Syllogism

Study Materials › Reasoning › Syllogism

You are given two or three statements and asked which conclusions follow. The statements are to be taken as true even when they contradict the real world — "All cats are birds" must be accepted for the purposes of the question. Draw circles; do not reason in your head.

About the examples on this page. Syllogism is a standard reasoning topic, but neither the 2022 Constable paper nor the 2022 SI paper contained one — those papers tested the related conclusions and arguments formats instead. The examples below are therefore constructed for practice and are marked as such.

The Four Statement Types

FormMeaningValid immediate conversion
All A are BA sits entirely inside BSome B are A
No A is BA and B do not overlapNo B is A
Some A are BA and B overlap partlySome B are A
Some A are not BPart of A lies outside Bnothing
"All A are B" never gives "All B are A". All police officers are citizens, but not all citizens are police officers. Reversing a universal statement is the single most common syllogism error.

Rules That Save Time

1. Two particular statements ("Some …") give no definite conclusion.

2. Two negative statements ("No …") give no definite conclusion.

3. If either statement is particular, the conclusion must be particular.

4. If either statement is negative, the conclusion must be negative.

5. A conclusion follows only if it holds in every possible diagram, not just the obvious one.

Either-Or Cases

When two conclusions individually fail but together cover every possibility, and they concern the same pair of terms, the answer is "either I or II follows". This arises most often with a complementary pair such as "Some A are B" and "No A is B".

Practice Examples

Constructed practice question

Statements: All pens are books. All books are tables.
Conclusions: I. All pens are tables.   II. All tables are pens.

Pens sit inside books, and books sit inside tables — so pens sit inside tables.
I follows.
II reverses a universal statement. Tables may contain much that is not a book, let alone a pen. II does not follow.
Answer: Only conclusion I follows.
Note: two universals chaining in the same direction is the one case that yields a clean universal conclusion.

Constructed practice question

Statements: Some doctors are engineers. All engineers are graduates.
Conclusions: I. Some doctors are graduates.   II. All doctors are graduates.

The doctors who are engineers must be graduates, since every engineer is one.
I follows.
II claims something about all doctors, but the statements say nothing about doctors who are not engineers. II does not follow.
Answer: Only conclusion I follows.
Rule 3 in action: one statement is particular ("Some"), so no universal conclusion is available.

Constructed practice question

Statements: All cups are plates. No plate is a spoon.
Conclusions: I. No cup is a spoon.   II. Some spoons are cups.

Cups lie entirely within plates, and plates are wholly separate from spoons. So no cup can be a spoon.
I follows.
II directly contradicts I, so it cannot follow. II does not follow.
Answer: Only conclusion I follows.
Rule 4 in action: one statement is negative, so the conclusion is negative.

Constructed practice question

Statements: Some keys are locks. Some locks are doors.
Conclusions: I. Some keys are doors.   II. No key is a door.

Both statements are particular, so by Rule 1 no definite conclusion is available — you can draw a diagram where some keys are doors and another where none are.
Neither I nor II follows on its own. But between them they exhaust the possibilities (either at least one key is a door, or none is), and they concern the same pair of terms.
Answer: Either I or II follows.
This is the case candidates miss. Having found that neither conclusion holds individually, check whether the two together cover every case before answering "neither follows".

Test a conclusion by trying to break it. Draw the diagram that would make it false. If no such diagram exists, the conclusion follows; if one does, it does not. This is faster and more reliable than trying to prove it true.

Also see: Inequalities · Assumptions & Conclusions · Non-Verbal Reasoning

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