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Syllogism

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"Epagoge" redirects here. For the genus of moth, see Epagoge (genus).

"Minor premise" redirects here. For the 2020 thriller film, see Minor Premise (film).

A syllogism (Greek: συλλογισμός, syllogismos, 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.

In its earliest form (defined by Aristotle in his 350 BCE book Prior Analytics), a syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across.[1] For example, knowing that all men are mortal (major premise) and that Socrates is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form.

"Socrates" at the Louvre

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. This article is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning in which facts are determined by repeated observations.

Within an academic context, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879). However, syllogisms remain useful in some circumstances, and for general-audience introductions to logic.

Basic structure

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A categorical syllogism consists of three parts:

  1. Major premise
  2. Minor premise
  3. Conclusion

Each part is a categorical proposition, and each categorical proposition contains two categorical terms. In Aristotle, each of the premises is in the form "All A are B," "Some A are B", "No A are B" or "Some A are not B", where "A" is one term and "B" is another:

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, this is the minor term (i.e., the subject of the conclusion). For example:

Major premise: All humans are mortal.
Minor premise: All Greeks are humans.
Conclusion: All Greeks are mortal.

Each of the three distinct terms represents a category. From the example above, humans, mortal, and Greeks: mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which is known as the middle term; in this example, humans. Both of the premises are universal, as is the conclusion.

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. Again, both premises are universal, hence so is the conclusion.

Types

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There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes the form (note: M – Middle, S – subject, P – predicate.):

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 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 affirmative All humans are mortal.
E No S are P universal negative No humans are perfect.
I Some S are P particular affirmative Some humans are healthy.
O Some S are not P particular negative Some humans are not clever.

In Prior Analytics, Aristotle uses mostly the letters A, B, and C (Greek letters alpha, beta, and gamma) as term place holders, rather than giving concrete examples. 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 so the categorical statements can be written succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic:

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

The convention here is that the letter S is the subject of the conclusion, P is the predicate of the conclusion, and 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 where it appears. The differing positions of the major, minor, and middle terms 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

(Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan—reject the fourth figure as a figure distinct from the first.)

Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though 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 syllogism BARBARA below is AAA-1, or "A-A-A in the first figure".

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. 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. All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from the premises.

Figure 1 Figure 2 Figure 3 Figure 4
Barbara Cesare Datisi Calemes
Celarent Camestres Disamis Dimatis
Darii Festino Ferison Fresison
Ferio Baroco Bocardo Calemos
Barbari Cesaro Felapton Fesapo
Celaront Camestros Darapti Bamalip
e

Fig. 1, treble clef. "A syllogism's letters can be best represented in music— take E, for example." -Marilyn Damord

The letters A, E, I, and 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.

Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is the middle term, the second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P).

The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii) could also be written as "Some kittens are pets" (MiS in Datisi).

In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate ("logical not") the result of that expression.

It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms.

Syllogistic fallacies

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See also: Syllogistic fallacy

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 things (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 in the structure of the syllogism invoked (i.e. III-1) the middle term is not distributed in either the major premise or in the minor premise, a pattern called the "fallacy of the undistributed middle". Because of this, it can be hard to follow formal logic, and a closer eye is needed in order to ensure that an argument is, in fact, valid.[2]

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:

References

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  1. John Stuart Mill, A System of Logic, Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence, and the Methods of Scientific Investigation, 3rd ed., vol. 1, chap. 2 (London: John W. Parker, 1851), 190.
  2. ^ Jump up to:a b Frede, Michael. 1975. "Stoic vs. Peripatetic Syllogistic." Archive for the History of Philosophy 56:99–124.
  3. ^ Hurley, Patrick J. 2011. A Concise Introduction to Logic. Cengage Learning. ISBN 9780840034175
  4. ^ Zegarelli, Mark. 2010. Logic for Dummies. John Wiley & Sons. ISBN 9781118053072.
  5. ^ Aristotle, Prior Analytics, 24b18–20
  6. ^ Bobzien, Susanne. [2006] 2020. "Ancient Logic." Stanford Encyclopedia of Philosophy. § Aristotle.
  7. ^
  8. ^ Jump up to:a b Bacon, Francis. [1620] 2001. The Great Instauration. – via Constitution Society. Archived from the original on 13 April 2019.
  9. ^ Boole, George. [1854] 2003. The Laws of Thought, with an introduction by J. Corcoran. Buffalo: Prometheus Books.
  10. ^ van Evra, James. 2004. "'The Laws of Thought' by George Boole" (review). Philosophy in Review 24:167–69.
  11. ^ Jump up to:a b Corcoran, John. 2003. "Aristotle's 'Prior Analytics' and Boole's 'Laws of Thought'." History and Philosophy of Logic 24:261–88.
  12. ^
  13. ^ 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; the first capitalized letter of each word is for universal, the second for particular'
  14. ^
  15. ^
  16. ^ See, e.g., Evans, J. St. B. T (1989). Bias in human reasoning. London: LEA.
  17. ^ Khemlani, S., and P. N. Johnson-Laird. 2012. "Theories of the syllogism: A meta-analysis." Psychological Bulletin 138:427–57.
  18. ^ Chater, N., and M. Oaksford. 1999. "The Probability Heuristics Model of Syllogistic Reasoning." Cognitive Psychology 38:191–258.

Sources

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  • Aristotle's Prior Analytics: the Theory of Categorical Syllogism an annotated bibliography on Aristotle's syllogistic
  • Fuzzy Syllogistic System
  • Development of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches
  • Comparison between the Aristotelian Syllogism and the Indian/Tibetan Syllogism
  • The Buddhist Philosophy of Universal Flux (Chapter XXIII – Members of a Syllogism (avayava))
  • Online Syllogistic Machine An interactive syllogistic machine for exploring all the fallacies, figures, terms, and modes of syllogisms.
  1. ^ Lundberg, Christian (2018). The Essential Guide to Rhetoric. Bedford/St.Martin's. p. 38.
  2. ^ Lundberg, Christian (2018). The Essential Guide to Rhetoric. Bedford/St. Martin's. p. 39.
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