A syllogism has two premises and three terms. The test is the one for a single premise: enumerate the patterns of empty and occupied regions the premises allow, and check the conclusion in each. Barbara, Celarent, Darii and Ferio pass; the undistributed middle and the illicit major do not.
Core ideas
Three circles make eight regions and two premises constrain several of them. What remains open is the whole question.
The conclusion joins the minor term S to the major term P. Each premise joins one of them to the middle term M, which does not appear in the conclusion. The premise with P is the major premise, and the one with S the minor.
The mood is the three letters of the premises and conclusion, such as AAA. The figure, one to four, says where M sits in the premises. Together they fix the form, and the valid forms carry traditional names; the four from the first figure are the ones learnt here, and the lab knows the rest.
A syllogism is valid when no allowed pattern makes the conclusion false. Whether the premises or the conclusion are true in the world plays no part, and the same holds for a one-premise inference.
The material
The test of the last two lessons, applied to three terms, with the names that go with the forms that pass.
A categorical syllogism has two premises and a conclusion, each a categorical statement, over three terms. The minor term S is the conclusion's subject, the major term P its predicate, and the middle term M occurs in both premises and not in the conclusion.
Three circles divide the plane into eight regions, so a pattern of empty and occupied regions is one of 256. The lab considers them all, keeps those in which both premises hold, and tests the conclusion in each. The count it reports is the number of patterns the premises allow, and the argument is valid when the conclusion is true in every one.
All M are P, All S are M, so All S are P. The first premise empties M outside P. The second empties S outside M. A thing of S outside P would have to be either inside M, which the first premise forbids, or outside it, which the second forbids. The region is empty in every allowed pattern, so the conclusion holds.
Barbara (AAA), Celarent (EAE), Darii (AII) and Ferio (EIO) are valid. Celarent is No M are P, All S are M, so No S are P. In Darii and Ferio the minor premise, Some S are M, occupies a region inside S and M. All M are P empties its part outside P, so the occupied region is inside P; No M are P empties its part inside P, so it is outside.
Two forms look like these and fail. In the second figure, All P are M and All S are M, so All S are P (AAA-2) is the undistributed middle. Both premises say only that a circle sits inside M. Nothing relates the two circles, and S may lie in M outside P. The lab draws that pattern.
The second failure is the illicit major: All M are P, No S are M, so No S are P. The conclusion makes a claim about every P, and the major premise gives no information about P outside M. The pattern with S and P overlapping outside M satisfies both premises and fails the conclusion.
The traditional rules, such as that the middle term must be distributed at least once, are summaries of these region facts. They are quick when they apply and opaque when they do not, while the patterns are always available. Where a rule and the lab seem to disagree, the lab has the definition.
Truth in the world does not decide any of this. All cats are mammals, all dogs are mammals, so all dogs are cats has true premises and a false conclusion, which is the signature of an invalid form. But No cats are fish, No dogs are fish, so no dogs are cats has true premises and a true conclusion and is invalid too: its form allows cats and dogs to overlap outside fish. A true conclusion is a coincidence of subject matter and is no evidence of form.
One limit, stated once. The lab takes at most three terms and defaults to the Boolean reading, on which the weakened moods that draw a particular conclusion from two universal premises are invalid. Switching to the Aristotelian reading restores them.
Interactive
Every pattern of empty and occupied regions
Method
Five steps. A diagram helps; the patterns decide.
The conclusion's subject is S and its predicate P. The remaining term, in both premises, is M.
Each universal premise empties a region of the three-circle diagram; each particular premise requires one occupied.
Mark the regions emptied. Note where a particular premise has more than one region it could use.
Ask whether each pattern the premises allow makes the conclusion true. One failure is the counterexample.
Write mood and figure. If it is one of Barbara, Celarent, Darii or Ferio, or the undistributed middle, you have a name; the verdict comes from the patterns.
“All cats are mammals. All dogs are mammals. So all dogs are cats.” Let S be the dogs, P the cats and M the mammals. The major premise is the one with the cats.
Both premises are satisfied by a pattern in which the only occupied region is the dogs that are mammals but not cats. In it the conclusion fails. Nothing about cats or dogs was used beyond the shape: the same pattern refutes the form whatever the three terms name, which is why the verdict can be read off before anyone asks whether the sentences are true.
Given a three-term syllogism, identify the major, minor and middle terms, state mood and figure, say whether the premises force the conclusion, and when they do not describe the counterexample pattern in plain words.
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