A syllogism must have exactly three terms. Real arguments often seem to have more: one premise says "students" and another "pupils", or one says "players" and another "non-players". Before calling it the fallacy of four terms, try to reduce the number of terms with tools you already have: synonyms, complements and obversion, and then put it in standard form.
Key idea
Replace synonyms with one word. When a term appears with and without "non-", rewrite a premise (by obversion, or by contraposition for A and O) so the "non-" disappears. If three terms remain, test the syllogism as usual.
Why it works
Rewriting is allowed because the new statement means exactly the same as the old one. Synonyms name the same class; obversion and valid contraposition give equivalent statements. Changing the words without changing the claims cannot change whether the argument is valid, so the reduced syllogism stands or falls with the original.
| Seems like | Rewrite | Why it is allowed |
|---|---|---|
| students / pupils | use "students" both times | synonyms name one class |
| No S are non-P | All S are P | obversion |
| All non-P are non-S | All S are P | contraposition (A) |
Worked examples
Worked example: Removing a complement
Reduce to three terms and test: "No bakers are non-anglers. Some sailors are not anglers. So some sailors are not bakers."
- Four apparent terms: bakers, non-anglers, sailors, anglers. But non-anglers is the complement of anglers.
- Obvert premise 1: change the quality and complement the predicate. "No bakers are non-anglers" becomes "All bakers are anglers".
- Now: All bakers are anglers / Some sailors are not anglers / ∴ Some sailors are not bakers. That is AOO-2 (Baroko).
Answer: three terms; valid
Worked example: Synonyms and contraposition
Reduce and test: "All non-runners are non-athletes. All pupils on the team are athletes. So all students on the team are runners."
- "Pupils on the team" and "students on the team" are synonyms: use one phrase.
- Contrapose premise 1 (an A statement): "All athletes are runners".
- Now: All athletes are runners / All students on the team are athletes / ∴ All students on the team are runners. That is AAA-1 (Barbara).
Answer: three terms; valid
Common mistake
Calling an argument invalid for four terms when two of its terms are synonyms or complements, or the reverse: treating two terms as the same when the meaning has shifted ("light" as in weight and "light" as in color). Check: do the two words name exactly the same class, or exact complements? Only then may you merge them.