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 or more propositions that are asserted or assumed to be true.

In a form, defined by Aristotle, from the combination of a general statement (the major premise) and a specific statement (the minor premise), a conclusion is deduced. 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:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.^{[1]} From the Middle Ages onwards, *categorical syllogism* and *syllogism* were usually used interchangeably. This article is concerned only with this traditional use. The syllogism was at the core of traditional deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning where 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Within an academic context, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his *Begriffsschrift* (*Concept Script*; 1879).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.^{[2]}^{[3]}

In antiqu

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.^{[1]} From the Middle Ages onwards, *categorical syllogism* and *syllogism* were usually used interchangeably. This article is concerned only with this traditional use. The syllogism was at the core of traditional deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations.

Within an academi

Within an academic context, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his *Begriffsschrift* (*Concept Script*; 1879).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.^{[2]}^{[3]}

In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.^{[1]}

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."^{[4]} Despite this very general definition, in *Prior Analytics*, Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.^{[5]}

The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Prior to 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 re-discovery, 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

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."^{[4]} Despite this very general definition, in *Prior Analytics*, Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.^{[5]}

The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle. Prior to the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as *Categories* and *Aristotle. Prior to 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 re-discovery, 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' commenta

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.

John 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.^{[6]}

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about how to get valid conclusion 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.^{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.[7] 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.[7] Yet, a full method to come to conclusions in nature is not the scope of logic or syllogism.
}

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.) Although there were alternative systems of logic elsewhere, such as Avicennian logic or Indian logic, 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, due to 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.^{[original research?]} 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.

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*.^{[8]}^{[9]} Corcoran also wrote a point-by-point comparison of *Prior Analytics* and *Laws of Thought*.^{[10]} 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:^{[10]}

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

More specifically,

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:

- Major premise
- Major premise
- Minor premise
- Conclusion
- "All A are B," and "No A are B" are termed 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. TEach 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

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 categorical syllogism consists of three parts:

Each part is a categorical proposition, and each categorical proposition contains two categorical terms.^{categorical proposition, and each categorical proposition contains two categorical terms.[11] 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:
}