Type of logical argument that applies deductive reasoning
A syllogism (, "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 BC book "Prior Analytics"), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. 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: 
 ;poem ;All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal. ;/poem ;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 predicted by repeated observations.
Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his "Begriffsschrift" ("Concept Script"; 1879). Syllogism, being a method of valid logical reasoning, will always be useful in most circumstances, and for general-audience introductions to logic and clear-thinking.
Early history.
In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.
Aristotle.
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." Despite this very general definition, in "Prior Analytics" Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.
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.
Medieval syllogism.
Boethius.
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.
Peter Abelard.
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.
Jean Buridan.
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.
Modern history.
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. 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. 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.
This led to the rapid development of sentential logic and first-order predicate logic, subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. The Aristotelian system is explicated in modern fora of academia primarily in introductory material and historical 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.
Boole's acceptance of Aristotle.
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". Corcoran also wrote a point-by-point comparison of "Prior Analytics" and "Laws of Thought". 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:
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."
Basic structure.
A categorical syllogism consists of three parts:
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 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:
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.
Polysyllogism.
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.
Types.
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 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:
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