Definitions

# syllogism

[sil-uh-jiz-uhm]
syllogism, a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years. Every syllogism is a sequence of three propositions such that the first two imply the third, the conclusion. There are three basic types of syllogism: hypothetical, disjunctive, and categorical. The hypothetical syllogism, modus ponens, has as its first premise a conditional hypothesis: If p then q; it continues: p, therefore q. The disjunctive syllogism, modus tollens, has as its first premise a statement of alternatives: Either p or q; it continues: not q, therefore p. The categorical syllogism comprises three categorical propositions, which must be statements of the form all x are y, no x is y, some x is y, or some x is not y. A categorical syllogism contains precisely three terms: the major term, which is the predicate of the conclusion; the minor term, the subject of the conclusion; and the middle term, which appears in both premises but not in the conclusion. Thus: All philosophers are men (middle term); all men are mortal; therefore, All philosophers (minor term) are mortal (major term). The premises containing the major and minor terms are named the major and minor premises, respectively. Aristotle noted five basic rules governing the validity of categorical syllogisms: The middle term must be distributed at least once (a term is said to be distributed when it refers to all members of the denoted class, as in all x are y and no x is y); a term distributed in the conclusion must be distributed in the premise in which it occurs; two negative premises imply no valid conclusion; if one premise is negative, then the conclusion must be negative; and two affirmatives imply an affirmative. John Venn, an English logician, in 1880 introduced a device for analyzing categorical syllogisms, known as the Venn diagram. Three overlapping circles are drawn to represent the classes denoted by the three terms. Universal propositions (all x are y, no x is y) are indicated by shading the sections of the circles representing the excluded classes. Particular propositions (some x is y, some x is not y) are indicated by placing some mark, usually an "X," in the section of the circle representing the class whose members are specified. The conclusion may then be read directly from the diagram.

Form of argument that, in its most commonly discussed instances, has two categorical propositions as premises and one categorical proposition as conclusion. An example of a syllogism is the following argument: Every human is mortal (every M is P); every philosopher is human (every S is M); therefore, every philosopher is mortal (every S is P). Such arguments have exactly three terms (human, philosopher, mortal). Here, the argument is composed of three categorical (as opposed to hypothetical) propositions, it is therefore a categorical syllogism. In a categorical syllogism, the term that occurs in both premises but not in the conclusion (human) is the middle term; the predicate term in the conclusion is called the major term, the subject the minor term. The pattern in which the terms S, M, and P (minor, middle, major) are arranged is called the figure of the syllogism. In this example, the syllogism is in the first figure, since the major term appears as predicate in the first premise and the minor term as subject of the second.

A syllogism, or logical appeal, (συλλογισμός — "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 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 a 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:

 code quantifier subject copula predicate type example A All S are P universal affirmatives All humans are mortal. E No S are P universal negatives No humans are perfect. I Some S are P particular affirmatives Some humans are healthy. O Some S are not P particular negatives Some humans are not clever.

(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:

 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 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.

 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

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 animals.
All animals are mortal.
All men are 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 are apples 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

Syllogism dominated Western philosophical thought until The Age of Enlightenment in the 17th Century. At that time, Sir Francis Bacon rejected the idea of syllogism and deductive reasoning by asserting that it was fallible and illogical. Bacon offered a more inductive approach to logic in which experiments were conducted and axioms were drawn from the observations discovered in them.

In the 19th Century, modifications to syllogism 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 Aristotelian logic more or less included everything about logic there was to know. Though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kant's opinion stood unchallenged in the West 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 does not follow from the parameters 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 (P - meaning the proposition is negative); however, the major premise does not account for them all (i e P is either an affirmative predicate or a particular subject there).
Illicit treatment of the minor term - Same as above, but for the minor term (S - meaning the proposition is universal) and minor premise (where S is either a particular subject or an affirmative predicate).
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, one school of thought says they do not imply the existence of any members of the terms. In this case, the conclusion cannot be existential; i.e. beginning with "Some". Another school of thought says that affirmative statements (universal or particular) do imply the subject's existence, but negatives do not. A third school of thought says that the any type of proposition may or may not involve the subject's existence, and although this may condition the conclusion it does not affect the form of the syllogism.

## References

• Aristotle, Prior Analytics. transl. Robin Smith (Hackett, 1989) ISBN 0-87220-064-7.
• Blackburn, Simon, 1996. "Syllogism" in the Oxford Dictionary of Philosophy. Oxford University Press. ISBN 0-19-283134-8.
• Broadie, Alexander, 1993. Introduction to Medieval Logic. Oxford University Press. ISBN 0-19-824026-0.
• Irving Copi, 1969. Introduction to Logic, 3rd ed. Macmillan Company.
• Hamblin, Charles L., 1970. Fallacies, Methuen : London, ISBN 0-416-70070-5. Cf. on validity of syllogisms: "A simple set of rules of validity was finally produced in the later Middle Ages, based on the concept of Distribution.“
• Jan Łukasiewicz, 1987 (1957). Aristotle's Syllogistic from the Standpoint of Modern Formal Logic. New York: Garland Publishers. ISBN 0824069242. OCLC 15015545.