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Syllogism-Based Data Sufficiency

Introduction

Syllogism-based Data Sufficiency problems ask whether the given statement(s) logically lead to a stated conclusion. Instead of proving the conclusion, your task is to decide whether each statement alone - or their combination - provides enough information to establish the conclusion with certainty.

This pattern is important because it trains precise logical inference: distinguishing what must be true from what may be true.

Pattern: Syllogism-Based Data Sufficiency

Pattern

Convert verbal statements into standard categorical relations (All, No, Some, Some not). Test the conclusion using formal syllogistic rules: if either statement alone guarantees the conclusion, mark it sufficient; if only both together guarantee it, mark them necessary; if neither does, mark insufficient.

Useful translations:
All A are B → A ⊂ B
No A are B → A ∩ B = ∅
Some A are B → A ∩ B ≠ ∅ (existence implied)
Some A are not B → not(A ⊂ B) and existence implied.

Step-by-Step Example

Question

Does the conclusion "All A are C" follow?
(I) All A are B.
(II) All B are C.

Options:
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) says All A are B ⇒ A ⊂ B. This does not connect B to C, so we cannot deduce All A are C. → (I) insufficient.

  2. Step 2: Analyze (II)

    (II) says All B are C ⇒ B ⊂ C. This does not tell us whether A ⊂ B (or even that A exists), so we cannot deduce All A are C from (II) alone. → (II) insufficient.

  3. Step 3: Combine

    From (I) A ⊂ B and (II) B ⊂ C, transitivity gives A ⊂ C ⇒ All A are C. Both statements together are sufficient.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Transitive chain A→B and B→C ⇒ A→C. Neither link alone gives the complete chain ✅

Quick Variations

1. Mix of universal and particular: e.g., All A are B and Some C are A - watch for existential import.

2. Negative premises: No A are B combined with All B are C has different implications - convert to set relations carefully.

3. Existential pitfalls: 'Some' implies existence; many syllogisms fail unless existence is explicit.

4. Chain reasoning: often you must link multiple universals (All) to reach a universal conclusion.

Trick to Always Use

  • Step 1: Translate each statement into set-relations (All / No / Some).
  • Step 2: Check for transitivity: All A→B and All B→C ⇒ All A→C.
  • Step 3: Watch existence: conclusions using "Some" require at least one existence-providing premise.
  • Step 4: If either statement alone guarantees the set relation in the conclusion, it’s sufficient; else check the combination.

Summary

Summary

  • Translate premises into standard categorical relations before testing the conclusion.
  • Use transitivity for universal premises: All A→B and All B→C ⇒ All A→C.
  • Be careful with "Some": it requires existential information; universal premises alone may not imply existence.
  • When in doubt, construct a counterexample to test insufficiency (one model where premises true but conclusion false).

Example to remember:
If (I) All A are B and (II) All B are C, then together we can conclude All A are C; neither statement alone suffices.

Practice

(1/5)
1. Does the conclusion 'All A are B' follow?<br>(I) All A are B.<br>(II) Some B are C.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) explicitly states All A are B ⇒ A ⊂ B, so the conclusion 'All A are B' follows directly from (I). → Sufficient.

  2. Step 2: Analyze (II)

    (II) Some B are C gives information about B and C only and does not tell us anything about A → Insufficient.

  3. Final Answer:

    Only (I) is sufficient → Option A

  4. Quick Check:

    (I) gives the required inclusion explicitly; (II) is unrelated to A → ✅
Hint: If a statement repeats the conclusion verbatim (All X are Y), it is immediately sufficient.
Common Mistakes: Looking for transitive links when the conclusion is already stated.
2. Does the conclusion 'All B are C' follow?<br>(I) Some A are B.<br>(II) All B are C.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) Some A are B only states an overlap between A and B; it gives no information about B and C → Insufficient.

  2. Step 2: Analyze (II)

    (II) directly states All B are C ⇒ B ⊂ C, so the conclusion follows from (II) alone → Sufficient.

  3. Final Answer:

    Only (II) is sufficient → Option B

  4. Quick Check:

    (II) is the exact universal inclusion required for the conclusion → ✅
Hint: A direct universal premise matching the conclusion is sufficient by itself.
Common Mistakes: Treating partial overlap (Some) as if it implies universal inclusion.
3. Does the conclusion 'All C are D' follow?<br>(I) All C are D.<br>(II) All C are D.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) states All C are D ⇒ C ⊂ D, so the conclusion follows from (I) alone → Sufficient.

  2. Step 2: Analyze (II)

    (II) also states All C are D ⇒ (II) alone is likewise sufficient.

  3. Final Answer:

    Each statement alone is sufficient → Option C

  4. Quick Check:

    Both premises give the same universal inclusion required by the conclusion → ✅
Hint: If each premise independently provides the same universal relation, both are individually sufficient.
Common Mistakes: Looking for combination when single-premise universals already suffice.
4. Does the conclusion 'Some A are C' follow?<br>(I) Some A are B.<br>(II) All B are C.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) Some A are B indicates overlap A-B but gives no information about C → Insufficient.

  2. Step 2: Analyze (II)

    (II) All B are C ⇒ B ⊂ C but gives no information about A → Insufficient.

  3. Step 3: Combine

    From (I) Some A are B and from (II) All B are C ⇒ those A that are B are also C ⇒ Some A are C → Sufficient together.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Overlap (Some) + inclusion (All) ⇒ overlap propagates to C → ✅
Hint: A 'Some' overlap combined with an 'All' inclusion gives the required 'Some' conclusion.
Common Mistakes: Assuming 'All' alone implies existence for 'Some' conclusions without an overlap premise.
5. Does the conclusion 'Some B are A' follow?<br>(I) All B are A.<br>(II) There exists at least one B.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) All B are A ⇒ B ⊂ A. This is a universal inclusion but does not guarantee that any B actually exists (universal statements do not imply existence). Therefore (I) alone is insufficient to conclude 'Some B are A'.

  2. Step 2: Analyze (II)

    (II) There exists at least one B ⇒ ensures existence of B but gives no information about A. Thus (II) alone is insufficient to conclude 'Some B are A'.

  3. Step 3: Combine

    Combining (I) and (II): (I) provides B ⊂ A and (II) provides that B is non-empty. Together these imply that at least one member of B exists and, since B ⊂ A, that member is in A. Hence 'Some B are A' follows from the pair of statements.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Universal inclusion + explicit existence ⇒ particular conclusion (Some) ✅
Hint: A universal inclusion (All X are Y) needs an existence assertion to conclude a 'Some' statement.
Common Mistakes: Assuming 'All B are A' implies there exists B; universal claims do not imply existence.

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