Categorical syllogism
from 350 BCE Aristotle's Prior Analytics; dominant until the 1870s
Two claims about what belongs to what, sharing a middle term, yielding a third claim in which that middle term no longer appears.
- Active
- from 350 BCE
- Where
- Athens
- Also
- syllogismos
- Status
- In use
- Tradition
- Forms of inference
What it is
A categorical syllogism has exactly three terms and three statements. Two premises relate a major and a minor term to a shared middle term; the conclusion relates major to minor directly, and the middle drops out.
Aristotle worked out which arrangements are valid and which are not, and the answer is that most are not. Of the 256 possible forms, twenty-four are valid, and knowing which is the whole of traditional term logic.
How it works
Identify the three terms. Check the quantity of each statement — all, some, none — and its quality, affirmative or negative. Check the arrangement against the valid forms.
Two rules catch most errors. The middle term must be distributed — taken universally — in at least one premise, or the two premises may be talking about different parts of it. And no term may be distributed in the conclusion unless it was distributed in a premise, or the conclusion claims more than was granted.
The form of the inference
- All M are P
- All S are M
- Therefore all S are P
The argument
Proposition This batch of meat has been inspected.
- Every carcass processed at a licensed abattoir is inspected before release.
- The meat in this consignment came from carcasses processed at a licensed abattoir.
- So the carcasses this meat came from are among those processed at a licensed abattoir.
- The middle term — carcasses processed at a licensed abattoir — appears in both premises and in neither the subject nor the predicate of what follows.
Therefore The meat in this consignment has been inspected.
This is the pattern Aristotle called Barbara, the first and clearest of the valid forms. The whole apparatus of quantity and distribution exists to tell this from the arrangements that look identical and are invalid.
What it is good for
Reasoning about classes and membership, which covers a great deal of law, regulation and classification. It is also the best training there is for noticing when a conclusion has quietly widened beyond its premises.
Where it fails
It cannot handle relations. *All horses are animals, therefore all horses' heads are animals' heads* is a valid inference that no categorical syllogism can represent, because the logic has no way to express a relation between two things. Aristotle knew of the problem and it defeated the tradition for two thousand years.
It also cannot manage multiple generality — statements like *every teacher recommends some book to every student* — which is precisely the kind of claim mathematics is full of. This is why term logic could not serve as a foundation for mathematics, and why Frege had to start again.
Common misconceptions
That it is what logic is. It was, for two millennia, and it is now understood as a small fragment of first-order predicate logic.
Also that valid means true. A syllogism can be impeccably valid with premises that are nonsense, and it will deliver nonsense with complete formal propriety.
Where it came from
Aristotle's Prior Analytics, which is the first formal system in the history of thought and remained essentially unimproved for longer than any other intellectual achievement on record.
What became of it
Superseded in the 1870s and 1880s when Frege and Peirce independently developed quantified predicate logic, which subsumes the syllogism as a special case and handles everything it could not.
Lasting influence
It gave the West its idea of what rigour looks like. The medieval curriculum was built on it, and the mnemonic names for its valid forms — Barbara, Celarent, Darii, Ferio — were memorised by every educated person for a thousand years.
Why it still matters
Historically enormous, practically now modest. It remains a good teaching instrument and a poor working tool.
Put another way
A well-made hand plane in a workshop full of power tools. Still cuts true, still limited to what it was shaped for.
Texts
Aristotle, Prior Analytics
Book I sets out the whole system. Dense, but the achievement is visible on the page.
Where to start
Learn Barbara and one invalid form that resembles it. The contrast teaches more than the full table of twenty-four.
The surprising bit
Kant declared in 1787 that logic had been unable to advance a single step since Aristotle and was therefore complete. Within a century it had been overtaken entirely.
In their own words
A syllogism is a discourse in which, certain things being stated, something other than what is stated follows of necessity
Who held it
the medieval logicians
Connections
- Influenced
- Formalisation
- Parallel to
- Hypothetical syllogism
The verdict
The most durable formal system ever built, and finally too narrow for the job.