The question format
You are given two or more statements, assumed true even if they sound odd, and some conclusions. A conclusion follows only if it is true in every possible situation the statements allow. "Possibly true" is not enough.
The Venn diagram method
Draw each category as a circle.
Statement | Diagram |
|---|---|
All A are B | circle A entirely inside circle B |
Some A are B | circles A and B overlap |
No A is B | circles A and B do not touch |
Some A are not B | part of A lies outside B |
Draw the least overlapping picture the statements allow. If a conclusion still holds, it follows.
Conversions that always work
All A are B → Some B are A. (Not "All B are A".)
Some A are B → Some B are A.
No A is B → No B is A.
"Some A are not B" cannot be converted.
Combining two statements
Statement 1 | Statement 2 | Definite conclusion |
|---|---|---|
All A are B | All B are C | All A are C |
All A are B | No B is C | No A is C |
Some A are B | All B are C | Some A are C |
Some A are B | No B is C | Some A are not C |
Some A are B | Some B are C | nothing definite about A and C |
No A is B | Some B are C | Some C are not A |
Worked example
Statements: Some books are pens. All pens are red.
I. Some books are red. The books that are pens are red, so this follows.
II. All red things are books. Red could include things that aren't books, so this does not follow.
Answer: only I.
Shortcuts and tricks
The standard way is to draw Venn diagrams that fit the statements and check whether each conclusion holds in every diagram. The rules below give the same answers for the common patterns. The statements must link through a shared middle term, in this order.
Rules for two statements
First statement | Second statement | Definite conclusion |
|---|---|---|
All A are B | All B are C | All A are C |
All A are B | No B is C | No A is C |
Some A are B | All B are C | Some A are C |
Some A are B | No B is C | Some A are not C |
No A is B | Some B are C | Some C are not A |
Some A are B | Some B are C | nothing definite |
All A are B | Some B are C | nothing definite |
Conversions
"All A are B" gives "Some B are A".
"Some A are B" gives "Some B are A".
"No A is B" gives "No B is A".
"Some A are not B" can't be turned round.
Worked comparison
Example: Some pens are blue. All blue things are bright. Does "some pens are bright" follow?
Standard way: draw blue inside bright and pens overlapping blue. The overlap is inside bright in every diagram, so yes.
Shortcut: Some + All gives Some: some pens are bright.
Exam tips
Never use real-world knowledge; only the statements count.
"Some" means "at least one, possibly all".
Two "some" statements never give a definite conclusion between the end terms.
When two conclusions are "either-or" (one of them must be true), the answer may be "either I or II".