similarly, \( q \to r \) is false when \( q \) is true and \( r \) is false.

Step 1: Analyze \( p \to q \) being true

If \( p \to q \) is true, then either:

• \( p \) is false (regardless of \( q \)), or
• \( p \) is true and \( q \) is true.

Step 2: Analyze \( q \to r \) being false

If \( q \to r \) is false, by the definition of a conditional, this means \( q \) is true and \( r \) is false (since \( q \to r \) is false only when \( q \land
eg r \)).

Step 3: Combine the two conditions

From \( q \to r \) being false, we know \( q \) is true and \( r \) is false. Now, substitute \( q = \text{true} \) into the analysis of \( p \to q \):

Since \( p \to q \) is true and \( q \) is true, \( p \) can be either true or false (because \(
eg p \lor q \) is true when \( q \) is true, regardless of \( p \)).

Now, we check \( p \to r \):

• A conditional \( p \to r \) is false only when \( p \) is true and \( r \) is false.
• We know \( r \) is false (from \( q \to r \) being false). So:
• If \( p \) is true, then \( p \to r \) is false (because true \( \to \) false is false).
• If \( p \) is false, then \( p \to r \) is true (because false \( \to \) false is true).

Thus, \( p \to r \) could be either true or false—there’s no guarantee it’s false. Now, we match this reasoning to the options:

• Option B states: “No, \( p \to r \) is not necessarily false. If \( q \to r \) is false, then \( q \) is true and \( r \) is false. This means that either \( p \) is true and \( r \) is false (so that \( p \to r \) is false), or \( p \) and \( r \) are both false (so that \( p \to r \) is true). Therefore, \( p \to r \) could be either true or false.”

Answer:

B. No, \( p \to r \) is not necessarily false. If \( q \to r \) is false, then \( q \) is true and \( r \) is false. This means that either \( p \) is true and \( r \) is false, so that \( p \to r \) is false, or \( p \) and \( r \) are both false, so that \( p \to r \) is true. Therefore, \( p \to r \) could be either true or false.