6.2 Syllogism

Why This Matters

Syllogism is a 3-5 mark fixture in the RRB Group D Reasoning section. It tests whether a conclusion follows necessarily from two statements — no guesswork, no real-world knowledge. Two methods work: the Venn diagram method (most reliable) and the shortcut inference rules (faster). The exam rewards students who spot the complementary pair trap and reject "possibility" conclusions that the statements do not force.

Four Canonical Statements

Every syllogism is built from four statement types:

The Venn Method

1. Draw each premise as a Venn diagram. "All A are B" puts circle A fully inside circle B. "No A are B" makes circles A and B non-touching. "Some A are B" puts a tick in the overlap. "Some A are not B" puts a tick in the A-only zone.
2. Combine the two premise diagrams into one.
3. Check each conclusion against the combined diagram. If the conclusion holds in every valid diagram, it follows. If it fails in even one valid diagram, it does not follow.

Shortcut Inference Rules

Valid combinations:

• All A are B + All B are C → All A are C.
• All A are B + No B are C → No A are C.
• Some A are B + All B are C → Some A are C.
• No A are B + Some B are C → Some C are not A.

Combinations that yield no definite conclusion — memorise these too, because RRB sets them as traps:

• All A are B + Some B are C → nothing definite about A and C. The B's that are C might all lie outside A. (Worked Example 4 below is exactly this shape: "All books are pens" plus "Some pens are pencils" does not give "Some books are pencils".) Note how this differs from the valid "Some A are B + All B are C" — order matters, and swapping the two premise types breaks the inference.
• No A are B + No B are C → nothing definite about A and C.
• Some A are B + Some B are C → nothing definite about A and C.

A useful negative rule: "Some A are B" never implies "All A are B." Particular statements do not upgrade to universal ones.

Complementary Pairs

Two conclusions form a complementary pair when, between them, they must cover all cases: one is true if the other is false. The two recognized pairs are:

• All A are B ↔ Some A are not B.
• No A are B ↔ Some A are B.

When the premises do not determine which member of the pair is true, mark "Either I or II follows." When the premises do determine one member, only that member follows — do not use the either/or shortcut.

Possibility Cases

A conclusion with "can be" or "possibility" is valid only when the statements do not deny the possibility. For example, from "Some A are B," the conclusion "All A can be B" is a valid possibility, because Some does not contradict All. But "All A are B" as a definite conclusion does not follow from "Some A are B."

Worked Example 1

Statements:

• All cats are dogs.
• All dogs are rats.

Conclusions:

• I. All cats are rats.
• II. Some rats are cats.
• III. Some rats are dogs.

Solution:

• All cats are dogs + All dogs are rats → All cats are rats (I follows).
• If all cats are rats, then some rats are cats (II follows).
• All dogs are rats → some rats are dogs (III follows).

All I, II, and III follow.

Worked Example 2

Statements:

• Some pens are pencils.
• No pencil is an eraser.

Conclusions:

• I. Some pens are not erasers.
• II. Some pens are erasers.
• III. No pen is an eraser.

Solution:

• Draw "Some pens are pencils" with an overlap. "No pencil is an eraser" makes the pencil and eraser circles non-touching.
• I: Pens that are pencils cannot be erasers, so "Some pens are not erasers" follows.
• II: We do not know about pens that are not pencils; they could be erasers, but the definite "Some pens are erasers" is not forced. Does not follow.
• III: "No pen is an eraser" is too strong — only the pen-pencil overlap is excluded. Does not follow.

Only I follows.

Worked Example 3 (Complementary Pair)

Statements:

• Some birds are crows.

Conclusions:

• I. All birds are crows.
• II. Some birds are not crows.

Solution: "Some birds are crows" does not decide between All and Some-not. The two conclusions form a complementary pair (All vs Some-not). Either I or II follows.

Worked Example 4 (Definite Particular)

Statements:

• All books are pens.
• Some pens are pencils.

Conclusions:

• I. Some books are pencils.
• II. Some pens are books.

Solution:

• I: All books are pens; some pens are pencils. The books could or could not overlap with the pencils. Not necessary. Does not follow (a possibility conclusion "Some books can be pencils" would follow, but the definite one does not).
• II: All books are pens → Some pens are books. Follows.

Only II follows.

Exam Traps

• "Some" never implies "All." A particular statement cannot be upgraded to a universal conclusion.
• "Some A are B" and "Some A are not B" are not a complementary pair — both can be true at the same time.
• A conclusion that uses a term not present in either statement cannot follow. Watch for terms that appear only in conclusions.
• "No A are B" plus "No B are C" tells you nothing about A and C — never conclude "No A are C."