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: Some circles are rectangles. Some rectangles are squares. All squares are triangles. Conclusions: I. No circle is a triangle. II. Some rectangles are triangles. III. All rectangles are triangles.

Correct — D. A conclusion follows only if it holds in every diagram the statements allow.

Rule: Some X are Y + All Y are Z gives Some X are Z.

Some rectangles are squares, and all squares are triangles, so at least one rectangle is inside the triangle region. Conclusion II follows.

Conclusion I asks for a certainty the statements do not give: a circle may overlap the rectangles at exactly the part that is square, and squares are triangles.

Conclusion III upgrades some to all — only some rectangles are squares, and the rest need not be triangles at all. II alone survives, which is option (d).

Why the others are wrong

• (a)
  Both conclusion I and II follow — II does follow, but I does not. No circle is a triangle is merely possible: nothing in the statements stops the circle-rectangle overlap from falling inside the square region.
• (b)
  Only conclusion I follows — This keeps the conclusion that cannot be proved and drops the one that can. I is a possibility; II is forced by the square-triangle link.
• (c)
  Only conclusion III follows — All rectangles are triangles would need all rectangles to be squares. The statements give only some, so III claims more than it is entitled to.

Syllogism is a test of certainty, not of plausibility.

Only a few statement pairs combine at all. Some X are Y plus All Y are Z is the one used here, and it yields Some X are Z.

Two particular statements, both starting with some, combine to nothing: some circles are rectangles and some rectangles are squares leave the circles and the squares completely unconnected.

A negative conclusion needs a negative statement somewhere in the set. All three statements here are affirmative, so no conclusion beginning No can follow.

Key facts

• Some X are Y combined with All Y are Z always yields Some X are Z.
• Two statements that both begin with Some yield no valid conclusion.
• A negative conclusion requires at least one negative statement, and all three statements here are affirmative.

Common traps

• Drawing one diagram, seeing conclusion I hold in it, and calling it proved.
• Chaining some circles are rectangles with some rectangles are squares.
• Reading conclusion III as a restatement of II rather than a stronger claim.

No directly related past PYQ was found.