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A syllogism is a structured logical argument in which a conclusion is reached as a necessary result of two premises. Aristotle defined a syllogism as, "a discourse in which, certain things having been supposed, something different from the things supposed results of necessity because these things are so."


Basic structure

A properly constructed syllogism consists of a major premise, a minor premise, and a conclusion. The conclusion has a subject (S) and a predicate (P) which are derived from the premises. The major premise addresses the predicate, the minor premise addresses the subject and the two premises share a minor (or middle) term (M) which connects them. For example:

Major premise: All M are P.
Minor premise: All S are M.
Conclusion: All S are P.


While the subject-predicate order of the conclusion is fixed, the premises can be structured with the middle term as either the subject or the predicate. The ability to swap terms in the premises results in 4 possible syllogistic configurations known as "figures". The following table shows the 4 possible figures:

Figure 1 Figure 2 Figure 3 Figure 4
Major premise: M–P P–M M–P P–M
Minor premise: S–M S–M M–S M–S
Conclusion: S–P S–P S–P S–P


While figures define the structured order of the argument, type defines the logical style or nature of each premise and the conclusion. There are four possible types, each of which has traditionally been associated with a vowel:

Vowel Type Example
A universal affirmative "all A are B"
I particular affirmative "some A are B"
E universal negative "no A are B"
O particular negative "some A are not B"

The 4 type designations result in 256 possible types of syllogisms, though most of them are invalid. When the type designation is applied to the major premise, the minor premise and the conclusion, in that order, we arrive at a "shorthand" definition of the argument, like AAA or AEO. The figure number can be appended to this designation (IOE-3) to further define the argument.

As the type designators are vowels, this shorthand has been used to form form mnemonic names for each of the valid forms. The following table shows the names associated with valid arguments, by figure:

Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Darapti Bramantip
Celarent Camestres Disamis Camenes
Darii Festino Datisi Dimaris
Ferio Baroco Felapton Fesapo
    Bocardo Fresison


An example syllogism of each type follows. [1]


All men are mortal.
Socrates is a man.
Socrates is mortal.


No reptiles have fur.
All snakes are reptiles.
No snakes have fur.


All kittens are playful.
Some pets are kittens.
Some pets are playful.


No homework is fun.
Some reading is homework.
Some reading is not fun.


No healthy food is fattening.
All cakes are fattening.
No cakes are healthy.


All horses have hooves.
No humans have hooves.
No humans are horses.


No lazy people pass exams.
Some students pass exams.
Some students are not lazy.


All informative things are useful.
Some websites are not useful.
Some websites are not informative.


All fruit is nutritious.
All fruit is tasty.
Some tasty things are nutritious.


Some mugs are beautiful.
All mugs are useful.
Some useful things are beautiful.


All the industrious boys in this school have red hair.
Some of the industrious boys are boarders.
Some boarders in this school have red hair.


No jug in this cupboard is new.
All jugs in this cupboard are cracked.
Some of the cracked items in this cupboard are not new.


Some cats have no tails.
All cats are mammals.
Some mammals have no tails.


No tree is edible.
Some trees are green.
Some green things are not edible.


All apples in my garden are wholesome.
All wholesome fruit is ripe.
Some ripe fruit is in my garden.


All coloured flowers are scented.
No scented flowers are grown indoors.
No flowers grown indoors are coloured.


Some small birds live on honey.
All birds that live on honey are colourful.
Some colourful birds are small.


No humans are perfect.
All perfect creatures are mythical.
Some mythical creatures are not human.


No competent person is always blundering.
Some people who are always blundering work here.
Some people who work here are incompetent.
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