Syllogism

A syllogism (συλλογισμός &mdash; "conclusion," "inference"), (usually the categorical syllogism) is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. In Aristotle's Prior Analytics, he 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." (24b18–20) Despite this very general definition, he limits himself first to categorical syllogisms (and later to modal syllogisms). The syllogism is at the core of deductive reasoning, where facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations.

Basic structure
A syllogism (henceforth categorical unless otherwise specified) consists of three parts: the major premise, the minor premise, and the conclusion. In Aristotle, each of the premises is in the form "Some/all A belong to B," where "Some/All A' is one term and "belong to B" is another, but more modern logicians allow some variation. Each of the premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate) of the conclusion; in a minor premise, it is the minor term (the subject) of the conclusion. For example:


 * Major premise: All humans are mortal.
 * Minor premise: Socrates is human.
 * Conclusion: Socrates is mortal.

Each of the three distinct terms represents a category, in this example, "human," "mortal," and "Socrates." "Mortal" is the major term; "Socrates," the minor term. The premises also have one term in common with each other, which is known as the middle term -- in this example, "human." Here the major premise is universal and the minor particular, but this need not be so. For example:


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

Here, the major term is "die," the minor term is "men," and the middle term is "[being] mortal things." Both of the premises are universal.

A sorites is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, if one argues that a given number of grains of sand does not make a heap and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a sorites argument.

Types of syllogism
Although there are infinitely many possible syllogisms, there are only a finite number of logically distinct types. We shall classify and enumerate them below. Note that 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 labelled by letters as follows.

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. The meaning of the letters is given by the table:

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

By definition, S is the subject of the conclusion, P is the predicate of the conclusion, M is the middle term, the major premise links M with P and the minor premise links M with S. However, the middle term 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:

Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, although this makes no difference logically). 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. Even some of these are sometimes considered to commit the existential fallacy, thus invalid. These controversial patterns are marked in italics.

The letters A, E, I, O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE etc.

A sample 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.
 * Some reading is homework.
 * Some reading is not fun.

Cesare
 * No healthful food is fattening.
 * All cakes are fattening.
 * No cakes are healthful.

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

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

'' Fesapo
 * No humans are perfect.
 * All perfect creatures are mythical.
 * Some mythical creatures are not human.''

Fresison
 * No competent people are people who always make mistakes.
 * Some people who always make mistakes are people who work here.
 * Some people who work here are not competent people.

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 19th century. Modifications were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Kant famously claimed that logic was the one completed science, and that Aristotle had more or less discovered everything about it there was to know. This opinion stood unchallenged until Frege invented first-order logic.

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. In the late 19th century, Charles Peirce's discovery of second-order logic revolutionized the field and the Aristotelian system has since been left to introductory material and historical study.

Everyday syllogistic mistakes
People often make mistakes when reasoning syllogistically.

For instance, given the following parameters: some A are B, some B are C, people tend to come to a definitive conclusion that therefore some A are C. However, this does not follow (for instance, while some cats (A) are black (B), and some black things (B) are televisions (C), it is false that some cats (A) are televisions (C)). This is because first, the mood of the syllogism invoked is illicit (III), and second, the supposition of the middle term is variable between that of the middle term in the major premise, and that of the middle term in the minor premise (not all "some" cats are by necessity of logic the same "some black things").

Determining the validity of a syllogism involves determining the distribution of each term in each statement, meaning whether all members of that term are accounted for.

In simple syllogistic patterns, the fallacies of invalid patterns are:


 * Undistributed middle - Neither of the premises accounts for all members of the middle term, which consequently fails to link the major and minor term.
 * Illicit treatment of the major term - The conclusion implicates all members of the major term; however, the major premise does not account for them all.
 * Illicit treatment of the minor term - Same as above, but for the minor term and minor premise.
 * Exclusive premises - Both premises are negative, meaning no link is established between the major and minor terms.
 * Affirmative conclusion from a negative premise - If either premise is negative, the conclusion must also be.
 * Existential fallacy - This is a more controversial one. If both premises are universal, i.e. "All" or "No" statements, they don't imply the existence of any members of the terms. In this case, the conclusion cannot be existential; i.e. beginning with "Some".