Venn diagrams test one syllogism at a time. Rules test by looking at distribution and quality, and they also name what went wrong. The first two rules are about distribution.
- Rule 1: the middle term must be distributed at least once. Breaking it: undistributed middle.
- Rule 2: a term distributed in the conclusion must be distributed in its premise. Breaking it: illicit major (for P) or illicit minor (for S).
Key idea
Mark the distributed terms (A: subject; E: both; I: neither; O: predicate). The middle term needs a mark in at least one premise, and every term marked in the conclusion needs a mark in its premise.
Why it works
Rule 1: if neither premise speaks of all of M, the premises may be about different parts of M, and then nothing links S to P. Rule 2: a conclusion that speaks of every P (or every S) claims more than a premise that spoke only of some of them. A conclusion can never say more about a class than the premises said.
Worked examples
Worked example: Undistributed middle
Which rule does this break? "All dogs are mammals. All cats are mammals. So all cats are dogs."
- Distributed terms: premise 1, dogs; premise 2, cats; conclusion, cats.
- The middle term mammals is distributed in neither premise.
- The dogs and the cats may be different parts of the mammals.
Answer: undistributed middle
Worked example: Illicit major
Which rule does this break? "All early risers are climbers. No singers are early risers. So no singers are climbers."
- The conclusion is E, so both singers and climbers are distributed in it.
- The major term climbers appears in the major premise as the predicate of an A statement: not distributed.
Answer: illicit major
Common mistake
Undistributed middle: "All dogs are animals; all cats are animals; so all cats are dogs." Both premises talk about only part of the animals. Check: underline the middle term each time it is distributed. If it has no underline, stop: the syllogism is invalid.