Question

If “All Philosophers are Logicians” is True, then which of the following propositions is false?

• A. Some Philosophers are Logicians
• B. No Philosophers are Logicians
• C. Some Logicians are Philosophers
• D. No Philosophers are non-Logicians

Hint:

The direct contradictory of “All S are P” is “Some S are not P”, but “No S are P” is clearly false when “All S are P” is true.

Answer 1

The Correct Option is B

Concept:
The given proposition is: \[ \text{All Philosophers are Logicians} \] This is an A-type universal affirmative proposition: \[ \text{All } S \text{ are } P \] Here: \[ S = \text{Philosophers} \] \[ P = \text{Logicians} \]

Step 1: Understand the meaning.
The statement means every philosopher belongs to the class of logicians. So, if someone is a philosopher, then that person is also a logician.

Step 2: Check option (A).
If all philosophers are logicians, then it is also true that: \[ \text{Some Philosophers are Logicians} \] provided philosophers exist. So, option (A) is not false.

Step 3: Check option (B).
Option (B) says: \[ \text{No Philosophers are Logicians} \] This directly contradicts: \[ \text{All Philosophers are Logicians} \] So, option (B) is false.

Step 4: Check option (C).
If some philosophers exist and all philosophers are logicians, then some logicians are philosophers. So, option (C) is not false.

Step 5: Check option (D).
If all philosophers are logicians, then no philosopher is a non-logician. So, option (D) is also consistent. Hence: \[ \boxed{\text{No Philosophers are Logicians}} \]