Syllogisms: how to solve them in reasoning tests

The four kinds of statement

Almost every syllogism is built from four statement types. Learn exactly what each one does and does not say.

• All A are B. Every A is inside B. It says nothing about whether every B is an A. "All auditors are accountants" does not mean all accountants are auditors.
• No A are B. A and B do not overlap at all. This one reverses safely: "No A are B" means the same as "No B are A".
• Some A are B. At least one A is a B. It may be just one, and it may be all of them. It also reverses safely: "Some A are B" means "Some B are A".
• Some A are not B. At least one A is outside B. It does not reverse: "Some A are not B" does not tell you that some B are not A.

"Some" is the word that costs the most marks. In everyday speech, "some managers are remote" suggests that others are not. In logic it does not. "Some A are B" is still true if all A are B, so you can never conclude "Some A are not B" from it.

Translating everyday wording

Test questions in logical reasoning and critical thinking practice rarely use the bare "All A are B" form. They use normal sentences, and you need to translate them before you sketch anything.

• "Every", "each", "any", "whoever is" all mean All.
• "None", "not one", "no one who" mean No.
• "At least one", "a few", "several", "many", "most" all mean Some for the purpose of the logic.
• "Not all A are B" means Some A are not B.
• "Only A are B" means All B are A. This is the reversal that catches most people. "Only certified staff operate the forklift" means every forklift operator is certified, not that every certified person operates it.
• "A are B unless they are C" usually means All A that are not C are B. Read these slowly.

Once a premise is translated, write it in short form (for example "All F are C") so you do not have to reread the sentence.

A reliable diagram method

Circle diagrams (often called Venn or Euler diagrams) are the most reliable way to solve syllogisms, because they force you to see every possibility.

1. Draw the first premise. "All A are B": a small A circle inside a big B circle. "No A are B": two separate circles. "Some A are B": two overlapping circles, with a dot or cross in the overlap to show something is definitely there.
2. Add the second premise to the same sketch, using the term the premises share (the middle term) to place the third circle.
3. Read the conclusion off the sketch.
4. Now try to break it. Draw a different sketch that still fits both premises. Move the third circle, make a circle bigger, or make two circles overlap more. If any valid sketch makes the conclusion false, the conclusion does not follow.

Step 4 is what separates people who get syllogisms right from people who guess. Most wrong options are conclusions that are true in the first sketch you drew, but not in every sketch.

Only mark what you know. If a premise says "Some A are B", put your cross in the overlap. Do not shade the rest of A as "not B", because you do not know that.

Worked examples

Example 1: follows. All managers are employees. All employees are badge holders. Conclusion: All managers are badge holders.

The managers circle sits inside employees, which sits inside badge holders. There is no way to draw a manager outside badge holders, so the conclusion follows.

Example 2: does not follow. All analysts are graduates. Some graduates are linguists. Conclusion: Some analysts are linguists.

Draw analysts inside graduates. The linguists circle must overlap graduates, but it can overlap the part of graduates outside analysts. That sketch fits both premises and breaks the conclusion, so it does not follow. (Some analysts could be linguists; they do not have to be.)

Example 3: follows. No contractors are shareholders. All directors are shareholders. Conclusion: No directors are contractors.

Directors sit inside shareholders, and shareholders do not touch contractors at all. So directors cannot touch contractors either. The conclusion follows.

Example 4: one follows, one does not. Some drivers are trainers. No trainers are smokers.

• "Some drivers are not smokers" follows: the drivers who are trainers cannot be smokers.
• "No drivers are smokers" does not follow: the drivers who are not trainers could be smokers.

Example 5: a negative conclusion. All auditors are accountants. Some consultants are not accountants. Conclusion: Some consultants are not auditors.

The consultants outside accountants are also outside auditors, because auditors sit inside accountants. The conclusion follows.

Must be true, could be true, cannot be true

Read the question stem carefully, because the same premises give different answers depending on what it asks.

• Must be true (or "follows"): true in every sketch that fits the premises.
• Could be true: true in at least one sketch. Most statements that do not contradict a premise could be true.
• Cannot be true: false in every sketch, usually because it directly contradicts a premise.

Worked example. All managers attended the training. Some people who attended the training work remotely.

• "Some managers work remotely" could be true but does not have to be. The remote group might overlap only the trainees who are not managers.
• "Some managers did not attend the training" cannot be true. It contradicts the first premise.
• "All managers attended the training" must be true. It is the first premise.

In a true, false or cannot say format, the first statement is "cannot say", the second is "false" and the third is "true". The "cannot say" answer is the one that candidates miss most often, because a statement that sounds likely feels true. Our deductive reasoning examples guide covers that wider format, with other kinds of argument besides syllogisms.

Common traps in syllogism questions

• Reversing "All". All A are B does not give All B are A. This is the single most common wrong answer.
• Two "some" premises. "Some A are B" and "Some B are C" never give a definite conclusion about A and C. The A that are B might not be the B that are C.
• Two negative premises. "No A are B" and "No B are C" tell you nothing definite about A and C. They might overlap or might not.
• The shared term never covers its whole group. "All A are B" and "All C are B" tell you nothing about A and C. Both sit inside B, but they can sit in different parts of it.
• Reading "some" as "only some". "Some reports are late" does not mean some reports are on time.
• Using real-world knowledge. If the premises say "All birds are swimmers", accept it. The question tests whether the conclusion follows, not whether it is true in real life. Equally, a conclusion that is true in real life but not supported by the premises does not follow.
• Mixing up "only". "Only A are B" means All B are A. Translate it before you draw.

How syllogisms appear in logical reasoning and critical thinking tests

Syllogisms show up in several formats in employer-style tests. The logic is the same; the answer format changes.

• Pick the conclusion that follows. Two premises and four or five conclusions, exactly one of which must be true. This is the classic syllogism test format, common in logical reasoning test practice.
• Follows or does not follow. A short passage, then several proposed conclusions, each to be judged on its own. This style appears in deduction sections of critical thinking test practice. The premises are usually written as everyday sentences, so translation matters more than diagrams.
• True, false or cannot say. The premises are stated as facts and you judge each conclusion. "Cannot say" means the conclusion could be true but does not have to be.
• Abstract letters or nonsense words. "All blips are frams" removes real-world knowledge on purpose, which makes the question purely about structure. Many candidates find these easier once they trust the method.

In every format, the answer that sounds most reasonable is not necessarily the one that follows. That gap is what these questions are designed to measure.

A speed method for syllogism questions

When the clock is running, use these rules to rule out options before you draw anything. They come from the standard rules of categorical logic:

1. Both premises "some"? Nothing definite follows about the outer terms.
2. Both premises negative? Nothing definite follows about the outer terms.
3. One premise negative? Any conclusion that follows must also be negative.
4. One premise "some"? Any conclusion that follows must also be a "some" statement.
5. Shared term only after "are" in two "All" premises? "All A are B" and "All C are B" give nothing definite about A and C.

If an option survives these checks, take 10 seconds to sketch it and try to break it. With practice, the sketch becomes a quick doodle of two or three circles, and most questions take well under a minute.