Syllogism

A syllogism, or logical appeal, (Greek: 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 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 [1] (and later to modal syllogisms). The syllogism was at the core of traditional deductive reasoning, where facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations. Syllogism was superseded by first-order predicate logic following the work of Frege, in particular 1879 Begriffsschrift (Concept Script) 1879.

Basic structure

A categorical syllogism consists of three parts: the major premise, the minor premise, and the conclusion, each part of which is a categorical proposition, and each categorical proposition containing two categorical terms [2]. In Aristotle, each of the premises is in the form "Some/all A belong to B," or "Some/all A is/are [not]B,"where "A" is one term and "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: Some animals are human.

Conclusion: Some animals are mortal.

Each of the three distinct terms represents a category, in this example, "human," "mortal," and "animal." "Mortal" is the major term; "animal," 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 mortals die.

Minor premise: All men are mortals.

Conclusion: All men die.

Here, the major term is "die", the minor term is "men," and the middle term is "mortals" 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 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.)

In Analytics, Aristotle mostly uses the letters A, B and C as term place holders, rather than giving concrete examples, an innovation at the time. It is traditional to use is rather than are as the copula, hence All A is B rather than All As are Bs It is traditional and convenient practice to use a,e,i,o as infix operators to enable the categorical statements to be written succinctly thus:

Form	Shorthand
All A is B	AaB
No A is B	AeB
Some A is B	AiB
Some A is not B	AoB

Hence the form BARABARA can be written neatly as BaC,AaB -> AaC

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. Given that in each case the conclusion is S-P, 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

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, meaning they are invalid if they mention an empty category. 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 animals are mortal.

All men are animals.

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.

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.

Some 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.

Terms in syllogism

We may, with Aristotle, distinguish singular terms such as Socrates and general terms such as Greeks. Aristotle further distinguished (a) terms which could be the subject of predication, and (b) terms which could be predicated of others by the use of the copula (is are). (Such a predication is known as a distributive as opposed to non-distributive as in Greeks are numerous.) It is clear that Aristotle’s syllogism works only for distributive predication for we cannot reason All Greeks are Animals, Animals are numerous, therefore All Greeks are numerous) In Aristotle’s view singular terms were of type (a) and general terms of type (b). Thus Men can be predicated of Socrates but Socrates cannot be predicated of anything. Therefore to enable a term to be interchangeable - that is to be to be either in the subject or predicate position of a proposition in a syllogism - the terms must be general terms, or categorical terms as they came to be called. Consequently the propositions of a syllogism should be categorical propositions (both terms general) and syllogism employing just categorical terms came to be called categorical syllogisms.

It is clear that nothing would prevent a singular term occurring in a syllogism - so long as it was always in the subject position - however such a syllogism, even if valid, would not be a categorical syllogism. An example of such would be Socrates is a man, All men are mortal, therefore Socrates is mortal. Intuitively this is as valid as All Greeks are men, all men are mortal therefore all Greeks are mortals. To argue that its validity can be explained by the theory of syllogism it would be necessary to show that Socrates is a man is the equivalent of a categorical proposition. It can be argued Socrates is a man is equivalent to All that are identical to Socrates are men, so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

Existential Import

If a statement includes a term so that the statement is false if the term has no instances (is not instantiated) then the statement is said to entail existential import with respect to that term. In particular, a universal statement of the form All A is B has existential import with respect to A if All A is B is false if there are no As.

The following problems arise (a) In natural language and normal use which statements of the forms All A is B, No A is B, Some A is B and Some A is not B have existential import and with respect to which terms (b) In the four forms of categorical statements used in syllogism which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms (c) What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition be valid (d) What existential imports must the forms AaB, AeB, AiB and AoB to preserve the validity of the traditionally valid forms of syllogisms (e) Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms Ahab, Abe, Ail and Alb

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further if it is accepted that AiB entail BiA, then AiB and AaB have existential import with respect to B as well. Similarly If AoB is false if there are no As, and AeB entail AoB, and AeB entails BeA (which in turn will entail BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

All flying horses are mythological is false if there are not flying horses If No men are fire eating rabbits is true, then there fire-eating dragon is false. and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC,AaB->AiC)

These problems and paradoxes arise in both natural language statements and statements form in syllogism because of ambiguity in particular ambiguity with respect to All. If Fred claims all his books were Nobel prize winners, is Fred claiming that he wrote any books? If not, then is what he says claims true? Suppose Jane says none of her friends are poor is that true if she has no friends. The first order predicate calculus avoids the problems of such ambiguity by using formulae which carry no existential import with respect to universal statements; existential claims have to be explicitly stated. Thus natural language statements of the forms All A is B, No A is B, Some A is B and Some A is not B can be exactly represented in first order predicate calculus in which any existential import with respect to terms A or B are both is made explicitly or not made at all. Consequently the four must forms AaB, AeB, AiB and AoB can be represented in first order predicate in every combination of existential import, so that it can be stabilised which construal if any would preserve the square of opposition and the validly of the traditionally valid syllogism. Strawson claims that such a construal is possible, but the results are such that, in his view, b the answer to question (a) above is no

Further reading: http://uk.geocities.com/frege@btinternet.com/cantor/Eximport.htm

The syllogism in the history of logic

Main article: 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 1879 when Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements - and statements which are not provided for in syllogism as well - by the use of quantifiers and variables. This led to the rapid development of sentential logic and first-order predicate logic subsuming syllogistic reasoning which was, therefore, after 2000 years, suddenly obsolete. 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, from the premises 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 according to the rules of classical logic. 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.

See also

• Venn diagram
• Syllogistic fallacy
• The False Subtlety of the Four Syllogistic Figures
• Enthymeme
• Forms of syllogism:
  • Disjunctive syllogism
  • Hypothetical syllogism
  • Polysyllogism
  • Prosleptic syllogism
  • Quasi-syllogism
  • Statistical 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.

External links

• Abbreviatio Montana article by Prof. R. J. Kilcullen of Macquarie University on the medieval classification of syllogisms.
• The Figures of the Syllogism is a brief table listing the forms of the syllogism.
• Syllogism Validator is a simple validator of basic syllogisms (with existential presupposition).
• Stanford Encyclopedia of Philosophy entry on Medieval Theories of Syllogisms