Syllogisms for AFCAT: Venn diagrams, definite conclusions and the either–or case

A syllogism question gives you two or three short statements, such as "All pilots are officers" and "Some officers are engineers", followed by two conclusions. You decide which conclusions must be true if the statements are true. The options are the familiar five: only I follows, only II follows, either I or II follows, neither follows, both follow.

These questions punish two habits: using what you know about the real world, and accepting a conclusion because it could be true. The cure is a quick Venn diagram and one question asked of every conclusion: is there any way to draw the statements so that this conclusion is false?

The four statement types

"Some" means at least one, possibly all. "Some A are B" does not rule out "All A are B". This single point decides a large share of questions.

In exam syllogisms, every group named is taken to have at least one member. That is why "All A are B" lets you say "Some B are A".

Conversion: what one statement gives you

Combining two statements

When two statements share a middle term (B), a definite conclusion about A and C exists only in certain combinations. Learn this table and you can answer many questions without drawing.

"Nothing definite about A and C" does not mean the answer is "neither follows". Check the conclusions for conversions of a single statement, and check for an either–or pair.

The drawing method

1. Draw the first statement in its tightest form.
2. Add the second statement, trying to place the new circle so that it breaks each conclusion.
3. If you can break a conclusion with a legal diagram, it does not follow.
4. If every attempt to break it fails, it follows.

Worked examples

Example 1.

• All pilots are officers.
• Some officers are engineers.

Conclusions: I. Some officers are pilots. II. Some pilots are engineers.

• I is the conversion of "All pilots are officers". It follows.
• For II, draw the engineers overlapping only the part of the officers circle that lies outside the pilots circle. Both statements hold, and no pilot is an engineer. II does not follow.
• Only I follows.

Example 2.

• No aircraft is a ship.
• All ships are vessels.

Conclusions: I. No aircraft is a vessel. II. Some vessels are not aircraft.

• For I, the vessels circle is large. It can reach out and overlap aircraft, so I can be false. It does not follow.
• For II, the ships are vessels, and no ship is an aircraft. So those vessels are not aircraft. II follows.
• Only II follows. This is the "No A is B, All B are C" row of the table.

Example 3 (either–or).

• Some cadets are athletes.
• Some athletes are swimmers.

Conclusions: I. Some cadets are swimmers. II. No cadet is a swimmer.

• Neither follows on its own. The cadets and swimmers can overlap or stay apart.
• But I and II cannot both be false: either some cadet is a swimmer or none is.
• Either I or II follows.

The either–or answer needs three things at once: neither conclusion follows by itself, both are about the same two groups, and together they cover every case. The usual pairs are "Some A are B" with "No A is B", and "All A are B" with "Some A are not B".

Example 4 (three statements).

• All pilots are officers.
• All officers are trained.
• Some trained people are instructors.

Conclusions: I. All pilots are trained. II. Some instructors are officers.

• Pilots sit inside officers, which sit inside trained. I follows.
• The instructors can overlap only the part of trained that lies outside officers. II does not follow.
• Only I follows.

Possibility conclusions

Some papers phrase conclusions as "All engineers being pilots is a possibility". Here the test flips. The conclusion is correct if you can draw at least one legal diagram where it is true.

• "All A are B, No B is C. Some A being C is a possibility." False: A sits inside B, and B shares nothing with C.
• "All A are B. All B being A is a possibility." True: A and B can be the same circle.

A possibility is ruled out only when a statement directly forbids it.

Practice set

For each question, decide: only I, only II, either I or II, neither, or both.

1. All pens are books. No book is a chair. I. No pen is a chair. II. Some books are pens.
2. Some flags are poles. All poles are rods. I. Some flags are rods. II. All rods are poles.
3. No cat is a dog. No dog is a rat. I. No cat is a rat. II. Some cats are rats.
4. Some jets are fast. Nothing fast is cheap. I. Some jets are not cheap. II. Some cheap things are jets.
5. All radars are sensors. Some sensors are not cheap. I. Some radars are not cheap. II. Some sensors are radars.
6. Some rivers are lakes. Some lakes are seas. I. Some rivers are seas. II. Some seas are lakes.
7. All cadets are officers. No officer is lazy. I. Some cadets being lazy is a possibility. II. All officers being cadets is a possibility.

Answers:

1. Both follow. All pens are books, and no book is a chair, so no pen is a chair. II is the conversion of the first statement.
2. Only I follows. Some flags are poles, and those poles are rods. The rods can include things that are not poles, so II fails.
3. Either I or II follows. Neither is forced, since cats and rats can overlap or stay apart. But one of them must be true.
4. Only I follows. The jets that are fast cannot be cheap. II fails because the cheap circle can sit apart from the jets altogether.
5. Only II follows. II is the conversion of "All radars are sensors". For I, the sensors that are not cheap may all lie outside radars.
6. Only II follows. II is the conversion of "Some lakes are seas". I fails because the river part and the sea part of the lakes circle can be different.
7. Only II. Cadets sit inside officers, and no officer is lazy, so a lazy cadet is impossible. But officers and cadets can be the same circle, so II is a possibility.

Each answer holds for every diagram that fits the statements, not just for one drawing.

What to do next

• Memorise the combination table, then test yourself by covering the last column.
• For every practice question you get wrong, draw the diagram that breaks the conclusion you accepted.
• Do 10 syllogisms a day for a week, including at least two either–or cases and two possibility cases.
• Revise alongside seating arrangement and classification, and see where syllogisms sit in the AFCAT reasoning plan.