Syllogism

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In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.^[3]

Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so."^[6] Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.^[7]

The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Before the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation, works that contributed heavily to the prevailing Old Logic, or logica vetus. The onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle developed his theory of the syllogism.

Prior Analytics, upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine", leaving very little for thinkers of the day to debate, and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to the concept over time. This theory of the syllogism would not enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid-14th century by the likes of John Buridan.

Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is, a premise containing the modal words necessarily, possibly, or contingently. Aristotle's terminology in this aspect of his theory was deemed vague, and in many cases unclear, even contradicting some of his statements from On Interpretation. His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, and would be replaced by new distinctions and new theories altogether.

Boethius (c. 475–526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on the categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions.

Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of the syllogism concept, and accompanying theory in the Dialectica—a discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model.

The French philosopher Jean Buridan (c. 1300 – 1361), whom some consider the foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica, in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that the primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous.^[8]

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions (axioms), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions.

In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature.^[9] Bacon proposed a more inductive approach to the observation of nature, which involves experimentation, and leads to discovering and building on axioms to create a more general conclusion.^[9] Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics.

In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables.

A noteworthy exception is the logic developed in Bernard Bolzano's work Wissenschaftslehre (Theory of Science, 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until the late 20th century, among other reasons, because of the intellectual environment at the time in Bohemia, which was then part of the Austrian Empire. In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study.

One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the Roman Rota, which still requires that any arguments crafted by Advocates be presented in syllogistic format.

George Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought.^[10][11] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.^[12] According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by:^[12]

1. providing it with mathematical foundations involving equations;
2. extending the class of problems it could treat, as solving equations was added to assessing validity; and
3. expanding the range of applications it could handle, such as expanding propositions of only two terms to those having arbitrarily many.

More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in the realm of foundations, Boole reduced Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle."

A categorical syllogism consists of three parts:

1. Major premise
2. Minor premise
3. Conclusion/Consequent

Each part is a categorical proposition, and each categorical proposition contains two categorical terms.^[13] In Aristotle, each of the premises is in the form "All S are P," "Some S are P", "No S are P" or "Some S are not P", where "S" is the subject-term and "P" is the predicate-term:

• "All S are P," and "No S are P" are termed universal propositions;
• "Some S are P" and "Some S are not P" are termed particular propositions.

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/Consequent: 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/Consequent: 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.

A polysyllogism, or 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, one might argue that all lions are big cats, all big cats are predators, and all predators are carnivores. To conclude that therefore all lions are carnivores is to construct a sorites argument.

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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/Consequent: All S are P.

The premises and conclusion of a syllogism can be any of four types, which are labeled by letters^[14] as follows. The meaning of the letters is given by the table:

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 are B	AaB	${\displaystyle \forall x.A(x)\rightarrow B(x)}$  or  ${\displaystyle \neg \exists x.A(x)\land \neg B(x)}$
No A are B	AeB	${\displaystyle \neg \exists x.A(x)\land B(x)}$  or  ${\displaystyle \forall x.A(x)\rightarrow \neg B(x)}$
Some A are B	AiB	${\displaystyle \exists x.A(x)\land B(x)}$
Some A are not B	AoB	${\displaystyle \exists x.A(x)\land \neg B(x)}$

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:

(Note, however, that following Aristotle's treatment of the figures, some logicians, such as Peter Abelard and Jean Buridan, do not regard the fourth figure as 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.

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.^[15]

M: men S: Greeks      P: mortal

   All men are mortal. (MaP)

   All Greeks are men. (SaM)

∴ All Greeks are mortal. (SaP)

M: reptile S: snake      P: fur

Similar: Cesare (EAE-2)

   No reptile has fur. (MeP)

   All snakes are reptiles. (SaM)

∴ No snake has fur. (SeP)

Camestres (AEE-2)
M: reptile S: fur      P: snake Camestres is essentially like Celarent with S and P exchanged. Similar: Calemes (AEE-4)    All snakes are reptiles. (PaM)    No fur bearing animal is a reptile. (SeM) ∴ No fur bearing animal is a snake. (SeP)

M: rabbit S: pet      P: fur

Similar: Datisi (AII-3)

   All rabbits have fur. (MaP)

   Some pets are rabbits. (SiM)

∴ Some pets have fur. (SiP)

Disamis (IAI-3)
M: rabbit S: fur      P: pet Disamis is essentially like Darii with S and P exchanged. Similar: Dimatis (IAI-4)    Some rabbits are pets. (MiP)    All rabbits have fur. (MaP) ∴ Some fur bearing animals are pets. (SiP)

M: homework S: reading      P: fun

Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)

   No homework is fun. (MeP)

   Some reading is homework. (SiM)

∴ Some reading is not fun. (SoP)

M: mammal S: pet      P: cat

   All cats are mammals. (PaM)

   Some pets are not mammals. (SoM)

∴ Some pets are not cats. (SoP)

M: cat S: mammal      P: pet

   Some cats are not pets. (MoP)

   All cats are mammals. (MaS)

∴ Some mammals are not pets. (SoP)

M: man S: Greek      P: mortal

   All men are mortal. (MaP)

   All Greeks are men. (SaM)

∴ Some Greeks are mortal. (SiP)

Bamalip (AAI-4)
M: man S: mortal      P: Greek Bamalip is exactly like Barbari with S and P exchanged:    All Greeks are men. (PaM)    All men are mortals. (MaS) ∴ Some (all) mortals are Greek. (SiP)

M: reptile S: snake      P: fur

Similar: Cesaro (EAO-2)

   No reptiles have fur. (MeP)

   All snakes are reptiles. (SaM)

∴ Some snakes have no fur. (SoP)

M: hooves S: human      P: horse

Similar: Calemos (AEO-4)

   All horses have hooves. (PaM)

   No humans have hooves. (SeM)

∴ Some humans are not horses. (SoP)

M: flower S: plant      P: animal

Similar: Fesapo (EAO-4)

   No flowers are animals. (MeP)

   All flowers are plants. (MaS)

∴ Some plants are not animals. (SoP)

M: square S: rhomb      P: rectangle

   All squares are rectangles. (MaP)

   All squares are rhombuses. (MaS)

∴ Some rhombuses are rectangles. (SiP)

This table shows all 24 valid syllogisms, represented by Venn diagrams. Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an existential assumption are dashed.

Table of all 24 valid syllogisms

	A ∧ A		A ∧ E				A ∧ I		A ∧ O		E ∧ I
1
2
3
4

People often make mistakes when reasoning syllogistically.^[16]

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.^[17][18] 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.^[19]

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.
• Negative conclusion from affirmative premises: If both premises are affirmative, the conclusion must also be.

• Disjunctive syllogism
• Hypothetical syllogism
• Legal syllogism
• Polysyllogism
• Prosleptic syllogism
• Quasi-syllogism
• Statistical syllogism

• Syllogistic fallacy
• Argumentation theory
• Buddhist logic
• Enthymeme
• Formal fallacy
• Logical fallacy
• The False Subtlety of the Four Syllogistic Figures
• Tautology (logic)
• Venn diagram
• Some decidable fragments of set theory are called syllogistics.