# Syllogism

Syllogism (Greek: συλλογισμός, meaning "conclusion" or "inference"), more correctly categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of certain forms. In his Prior Analytics, Aristotle defines 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." Despite this very general definition, he initially limits himself to categorical syllogisms (and later to modal syllogisms). The study of logical structures of syllogisms have been the main issue in the area of logic until the nineteenth century, when first-order logic emerged. Frege greatly shaped this new logic, which was the first radical transformation of logic since Aristotle. Aristotelian logic has since then been subsumed by modern logic.

## Basic structure

A syllogism consists of three parts—a major premise, a minor premise, and a conclusion. Each of the premises has one term in common with the conclusion. In the case of the major premise this is the major term, or the predicate of the conclusion; in the case of the minor premise it is the minor term, the subject of the conclusion. For example:

Major premise: All men are mortal.
Minor premise: Socrates is a man.
Conclusion: Socrates is mortal.

"Being mortal" is the major term and "Socrates" the minor term. The premises also have one term in common with each other, which is known as the middle term, in this case "being a man." Here the major premise is general and the minor particular, but this needn't be the case. For example:

Major premise: All mortal things die.
Minor premise: All men are mortal.
Conclusion: All men die.

Here, the major term is "die," the minor term is "all men," and the middle term is "being mortal."

## Types of syllogism

Syllogisms are categorized into logically distinct types. For instance, both of the syllogisms above share the same abstract form:

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

The premises and conclusion of a syllogism can be any of four types, which are labeled by letters[1] as follows. For instance, in the syllogisms above, only universal affirmatives (A) are used.

A. universal affirmatives ("All X are Y." e.g., "all humans are mortal")
I. particular affirmatives ("Some X are Y." e.g., "some humans are healthy")
E. universal negatives ("No X are Y." e.g., "no humans are perfect")
O. particular negatives ("Some X are not Y." e.g., "some humans are not clever")

(See Square of opposition for a discussion of the logical relationships between these types of propositions.)

Let us denote the subject of the conclusion as S, the predicate of the conclusion as P, the middle term as M. Then the major premise links M with P and the minor premise links M with S. However, M can be either the subject or the predicate of each premise that it appears in. This gives rise to another classification of syllogisms known as the figure. The four figures are:

 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

Putting it all together, there are 256 possible types of syllogism. Each premise and the conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogisms above are AAA-1.

Of course, the vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from the premises). The table below shows the valid forms of syllogism. The letters standing for the types of proposition (A, E, I, O) have been used since the Medieval Schools to form mnemonic names for the forms:

 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 Ferison

An example syllogism of each type follows.

Barbara

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

Celarent

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

Darii

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

Ferio

No homework is fun.

Cesare

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

Camestres

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

Festino

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

Baroco

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

Darapti

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

Disamis

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

Datisi

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

Felapton

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.

Bocardo

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

Ferison

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

Bramantip

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

Camenes

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

Dimaris

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

'Fesapo

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

Fresison

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

Forms can be converted to other forms, following certain rules, and all forms can be converted into one of the first-figure forms.

## The syllogism in the history of logic

Logic was dominated by syllogistic reasoning until the nineteenth century.[2] Modifications were incorporated to deal with disjunctive ("A or B") and conditional statements ("if A then B"). Kant infamously claimed that logic was the one completed science, and that Aristotle had more or less discovered everything about it there was to know.

Still, it was cumbersome and very limited in its ability to reveal the logical structure of complex sentences. For example, it was unable to express the claim that the real line is a dense order.[3] In the late ninteenth century, Frege's invention of first order logic revolutionized the field and the Aristotelian system has since been left to introductory materials and historical studies of logic.

## Notes

1. According to Copi, p. 127: 'The letter names are presumed to come from the Latin words "AffIrmo" and "nEgO," which mean "I affirm" and "I deny," respectively.
2. A prominent example is the Port-Royal Logic, a 1662 logic textbook by Antoine Arnauld and Pierre Nicole.
3. Michael Friedman emphasizes this in his Kant and the Exact Sciences (1992).

## ReferencesISBN links support NWE through referral fees

• Aristotle. Prior Analytics. transl. Robin Smith. Hackett, 1989. ISBN 0-87220-064-7
• Blackburn, Simon. Oxford Dictionary of Philosophy. Oxford University Press, 1996. ISBN 0-19-283134-8
• Broadie, Alexander. Introduction to Medieval Logic. Oxford University Press, 1993. ISBN 0-19-824026-0
• Copi, Irving M. Introduction to Logic. Third edition, Macmillan Company, 1969.