The distribution rules are two of six. The next three are about quality, that is, about the negative statements (E and O).
- Rule 3: two negative premises are not allowed. Breaking it: exclusive premises.
- Rule 4: a negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. Breaking it: affirmative conclusion from a negative premise, or negative conclusion from affirmative premises.
Key idea
Count the negatives. Zero negative premises: the conclusion must be affirmative. Exactly one: the conclusion must be negative. Two: the syllogism is invalid.
Why it works
A negative premise says that M excludes something. If both premises only say what M excludes, they never say what M includes, so S and P are never brought together and nothing follows about them. With one negative premise, one of S or P is linked to M and the other is shut out of M, so the only link that can follow is an exclusion: a negative conclusion. And affirmative premises only include classes in one another; they cannot produce an exclusion.
Worked examples
Worked example: Exclusive premises
Is this valid? "No fish are mammals. No whales are fish. So no whales are mammals."
- Both premises are negative (E, E).
- They say only what the fish exclude, so they never connect whales and mammals.
- Here the premises are true and the conclusion false, which no valid form allows.
Answer: invalid: exclusive premises
Worked example: Affirmative from a negative
Which quality rule does this break? "No poets are hikers. All hikers are skaters. So some skaters are poets."
- One negative premise (E).
- The conclusion (I) is affirmative.
Answer: affirmative conclusion from a negative premise
Common mistake
Thinking two negative premises can still link the terms ("No cats are dogs; no dogs are birds; so no cats are birds" sounds right). If neither premise says what the middle term includes, the terms never meet. Check: count the E and O premises first; two means stop.