To test a syllogism, set aside whether the statements are true and ask one question: can both premises be true while the conclusion is false? If they can, the syllogism is invalid. "Valid" describes the reasoning and "true" describes the facts, which is why a valid syllogism can still end in a false conclusion.
What a syllogism is
A syllogism has two premises and a conclusion, and it links three terms. The form was first set out systematically by Aristotle in the Prior Analytics, and it still sits at the core of reasoning tests.
Take this example: "All pilots are trained in navigation. Everyone trained in navigation can read charts. So all pilots can read charts." The term that appears in both premises but not in the conclusion, "trained in navigation", is the middle term. It is the bridge, and most errors happen at the bridge.
The four statement types, read literally
Logic reads quantity words more strictly than everyday speech does. Learn these four readings and half of the traps disappear.
- All A are B. Every A is a B. It says nothing about whether every B is an A, so it does not flip around.
- No A are B. The two groups do not overlap, which also means no B is an A. This one does flip.
- Some A are B. At least one A is a B, and possibly all of them. It also flips: some B are A.
- Some A are not B. At least one A lies outside B. It does not say that some A are B.
The circle method
Draw one circle per term. "All A are B" puts the A circle inside the B circle. "No A are B" keeps the circles apart. "Some A are B" overlaps them, with a mark in the overlap.
Now draw the picture that is hardest on the conclusion, and check whether it still fits the premises. Try "All sparrows are birds. All crows are birds. So all sparrows are crows." Both small circles sit inside the bird circle, but nothing stops them from being apart, so the conclusion fails.
Four quick checks that catch most invalid syllogisms
- Two "some" premises give nothing. "Some doctors are parents. Some parents are tired" tells you nothing about doctors and tiredness.
- Two "no" premises give nothing. Knowing that two groups both avoid a third tells you nothing about how they relate to each other.
- Negatives travel together. If a premise is negative, the conclusion must be negative. If the conclusion is negative, one premise must be negative.
- The middle term must be covered. At least one premise has to speak about the whole of the middle term's group. In the sparrows and crows case, "birds" is never covered, so the bridge is missing.
One small catch: textbooks disagree on whether "All A are B" guarantees that A has any members. A "some" conclusion drawn from two "all" premises is therefore the one case to handle with care, and the wording of the test instructions should settle it.
A full check, start to finish
Try this one: "No reptiles are mammals. Some pets are mammals. So some pets are not reptiles."
Draw the reptile and mammal circles apart, as the first premise says. At least one pet sits inside the mammal circle, and that spot is outside the reptile circle. So at least one pet is not a reptile, and the conclusion follows.
The quick checks agree. There is one negative premise and the conclusion is negative. The middle term, "mammals", is covered, because the "no" premise speaks about every mammal.
Valid is not the same as true
Because validity and truth are independent, there are four combinations, and a reasoning test may use any of them.
- Valid, true premises, true conclusion. "All squares are rectangles. All rectangles have four sides. So all squares have four sides."
- Valid, one false premise, false conclusion. "All birds can fly. Penguins are birds. So penguins can fly." The reasoning is perfect and the first premise is wrong.
- Invalid, true premises, false conclusion. "All oaks are trees. All pines are trees. So all oaks are pines."
- Invalid, true premises, true conclusion. "All tigers are mammals. All lions are mammals. So no tiger is a lion." The conclusion happens to be right, but the premises do not force it.
The last case is the dangerous one, because a correct-looking conclusion rewards a wrong method.
Why believable conclusions fool us
Experiments on belief bias show that people accept a conclusion more readily when it fits what they already believe, whether or not it follows. Jonathan Evans, Julie Barston and Paul Pollard reported this conflict between logic and belief in 1983, and Stephen Newstead and colleagues later investigated where the effect comes from.
Researchers also keep comparing competing theories of how people solve these problems. A 2012 meta-analysis by Sangeet Khemlani and Philip Johnson-Laird set a range of those theories against published data.
The practical remedy is to swap meaningful words for nonsense ones. "All blorps are zinks. All zinks are quells. So all blorps are quells" is easy to judge because there is nothing to believe or disbelieve.
Where this fits
This guide is about checking validity, which is the heart of any deductive question. For how syllogisms look inside verbal tests and the vocabulary around them, see syllogisms in verbal reasoning tests. To avoid the typical slips, read common deductive reasoning mistakes.
If you want practice, the JobCannon Deductive Reasoning Test includes syllogisms alongside if-then rules and ranking puzzles, all framed as "follows / does not follow" conclusions. JobCannon's older logical reasoning test is a broader mix that adds analytical puzzles. Both sets of questions are written in-house, so treat results as practice feedback.