A form used in logical argument. A categorical syllogism consists of two premises which contain a common idea, one being universal. From these, a conclusion can be drawn.

One premise relates the major term (P) with the middle term (M). The other relates the minor term (S) and the middle term. The conclusion relates the minor and major terms. The syllogism may take any of the following forms:

```Major term:  M--P  P--M  M--P  P--M
Minor term:  S--M  S--M  M--S  M--S
Conclusion:  S--P  S--P  S--P  S--P
```
Note that the conclusion cannot contain the middle term. In addition, the middle term must be a universal at least once, and the major and minor terms may not be used as universals in the conclusion if they were particulars in the premises. At least one premise must be affirmative; if one is negative, the conclusion must be negative, and if both are affirmative the conclusion must be as well. In addition, at least one must be universal; the conclusion may be universal only if both premises are. If any of these rules are violates, a logical fallacy results.

Note that in a categorical syllogism each premise and the conclusion have to be in the form of a general categorical statement, of which there are four kinds (A, E, I, O).

Since each figure (each of the columns above is called a "figure") has two premises and a conclusion, each figure has 43 = 64 possible syllogistic forms. With four figures, that means 64 x 4 = 256 possible syllogistic forms.

By following certain syllogistic rules, we can rule out most of the syllogistic forms as invalid, resulting in 24 valid syllogistic forms (in the traditional interpretation -- in the modern interpretation, this is further pared down to 15). They are:

```First         Second        Third        Fourth
figure        Figure        Figure       Figure

AAA           AEE           AAI          AAI
AII           AOO           AII          AEE
EAE           EAE           EAO          EAO
EIO           EIO           EIO          EIO
AAI*          AEO*          IAI          IAI
EAO*          EAO*          OAO          AEO*
```

The forms with the asterisk are called weakened forms, since a stronger, more universal conclusion can be obtained. For example, the AAI form in the first figure:
All dogs are mammals
All poodles are dogs
therefore Some poodles are mammals
The conclusion, while valid (again, in the traditional interpretation -- it is invalid in the modern interpretation), can be strengthened to say "All poodles are mammals" (AAA).

Further, all valid forms can be reduced to forms in the first figure, by applying immediate inference rules. Because of the difficult task of remembering all these rules, medieval scholars devised a clever mnemonic device that started with Barbara Celarent... to remember all these.

It was much later that diagrams were used to represent these rules. Euler was the first to use diagrams. Lewis Carrol had his own diagrammatic representation, but it was the Venn Diagram that came into popular use.

Reference: Harlan Miller, course notes, Philo 5505

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