Syllogism for RRB Group D

A syllogism question gives you two or three short statements and two conclusions. You must decide which conclusions definitely follow. The statements may sound silly ("All chairs are rivers"), and that is on purpose. You must treat them as true and ignore what you know about the real world.

There are only four kinds of statement, and each has a fixed picture. Draw the picture, test each conclusion, and the answer is almost mechanical.

The four statement types

"Some" means at least one. It does not mean "not all". So "Some A are B" is still true when all A are B.

The method

1. Draw the simplest diagram that fits all the statements.
2. For each conclusion, ask: "Is this true in every diagram I could draw from these statements?"
3. Try to draw a second diagram, still fitting the statements, in which the conclusion is false. If you can, the conclusion does not follow.
4. If no such diagram exists, the conclusion follows.

Rules that save time

Two more rules help with two statements joined in a chain:

• All A are B + All B are C gives All A are C.
• All A are B + No B is C gives No A is C.
• Some A are B + No B is C gives Some A are not C (the A's that are B cannot be C).

The either–or case

Sometimes neither conclusion is certain, but one of them must always be true. Then the answer is "either I or II follows". Check for three things:

1. Neither conclusion follows on its own.
2. Both conclusions are about the same two groups.
3. They are an "opposite pair": Some + No (Some A are B / No A is B), or All + Some not (All A are B / Some A are not B).

Worked examples

Example 1: All cats are animals. Some animals are pets.

• I. Some cats are pets. II. All animals are cats.
• Draw cats inside animals, and pets overlapping animals. You can draw pets overlapping only the part of animals outside cats. So I is not certain. II reverses the first statement.
• Answer: neither follows.

Example 2: All pens are tools. All tools are useful.

• I. All pens are useful. II. Some useful things are pens.
• Pens inside tools, tools inside useful things. I follows by the chain rule. II follows because pens exist and every pen is useful.
• Answer: both follow.

Example 3: Some books are bags. No bag is a box.

• I. Some books are not boxes. II. No book is a box.
• The books that are bags cannot be boxes, so I follows. But other books, outside bags, could be boxes, so II is not certain.
• Answer: only I follows.

Example 4: Some A are B.

• I. All A are B. II. Some A are not B.
• Neither is certain. Both are about A and B, and they are an All + Some-not pair. Either every A is a B, or at least one is not.
• Answer: either I or II follows.

Example 5: No X is Y. All Y are Z.

• I. Some Z are not X. II. No Z is X.
• The Y's are inside Z, and no Y is an X. So those Z's are not X: I follows. X could still overlap the part of Z outside Y, so II is not certain.
• Answer: only I follows.

Common mistakes

• Using real-world knowledge instead of the statements.
• Turning "All A are B" into "All B are A".
• Treating "possible" as "follows".
• Reading "some" as "only some". Some can mean all.

Practice set

For each, say which conclusions follow: only I, only II, both, neither, or either I or II.

1. All trees are plants. No plant is a stone. I. No tree is a stone. II. Some plants are trees.
2. Some cars are red. All red things are bright. I. Some cars are bright. II. All bright things are cars.
3. All doctors are graduates. I. All graduates are doctors. II. Some graduates are doctors.
4. Some chairs are tables. Some tables are desks. I. Some chairs are desks. II. No chair is a desk.
5. No fish is a bird. Some birds are pets. I. Some pets are not fish. II. Some fish are pets.
6. All buses are vans. Some vans are trucks. I. Some buses are trucks. II. No bus is a truck.
7. All keys are locks. All locks are doors. I. All keys are doors. II. All doors are keys.
8. Some pens are pencils. All pencils are erasers. I. Some erasers are pens. II. Some pens are not erasers.

Answers:

1. Both. Trees inside plants, and plants apart from stones, so no tree is a stone. Trees exist, so some plants are trees.
2. Only I. The red cars are bright. Bright things can include much more than cars.
3. Only II. All doctors are graduates, so some graduates are doctors. The reverse of "all" does not follow.
4. Either I or II. Chairs and desks are never linked, so neither is certain. They are a Some + No pair about the same groups.
5. Only I. The pets that are birds cannot be fish. Fish and pets are never linked, so II is not certain.
6. Either I or II. Buses and trucks are not linked. The two conclusions are a Some + No pair.
7. Only I. Chain rule. II reverses "all".
8. Only I. The pens that are pencils are erasers. Every pen could still be an eraser, so II is not certain.

What to do next

• Draw the four statement pictures from memory.
• Copy the "rules that save time" table into your notebook and learn it.
• Solve 10 syllogisms a day for a week, drawing a second diagram before you mark anything as following.
• Next, study Venn diagrams, which use the same circles, and statement and conclusion questions.