A syllogism (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 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." (24b18–20) Despite this very general definition, however, 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 experimenting on the world. In logic, an argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. ... Proposition is a term used in logic to describe the content of assertions. ... Inference is the act or process of deriving a conclusion based solely on what one already knows. ... In discourse, a premise (also premiss in British usage) is a claim which is part of a reason or objection. ... Prior Analytics is Aristotles work on deductive reasoning, part of his Organon, the organ of logical and scientific methods. ... Aristotle (Greek: AristotélÄ“s) (384 BC – March 7, 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... In philosophical logic, a modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ... Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). ... Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ...

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 the case of the major premise this is the major term, i.e., the predicate of the conclusion; in the case of the minor premise it is the minor term, the subject of the conclusion. For example: The major premise in a categorical syllogism is the premise whose terms are the syllogisms major term and middle term. ... In a categorical syllogism, the minor premise is the premise whose terms are the syllogisms minor term and middle term. ... In linguistics and logic, a predicate is an expression that can be true of something. ...

Major premise: All men are mortal.

Minor premise: Some philosophers are men.

Conclusion: Some philosophers are mortal.

"Being mortal" is the major term and "philosophers" the minor term. The premises also have one term in common with each other, which is known as the middle term, in this case "are men". 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 things.

Conclusion: All men die.

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

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. Sorites may refer to: Paradox of the heap Polysyllogism ...

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 both 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^[1] 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: Scholasticism comes from the Latin word scholasticus, which means that [which] belongs to the school, and is the school of philosophy taught by the academics (or schoolmen) of medieval universities circa 1100–1500. ... Wikiquote has a collection of quotations related to: English mnemonics A mnemonic (pronounced in Received Pronunciation) is a memory aid, and most serve an educational purpose. ...

(See Square of opposition for a discussion of the logical relationships between these types of propositions.) The Square of Opposition is a term from the study of Aristotelian logic or Term Logic in which the logical relationship between various types of sentences is spelled out. ...

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 existential fallacy is a logical fallacy committed in a categorical syllogism that is invalid because it has two universal premises and a particular conclusion. ...

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. Scholasticism comes from the Latin word scholasticus, which means that [which] belongs to the school, and is the school of philosophy taught by the academics (or schoolmen) of medieval universities circa 1100–1500. ... Wikiquote has a collection of quotations related to: English mnemonics A mnemonic (pronounced in Received Pronunciation) is a memory aid, and most serve an educational purpose. ...

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 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

Main article: History of Logic

Logic was dominated by syllogistic reasoning until the 19th century^[2]. 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. The history of logic documents the development of logic as it occurs in various rival cultures and traditions in history. ... Immanuel Kant (22 April 1724 – 12 February 1804), was a German philosopher from Königsberg in East Prussia (now Kaliningrad, Russia). ... Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

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 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. In mathematics, the real line is simply the set of real numbers. ... In mathematics, a partial order ≤ on a set X is said to be dense (or dense-in-itself) if, for all x and y in X for which x < y, there is a z in X such that x < z < y. ... Charles Sanders Peirce (IPA: /pɝs/), (September 10, 1839 – April 19, 1914) was an American polymath, physicist, and philosopher, born in Cambridge, Massachusetts. ... In mathematical logic, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. ...

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. Distribution of terms is a theory in logic that says that, in a categorical syllogism, any term distributed in the conclusion must be distributed in either premise. ...

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".

The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism isnt distributed. ... This article may be too technical for most readers to understand. ... Illicit minor is a logical fallacy committed in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion. ... The fallacy of exclusive premises is a formal fallacy committed in a categorical syllogism that is invalid because both of its premises are negative. ... Affirmative conclusion from a negative premise is a logical fallacy that is committed when a categorical syllogism has a positive conclusion, but one or two negative premises. ... The existential fallacy is a logical fallacy committed in a categorical syllogism that is invalid because it has two universal premises and a particular conclusion. ...

References

• 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).

Aristotle (Greek: Aristotélēs) (384 BC – March 7, 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Prior Analytics is Aristotles work on deductive reasoning, part of his Organon, the organ of logical and scientific methods. ... Robin Smith is also the name of a cricketer who played for England. ... Simon Blackburn (born 1944) is a British academic philosopher also known for his efforts to popularise philosophy. ...

See also

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

A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ... Syllogistic fallacies are logical fallacies that occur in syllogisms. ... The False Subtlety of the Four Syllogistic Figures Proved (Die falsche Spitzfindigkeit der vier syllogistischen Figuren erwiesen) was an essay published by Immanuel Kant in 1762. ... A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ... In logic, a hypothetical syllogism has two uses. ... A polysyllogism, sometimes called multi-premise syllogism, is a string of any number of syllogisms such that the conclusion of one is a premise for the next, and so on. ... Quasi-syllogism is a term that is sometimes used to describe what might be otherwise called a categorical syllogism but where one of the premises is singular, and thus not a categorical statement. ... A statistical syllogism is an inductive syllogism. ... Please wikify (format) this article or section as suggested in the Guide to layout and the Manual of Style. ...

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.
• Stanford Encyclopedia of Philosophy entry on Medieval Theories of Syllogisms

References