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: All men are mortal. Socrates is a man. Therefore, Socrates is mortal.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.
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 Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift (Concept Script; 1879).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.Hurley, Patrick J. 2011. A Concise Introduction to Logic. Cengage Learning. Zegarelli, Mark. 2010. Logic for Dummies. John Wiley & Sons. .
Early history
In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.Frede, Michael. 1975. "Stoic vs. Peripatetic Syllogistic." Archive for the History of Philosophy 56:99–124.
Aristotle
Aristotle defines the syllogism as Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions, including categorical modal syllogisms.Bobzien, Susanne. 2020. "Ancient Logic." Stanford Encyclopedia of Philosophy. § Aristotle.
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, Francis. 2001. The Great Instauration. – via Constitution Society. Archived from the original on 13 April 2019. 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. 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.Boole, George. 2003. The Laws of Thought, with an introduction by J. Corcoran. Buffalo: Prometheus Books.van Evra, James. 2004. "'The Laws of Thought' by George Boole" (review). Philosophy in Review 24:167–69. Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.Corcoran, John. 2003. "Aristotle's 'Prior Analytics' and Boole's 'Laws of Thought'." History and Philosophy of Logic 24:261–88. 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:
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, 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."
Types A syllogism consists of three parts: two premises and a conclusion. The interactions of these three parts vary depending on the type of syllogism. The three main types of syllogisms are categorical syllogisms, hypothetical syllogisms, and disjunctive syllogisms. A polysyllogism is an extended argument consisting of multiple, overlapping syllogisms in which the conclusion of one syllogism becomes a premise of a subsequent syllogism.
Categorical The most common type of syllogism is a categorical syllogism, and it consists of three parts: Major premise Minor premise 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 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: