Identify the appropriate conclusion based on the premises

Premise 1: No philosophers are mathematicians.

Premise 2: All logicians are philosophers.

Conclusion:

Premise 1: No philosophers are mathematicians.

Premise 2: All logicians are philosophers.

Think of three circles:

• The "Logicians" circle sits entirely inside the "Philosophers" circle (because all logicians are philosophers).
• The "Philosophers" circle has absolutely no overlap with the "Mathematicians" circle (because no philosophers are mathematicians).

So if logicians are inside the philosophers group, they too have zero overlap with mathematicians.

This gives us: No logician is a mathematician.

This matches Option 2.

Let's quickly verify the other options:

• Option 1: Says all logicians are mathematicians — the exact opposite of what we just found.
• Option 3: Says some philosophers are mathematicians — directly contradicts Premise 1.
• Option 4: Says all mathematicians are philosophers — Premise 1 says no philosophers are mathematicians, which also means no mathematicians are philosophers. So this is wrong too.

This is a classic syllogism — a logical argument where a conclusion is drawn from two premises:

All logicians are philosophers → No philosophers are mathematicians → Therefore, no logicians are mathematicians → correct