Three Statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: All peas are vegetables. All carrots are vegetables. All vegetables are greens. Conclusions: I. All peas are greens. II. All greens are carrots. III. Some greens are peas.

Correct — C. Chain the statements: peas inside vegetables, carrots inside vegetables, vegetables inside greens.

I. All peas are greens — peas sit inside vegetables and vegetables inside greens, so every pea is a green. I follows.

III. Some greens are peas — this is the particular that a universal affirmative always yields, since peas is treated as a non-empty class. III follows.

II. All greens are carrots — greens is the outermost class here and nothing shrinks it to carrots.

I and III together give option (c).

Why the others are wrong

• (a)
  Both conclusions II and III follow. — II and III keeps the wrong member of the pair. II would make greens no larger than carrots, but the statements only place carrots inside greens, never the reverse.
• (b)
  All conclusions follow. — All conclusions follow needs II as well. Containment runs one way, so vegetables inside greens says nothing about the greens that are neither peas nor carrots.
• (d)
  Both conclusions I and II follow. — I and II trades the sound conclusion III for the unsound II. III is the converse particular of I, and it holds automatically once all peas are greens.

A three-statement chain reads most easily as nested boxes. Peas and carrots are separate boxes inside vegetables, and vegetables sits inside greens.

Once nested, a definite conclusion is one you can read straight off the nesting. All peas are greens is visible at once. So is some greens are peas, because the pea box lies inside the green box and is not empty.

What you can never read off is a claim that the outer box lies inside an inner one, which is exactly what all greens are carrots asks for.

Key facts

• The nesting is peas inside vegetables, carrots inside vegetables, vegetables inside greens.
• All A are B always yields some B are A in this format, because A is taken as non-empty.
• Peas and carrots both sit inside vegetables, but nothing in the statements links them to each other.

Common traps

• Rejecting some greens are peas as too weak, when being weaker is exactly what makes it safe.
• Reading vegetables inside greens as though greens were inside vegetables.
• Assuming peas and carrots must overlap because both sit inside vegetables.

No directly related past PYQ was found.