Syllogisms

Syllogisms

Concept

A syllogism is a logical argument where you derive a conclusion from two premises. In GATE’s logical reasoning section, the premises are categorical statements about groups (sets), and the conclusion is either a definite statement that MUST be true or a possible statement that COULD be true based on the information given.

The key insight is that logic distinguishes between what we know for certain (definite conclusions) and what could be true (possibilities). Sometimes the information given is rich enough to pin down exactly what’s true. Sometimes it only tells us what might be true. And sometimes the premises don’t give us enough information to draw any meaningful conclusion at all.

Understanding this distinction is half the battle in syllogism questions. The other half is knowing how to represent categorical statements visually using Venn diagrams — a technique that transforms abstract logical relationships into concrete spatial ones.

Types of Categorical Statements

There are four standard forms, each with specific Venn diagram representations:

All A are B — Every member of A is also a member of B. Visual: Circle A is entirely inside Circle B.

No A is B — No member of A is a member of B. Visual: Two circles that don’t overlap at all.

Some A are B — At least one member of A is also a member of B. Visual: The overlapping region between A and B is shaded to indicate it exists.

Some A are not B — At least one member of A is not a member of B. Visual: Part of Circle A outside Circle B is shaded.

The words “some” in logic means “at least one” — not necessarily many, just one or more.

Types & Approach

Type 1: Definite Conclusion Questions These ask: “Which of the following conclusions definitely follows?”

• Approach: Draw Venn diagrams for both premises. Look for what is definitely true in ALL possible arrangements consistent with the premises.
• If you can draw the conclusion with certainty from the diagram, it’s a valid definite conclusion.

Type 2: Possible Conclusion Questions These ask: “Which of the following could be true?” or “Which is a possible conclusion?”

• Approach: Draw Venn diagrams. Check if the conclusion COULD be true — that is, if there’s at least one valid arrangement consistent with the premises where the conclusion holds.
• If yes, it’s a possible conclusion. You don’t need it to be always true.

Type 3: Invalid/Conclusion Questions These ask which conclusion does NOT follow or is invalid.

• Approach: Draw the Venn diagram, then check each conclusion. The ones that definitely don’t follow are your answers.

Step-by-Step Example

Q: Premise 1: All roses are flowers. Premise 2: Some flowers are red. Conclusion: ?

Approach: Step 1 → Draw two circles: one for “roses” inside one for “flowers” (All roses are flowers). Step 2 → Add the second premise: some flowers are red. This means there’s an overlap between flowers and red things. The red circle overlaps with the flowers circle. Step 3 → Check definite conclusions: Can we say “All roses are red”? No — we only know some flowers are red, not which ones. The roses could be outside the red region. Step 4 → Check possible conclusions: Could “Some roses are red” be true? Yes, it’s possible (the roses that are within the “some flowers” area could overlap with red). Step 5 → Check: Can we say “Some roses are not red”? Yes, also possible.

Answer: Only a possible conclusion (like “Some roses could be red”) follows definitely? Actually no definite conclusion follows about the relationship between roses and red. So the answer is: “No definite conclusion follows.”

Common Mistakes

• Confusing “some” with “all” → Fix: “Some” only means at least one. It doesn’t mean all. So “Some A are B” doesn’t tell you whether all A are B.**
• Assuming converse is true (if All A are B, assuming All B are A) → Fix: This is the converse fallacy. All cats are animals, but NOT all animals are cats.**
• Drawing only one Venn diagram configuration → Fix: For possibility questions, draw ALL possible configurations consistent with premises. Check if the conclusion holds in any of them.**
• The undistributed middle fallacy → Fix: When both premises are “All A are B” type but they don’t share a common element to connect, no conclusion can be drawn. Example: All A are C, All B are C. We can’t conclude anything about A and B’s relationship.**