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{{Short description|Type of logical argument that applies deductive reasoning}}
{{Redirect|Epagoge|the genus of moth|Epagoge (moth)}}
{{Redirect|Minor premise|the 2020 thriller film|Minor Premise (film){{!}}''Minor Premise'' (film)}}
{{Multiple issues|{{More citations needed|date=February 2023}}{{Essay-like|date=February 2023}}}}

A '''syllogism''' ({{langx|grc|συλλογισμός}}, ''syllogismos'', 'conclusion, inference') is a kind of [[Argument|logical argument]] that applies [[deductive reasoning]] to arrive at a [[Logical consequence|conclusion]] based on two [[proposition]]s that are asserted or assumed to be true.

[[File:Socrates_Louvre.jpg|thumb|"Socrates" at the Louvre]]
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.<ref>{{Cite book|last=Lundberg|first=Christian|title=The Essential Guide to Rhetoric|publisher=Bedford/St.Martin's|year=2018|pages=38}}</ref> 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: 
	
<blockquote><poem>All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.<ref>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.</ref></poem></blockquote>In antiquity, two rival syllogistic theories existed: [[Term logic|Aristotelian syllogism]] and [[Stoic logic|Stoic syllogism]].<ref name=":0" /> 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Within some academic contexts, syllogism has been superseded by [[First-order logic|first-order predicate logic]] following the work of [[Gottlob Frege]], in particular his ''[[Begriffsschrift]]'' (''Concept Script'';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.<ref>Hurley, Patrick J. 2011. ''A Concise Introduction to Logic''. Cengage Learning. {{ISBN|9780840034175}}</ref><ref>Zegarelli, Mark. 2010. ''Logic for Dummies''. John Wiley & Sons. {{ISBN|9781118053072}}.</ref>

==Early history==

{{Main|History of logic}}
In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism.<ref name=":0">[[Michael Frede|Frede, Michael]]. 1975. "Stoic vs. Peripatetic Syllogistic." ''Archive for the History of Philosophy'' 56:99–124.</ref>

===Aristotle===
{{main|Term logic}}
[[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."<ref>[[Aristotle]], ''[[Prior Analytics]]'', 24b18–20</ref> Despite this very general definition, in ''[[Prior Analytics]]'' Aristotle limits himself to categorical syllogisms that consist of three [[categorical proposition]]s, including categorical [[modal logic|modal]] syllogisms.<ref>[[Susanne Bobzien|Bobzien, Susanne]]. [2006] 2020. "[http://plato.stanford.edu/entries/logic-ancient/ Ancient Logic]." ''[[Stanford Encyclopedia of Philosophy]]''. § [https://plato.stanford.edu/entries/logic-ancient/#Ari Aristotle].</ref>

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 (Aristotle)|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 [[Modal logic|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&nbsp;– 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.<ref>{{cite encyclopedia|last=Lagerlund|first=Henrik|title=Medieval Theories of the Syllogism|url=http://plato.stanford.edu/archives/win2012/entries/medieval-syllogism/|encyclopedia=The Stanford Encyclopedia of Philosophy|date=2 February 2004 |publisher=Edward N. Zalta|access-date=17 February 2014}}</ref>

==Modern history==

The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself is about drawing valid conclusions from assumptions ([[axiom]]s), 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.<ref name="instauration">[[Francis Bacon|Bacon, Francis]]. [1620] 2001. ''[https://web.archive.org/web/20190413000330/http://www.constitution.org/bacon/instauration.htm The Great Instauration]''. – via ''Constitution Society''. Archived from the [http://www.constitution.org/bacon/instauration.htm original] on 13 April 2019.</ref> 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.<ref name="instauration" /> 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 syllogism|disjunctive]] ("A or B") and [[Conditional syllogism|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 ''[[Bernard Bolzano#Wissenschaftslehre (Theory of Science)|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.{{original research inline|date=January 2013}} 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.
 <!--
What follows is at best *very* obscure and desultory. I'll have a go at rewriting it when I can.
You can simply put list of rules for validity&nbsp;— as in Russian wp -->

===Boole's acceptance of Aristotle===
[[George Boole]]'s unwavering acceptance of Aristotle's logic is emphasized by the historian of logic [[John Corcoran (logician)|John Corcoran]] in an accessible introduction to ''[[Laws of Thought]]''.<ref>[[George Boole|Boole, George]]. [1854] 2003. ''[[The Laws of Thought]]'', with an introduction by J. Corcoran. Buffalo: [[Prometheus Books]].</ref><ref>van Evra, James. 2004. "'The Laws of Thought' by George Boole" (review). ''[[Philosophy in Review]]'' 24:167–69.</ref> Corcoran also wrote a point-by-point comparison of ''[[Prior Analytics]]'' and ''[[The Laws of Thought|Laws of Thought]]''.<ref name=":1">[[John Corcoran (logician)|Corcoran, John]]. 2003. "Aristotle's 'Prior Analytics' and Boole's 'Laws of Thought'." ''History and Philosophy of Logic'' 24:261–88.</ref> 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:<ref name=":1" />

# providing it with mathematical foundations involving equations;
# extending the class of problems it could treat, as solving equations was added to assessing [[Validity (logic)|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."

==Basic structure==
A categorical syllogism consists of three parts:
# Major premise
# Minor premise
# Conclusion/Consequent

Each part is a [[categorical proposition]], and each categorical proposition contains two categorical terms.<ref>{{cite web|url=http://www.philosophypages.com/dy/c.htm#capro |title=Philosophical Dictionary: Caird-Catharsis |publisher=Philosophypages.com |date=2002-08-08 |access-date=2009-12-14}}</ref> 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 proposition|''universal'' propositions]];
* "Some S are P" and "Some S are not P" are termed [[Particular proposition|''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 (grammar)|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 ===
{{Main|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<!--linked from 'Term logic'-->==
{{Original research section|date=July 2020}}
[[File:Square of opposition, set diagrams.svg|thumb|Relationships between the four types of propositions in the [[square of opposition]]<br /><br />(Black areas are empty,<br />red areas are nonempty.)]]
{{Further|List of valid argument forms}}

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&nbsp;– Middle, S&nbsp;– subject, P&nbsp;– 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<ref>According to [[Copi]], p. 127: 'The letter names are presumed to come from the Latin words "'''''A'''''ff'''''I'''''rmo" and "n'''''E'''''g'''''O'''''," which mean "I affirm" and "I deny," respectively; the first capitalized letter of each word is for universal, the second for particular'</ref> as follows. The meaning of the letters is given by the table:

{| class="wikitable"
|-
! 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 old.
|}

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 (linguistics)|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 notation|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:

{| class="wikitable"
|-
! Form
! Shorthand
! Predicate logic
|-
| All A is B
| AaB
| <math>\forall x (A(x) \rightarrow B(x))</math>&nbsp; ''or'' &nbsp;<math>\neg \exist x (A(x) \land \neg B(x))</math>
|-
|No A is B
|AeB
| <math>\neg \exist x (A(x) \land B(x))</math>&nbsp; ''or'' &nbsp;<math>\forall x (A(x) \rightarrow \neg B(x))</math>
|-
|Some A is B
|AiB
| <math>\exist x (A(x) \land B(x))</math>
|-
|Some A is not B
|AoB
| <math>\exist x (A(x) \land \neg B(x))</math>
|}

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:

{| class="wikitable"
|
! 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 [[logical consequence|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.

{| class="wikitable"
|-
! Figure 1
! Figure 2
! Figure 3
! Figure 4
|-
|B'''a'''rb'''a'''r'''a'''
|C'''e'''s'''a'''r'''e'''
|D'''a'''t'''i'''s'''i'''
|C'''a'''l'''e'''m'''e'''s
|-
|C'''e'''l'''a'''r'''e'''nt
|C'''a'''m'''e'''str'''e'''s
|D'''i'''s'''a'''m'''i'''s
|D'''i'''m'''a'''t'''i'''s
|-
|D'''a'''r'''ii'''
|F'''e'''st'''i'''n'''o'''
|F'''e'''r'''i'''s'''o'''n
|Fr'''e'''s'''i'''s'''o'''n
|-
|F'''e'''r'''io'''
|B'''a'''r'''o'''c'''o'''
|B'''o'''c'''a'''rd'''o'''
| ''C'''a'''l'''e'''m'''o'''s''
|-
| ''B'''a'''rb'''a'''r'''i'''''
| ''C'''e'''s'''a'''r'''o'''''
| ''F'''e'''l'''a'''pt'''o'''n''
| ''F'''e'''s'''a'''p'''o'''''
|-
| ''C'''e'''l'''a'''r'''o'''nt''
| ''C'''a'''m'''e'''str'''o'''s''
| ''D'''a'''r'''a'''pt'''i'''''
| ''B'''a'''m'''a'''l'''i'''p''
|}

The letters A, E, I, and O have been used since the [[Scholasticism|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 (AII-1)|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 [[graph (discrete mathematics)|graphs]] (consisting of vertices and edges) to evaluate syllogisms.<ref>{{Cite web|url=https://www.youtube.com/watch?v=MXRwmOpgqLw| archive-url=https://ghostarchive.org/varchive/youtube/20211211/MXRwmOpgqLw| archive-date=2021-12-11 | url-status=live|title=Syllogisms Made Easy| date=10 December 2019|via=www.youtube.com}}{{cbignore}}</ref>

===Examples===

{{SyllogismImages|Barbara|Greeks|men|mortal}}

====Barbara (AAA-1){{anchor|Modus Barbara}}====
{{SyllogismSentences|All men are mortal. (MaP)|All Greeks are men. (SaM)|All Greeks are mortal. (SaP)}}

{{SyllogismImages|Celarent|snake|reptile|fur}}

====Celarent (EAE-1){{anchor|Modus Celarent}}====
Similar: Cesare (EAE-2)
{{SyllogismSentences|No reptile has fur. (MeP)|All snakes are reptiles. (SaM)|No snake has fur. (SeP)}}

{{clear}}
{{anchor|Modus Camestres}}
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
! bgcolor="#ccccff"| Camestres (AEE-2) 
|-
|
{{SyllogismImages|Camestres|fur|reptile|snake}}
Camestres is essentially like Celarent with S and P exchanged.<br />
Similar: Calemes (AEE-4)
{{SyllogismSentences|All snakes are reptiles. (PaM)|No fur bearing animal is a reptile. (SeM)|No fur bearing animal is a snake. (SeP)}}
|}

{{SyllogismImages|Darii|pet|rabbit|fur}}

====Darii (AII-1){{anchor|Modus Darii}}====
Similar: Datisi (AII-3) 
{{SyllogismSentences|All rabbits have fur. (MaP)|Some pets are rabbits. (SiM)|Some pets have fur. (SiP)}}

{{clear}}
{{anchor|Modus Disamis}}
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
! bgcolor="#ccccff"| Disamis (IAI-3)
|-
|
{{SyllogismImages|Disamis|fur|rabbit|pet}}
Disamis is essentially like Darii with S and P exchanged.<br />
Similar: Dimatis (IAI-4)
{{SyllogismSentences|Some rabbits are pets. (MiP)|All rabbits have fur. (MaP)|Some fur bearing animals are pets. (SiP)}}
|}

{{SyllogismImages|Ferio|reading|homework|fun}}

====Ferio (EIO-1){{anchor|Modus Ferio}}====
Similar: Festino (EIO-2), Ferison (EIO-3), Fresison (EIO-4)
{{SyllogismSentences|No homework is fun. (MeP)|Some reading is homework. (SiM)|Some reading is not fun. (SoP)}}

{{SyllogismImages|Baroco|pet|mammal|cat}}

====Baroco (AOO-2){{anchor|Modus Baroco}}====
{{SyllogismSentences|All cats are mammals. (PaM)|Some pets are not mammals. (SoM)|Some pets are not cats. (SoP)}}

{{SyllogismImages|Bocardo|mammal|cat|pet}}

====Bocardo (OAO-3){{anchor|Modus Bocardo}}====
{{SyllogismSentences|Some cats are not pets. (MoP)|All cats are mammals. (MaS)|Some mammals are not pets. (SoP)}}

{{clear}}
----

{{SyllogismImages|Barbari|Greek|man|mortal}}

====''Barbari (AAI-1)''{{anchor|Modus Barbari}}====
{{SyllogismSentences|All men are mortal. (MaP)|All Greeks are men. (SaM)|Some Greeks are mortal. (SiP)}}

{{clear}}
{{anchor|Modus Bamalip}}
{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
! bgcolor="#ccccff"| ''Bamalip (AAI-4)'' 
|-
|
{{SyllogismImages|Bamalip|mortal|man|Greek}}
''Bamalip'' is exactly like ''Barbari'' with S and P exchanged: 
{{SyllogismSentences|All Greeks are men. (PaM)|All men are mortals. (MaS)|Some mortals are Greek. (SiP)}}
|}

{{SyllogismImages|Celaront|snake|reptile|fur}}

====''Celaront (EAO-1)''{{anchor|Modus Celaront}}====
Similar: ''Cesaro (EAO-2)''
{{SyllogismSentences|No reptiles have fur. (MeP)|All snakes are reptiles. (SaM)|Some snakes have no fur. (SoP)}} <!--- "Some" in the conclusion is correct! --->

{{SyllogismImages|Camestros|human|hooves|horse}}

====''Camestros (AEO-2)''{{anchor|Modus Camestros}}====
Similar: ''Calemos (AEO-4)''
{{SyllogismSentences|All horses have hooves. (PaM)|No humans have hooves. (SeM)|Some humans are not horses. (SoP)}} <!--- "Some" in the conclusion is correct! --->

{{SyllogismImages|Felapton|plant|flower|animal}}

====''Felapton (EAO-3)''{{anchor|Modus Felapton}}====
Similar: ''Fesapo (EAO-4)''
{{SyllogismSentences|No flowers are animals. (MeP)|All flowers are plants. (MaS)|Some plants are not animals. (SoP)}} 

{{SyllogismImages|Darapti|rhomb|square|rectangle}}

====''Darapti (AAI-3)''{{anchor|Modus Darapti}}====
{{SyllogismSentences|All [[Square (geometry)|squares]] are [[rectangle]]s. (MaP)|All squares are [[rhombus]]es. (MaS)|Some rhombuses are rectangles. (SiP)}}

===Table of all syllogisms===
This table shows all 24 valid syllogisms, represented by [[Venn diagram]]s. Columns indicate similarity, and are grouped by combinations of premises. Borders correspond to conclusions. Those with an existential assumption are dashed. 
<!--

If someone knows how to edit it, to be clear, file:Modus_Bamalip.svg should be updated to add after line 3, insert: => ∃x: Px∧Sx PiS thus some P is S [and next line]
=> ∃x: Sx thus some S exists [and then, the same conclusion SiP line as before]

-->
{| class="wikitable mw-collapsible" style="background: #FFF; text-align:center; vertical-align:middle;"
|-
!style="background: #AAA;"|figure
!colspan=2 style="border-left:2px solid #999;"|A ∧ A
!colspan=4 style="border-left:2px solid #999;"|A ∧ E
!colspan=2 style="border-left:2px solid #999;"|A ∧ I
!colspan=2 style="border-left:2px solid #999;"|A ∧ O
!style="border-left:2px solid #999;" |E ∧ I
|-
!style="background: #AAA;"|1	
|style="outline-offset:-5px;outline:2px solid #8F8;border-left:2px solid #999;"|[[File:Modus Barbara.svg|80px|thumb|none|Barbara]]
|style="outline-offset:-5px;outline:2px dashed #88F;"|[[File:Modus Barbari.svg|76px|thumb|none|''Barbari'']]
|style="border-left:2px solid #999;"|
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|style="outline-offset:-5px;outline:2px solid #F88;"|[[File:Modus Celarent.svg|80px|thumb|none|Celarent]]
|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Celaront.svg|76px|thumb|none|''Celaront'']]
|style="outline-offset:-5px;outline:2px solid #88F; border-left:2px solid #999;"| [[File:Modus Darii.svg|80px|thumb|none|Darii]]
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|style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"|[[File:Modus Ferio.svg|80px|thumb|none|Ferio]]
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|style="border-left:2px solid #999;"|
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|style="outline-offset:-5px;outline:2px solid #F88;border-left:2px solid #999;"|[[File:Modus Camestres.svg|80px|thumb|none|Camestres]]
|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Camestros.svg|76px|thumb|none|''Camestros'']]
|style="outline-offset:-5px;outline:2px solid #F88;"|[[File:Modus Cesare.svg|80px|thumb|none|Cesare]]
|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Cesaro.svg|80px|thumb|none|''Cesaro'']]
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|style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"|[[File:Modus Baroco.svg|80px|thumb|none|Baroco]]
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|style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"|[[File:Modus Festino.svg|80px|thumb|none|Festino]]
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|style="outline-offset:-5px;outline:2px dashed #88F;"|[[File:Modus Darapti.svg|66px|thumb|none|''Darapti'']]
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|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Felapton.svg|66px|thumb|none|''Felapton'']]
|style="outline-offset:-5px;outline:2px solid #88F;border-left:2px solid #999;"|[[File:Modus Datisi.svg|80px|thumb|none|Datisi]]
|style="outline-offset:-5px;outline:2px solid #88F;"|[[File:Modus Disamis.svg|80px|thumb|none|Disamis]]
|style="border-left:2px solid #999;"| 
|style="outline-offset:-5px;outline:2px solid #FC4;"|[[File:Modus Bocardo.svg|80px|thumb|none|Bocardo]]
|style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"|[[File:Modus Ferison.svg|80px|thumb|none|Ferison]]
|-
!style="background: #AAA;"|4
|style="border-left:2px solid #999;"| 
|style="outline-offset:-5px;outline:2px dashed #88F;"|[[File:Modus Bamalip.svg|76px|thumb|none|''Bamalip'']]
|style="outline-offset:-5px;outline:2px solid #F88;border-left:2px solid #999;"|[[File:Modus Calemes.svg|80px|thumb|none|Calemes]]
|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Calemos.svg|76px|thumb|none|''Calemos'']]
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|style="outline-offset:-5px;outline:2px dashed #FC4;"|[[File:Modus Fesapo.svg|66px|thumb|none|''Fesapo'']]
|style="border-left:2px solid #999;"| 
|style="outline-offset:-5px;outline:2px solid #88F;"|[[File:Modus Dimatis.svg|80px|thumb|none|Dimatis]]
|style="border-left:2px solid #999;"| 
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|style="outline-offset:-5px;outline:2px solid #FC4;border-left:2px solid #999;"|[[File:Modus Fresison.svg|80px|thumb|none|Fresison]]
|}

==Terms in syllogism==
{{Multiple issues|{{Original research section|date=July 2020}}
{{Unreferenced section|date=July 2020}}|section=yes}}
With Aristotle, we may distinguish [[singular term]]s, such as ''Socrates'', and general terms, such as ''Greeks''. Aristotle further distinguished types (a) and (b):

{{Ordered list|terms that could be the subject of predication; and|terms that could be predicated of others by the use of the copula ("is a").|type=lower-alpha}}

Such a predication is known as a [[Distributive (linguistics)|distributive]], as opposed to non-distributive as in ''Greeks are numerous''. It is clear that Aristotle's syllogism works only for distributive predication, since we cannot reason ''All Greeks are animals, animals are numerous, therefore all Greeks are numerous''. In Aristotle's view singular terms were of type (a), and general terms of type (b). Thus, ''Men'' can be predicated of ''Socrates'' but ''Socrates'' cannot be predicated of anything. Therefore, for a term to be interchangeable—to be either in the subject or predicate position of a proposition in a syllogism—the terms must be general terms, or ''categorical terms'' as they came to be called. Consequently, the propositions of a syllogism should be categorical propositions (both terms general) and syllogisms that employ only categorical terms came to be called ''categorical syllogisms''.

It is clear that nothing would prevent a singular term occurring in a syllogism—so long as it was always in the subject position—however, such a syllogism, even if valid, is not a categorical syllogism. An example is ''Socrates is a man, all men are mortal, therefore Socrates is mortal.'' Intuitively this is as valid as ''All Greeks are men, all men are mortal therefore all Greeks are mortals''. To argue that its validity can be explained by the theory of syllogism would require that we show that ''Socrates is a man'' is the equivalent of a categorical proposition. It can be argued ''Socrates is a man'' is equivalent to ''All that are identical to Socrates are men'', so our non-categorical syllogism can be justified by use of the equivalence above and then citing BARBARA.

==Existential import==
{{Multiple issues|{{Original research section|date=July 2020}}
{{Unreferenced section|date=July 2020}}|section=yes}}
If a statement includes a term such that the statement is false if the term has no instances, then the statement is said to have ''existential import'' with respect to that term. It is ambiguous whether or not a universal statement of the form ''All A is B'' is to be considered as true, false, or even meaningless if there are no As. If it is considered as false in such cases, then the statement ''All A is B'' has existential import with respect to A.

It is claimed Aristotle's logic system does not cover cases where there are no instances. Aristotle's goal was to develop a logic for science. He relegates fictions, such as mermaids and unicorns, to the realms of poetry and literature. In his mind, they exist outside the ambit of science, which is why he leaves no room for such non-existent entities in his logic. This is a thoughtful choice, not an inadvertent omission. Technically, Aristotelian science is a search for definitions, where a definition is "a phrase signifying a thing's essence." Because non-existent entities cannot be anything, they do not, in Aristotle's mind, possess an essence. This is why he leaves no place for fictional entities like goat-stags (or unicorns).<ref>{{Cite web |url=http://www.iep.utm.edu/aris-log/ |title=Groarke, Louis F., "Aristotle: Logic", section 7. (Existential Assumptions), Internet Encyclopedia of Philosophy |access-date=2017-03-07 |archive-url=https://web.archive.org/web/20170204235013/http://www.iep.utm.edu/aris-log/ |archive-date=2017-02-04 |url-status=dead }}</ref>

However, many logic systems developed since ''do'' consider the case where there may be no instances. Medieval logicians were aware of the problem of existential import and maintained that negative propositions do not carry existential import, and that positive propositions with subjects that do not [[supposition theory|supposit]] are false.

The following problems arise:

{{Ordered list
| In natural language and normal use, which statements of the forms, All A is B, No A is B, Some A is B, and Some A is not B, have existential import and with respect to which terms?
|In the four forms of categorical statements used in syllogism, which statements of the form AaB, AeB, AiB and AoB have existential import and with respect to which terms?
|What existential imports must the forms AaB, AeB, AiB and AoB have for the square of opposition to be valid?
|What existential imports must the forms AaB, AeB, AiB and AoB have to preserve the validity of the traditionally valid forms of syllogisms?
|Are the existential imports required to satisfy (d) above such that the normal uses in natural languages of the forms All A is B, No A is B, Some A is B and Some A is not B are intuitively and fairly reflected by the categorical statements of forms AaB, AeB, AiB and AoB?
| type = lower-alpha
}}

For example, if it is accepted that AiB is false if there are no As and AaB entails AiB, then AiB has existential import with respect to A, and so does AaB. Further, if it is accepted that AiB entails BiA, then AiB and AaB have existential import with respect to B as well. Similarly, if AoB is false if there are no As, and AeB entails AoB, and AeB entails BeA (which in turn entails BoA) then both AeB and AoB have existential import with respect to both A and B. It follows immediately that all universal categorical statements have existential import with respect to both terms. If AaB and AeB is a fair representation of the use of statements in normal natural language of All A is B and No A is B respectively, then the following example consequences arise:

:"All flying horses are mythical" is false if there are no flying horses.
:If "No men are fire-eating rabbits" is true, then "There are fire-eating rabbits" is true; and so on.

If it is ruled that no universal statement has existential import then the square of opposition fails in several respects (e.g. AaB does not entail AiB) and a number of syllogisms are no longer valid (e.g. BaC, AaB->AiC).

These problems and paradoxes arise in both natural language statements and statements in syllogism form because of ambiguity, in particular ambiguity with respect to All. If "Fred claims all his books were Pulitzer Prize winners", is Fred claiming that he wrote any books? If not, then is what he claims true? Suppose Jane says none of her friends are poor; is that true if she has no friends?

The first-order predicate calculus avoids such ambiguity by using formulae that carry no existential import with respect to universal statements. Existential claims must be explicitly stated. Thus, natural language statements—of the forms ''All A is B, No A is B'', ''Some A is B'', and ''Some A is not B''—can be represented in first order predicate calculus in which any existential import with respect to terms A and/or B is either explicit or not made at all. Consequently, the four forms ''AaB, AeB, AiB'', and ''AoB'' can be represented in first order predicate in every combination of existential import—so it can establish which construal, if any, preserves the square of opposition and the validity of the traditionally valid syllogism. Strawson claims such a construal is possible, but the results are such that, in his view, the answer to question (e) above is ''no''.

==Syllogistic fallacies==
People often make mistakes when reasoning syllogistically.<ref>See, e.g., Evans, J. St. B. T (1989). ''Bias in human reasoning''. London: LEA.</ref>

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.<ref>Khemlani, S., and P. N. Johnson-Laird. 2012. "Theories of the syllogism: A meta-analysis." ''[[Psychological Bulletin]]'' 138:427–57.</ref><ref>Chater, N., and M. Oaksford. 1999. "The Probability Heuristics Model of Syllogistic Reasoning." ''[[Cognitive Psychology (journal)|Cognitive Psychology]]'' 38:191–258.</ref> 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.<ref>{{Cite book|last=Lundberg|first=Christian|title=The Essential Guide to Rhetoric|publisher=Bedford/St. Martin's|year=2018|pages=39}}</ref>

Determining the validity of a syllogism involves determining the [[distribution of terms|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:

*[[Fallacy of the undistributed middle|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 major|Illicit treatment of the major term]]: The conclusion implicates all members of the major term (P&nbsp;– 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 minor|Illicit treatment of the minor term]]: Same as above, but for the minor term (S&nbsp;– meaning the proposition is universal) and minor premise (where S is either a particular subject or an affirmative predicate).
*[[Fallacy of exclusive premises|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.

== Other types ==

*[[Disjunctive syllogism]]
*[[Hypothetical syllogism]]
*[[Legal syllogism]]
*[[Polysyllogism]]
*[[Prosleptic syllogism]]
*[[Quasi-syllogism]]
*[[Statistical syllogism]]

==See also==
{{Portal|Philosophy}}
*[[Syllogistic fallacy]]
* [[Argumentation theory#Psychological aspects|Argumentation theory]]
* [[Buddhist logic]]
* [[Enthymeme]]
* [[Formal fallacy]]
* [[Logical fallacy]]
* [[The False Subtlety of the Four Syllogistic Figures]]
* [[Tautology (logic)]]
* [[Venn diagram]]

==References==
{{Reflist}}

===Sources===
*[[Aristotle]], [c. 350 BCE] 1989. ''[[Prior Analytics]]'', translated by R. Smith. Hackett. {{ISBN|0-87220-064-7}}
*[[Simon Blackburn|Blackburn, Simon]]. [1994] 1996. "Syllogism." In [[The Oxford Dictionary of Philosophy|''The'' ''Oxford Dictionary of Philosophy'']]. Oxford University Press. {{ISBN|0-19-283134-8}}.
* Broadie, Alexander. 1993. ''Introduction to Medieval Logic''. Oxford University Press. {{ISBN|0-19-824026-0}}.
*[[Irving Copi|Copi, Irving]]. 1969. ''Introduction to Logic'' (3rd ed.). Macmillan Company.
*[[John Corcoran (logician)|Corcoran, John]]. 1972. "Completeness of an ancient logic." ''[[Journal of Symbolic Logic]]'' 37:696–702.
* — 1994. "The founding of logic: Modern interpretations of Aristotle's logic." ''[[Ancient Philosophy (journal)|Ancient Philosophy]]'' 14:9–24.
* Corcoran, John, and Hassan Masoud. 2015. "Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions." ''History and Philosophy of Logic'' 36(1):39–61.
* Englebretsen, George. 1987. ''The New Syllogistic''. Bern: [[Peter Lang (publisher)|Peter Lang]].
*[[Charles Leonard Hamblin|Hamblin, Charles Leonard]]. 1970. ''Fallacies''. London: [[Methuen Publishing|Methuen]]. {{ISBN|0-416-70070-5}}. 
**[[Cf.]] on validity of syllogisms: "A simple set of rules of validity was finally produced in the later Middle Ages, based on the concept of Distribution."
*[[Jan Łukasiewicz|Łukasiewicz, Jan]]. [1957] 1987. ''Aristotle's Syllogistic from the Standpoint of Modern Formal Logic''. New York: Garland Publishers. {{ISBN|0-8240-6924-2}}. {{OCLC|15015545}}.
* Malink, Marko. 2013. ''Aristotle's Modal Syllogistic''. Cambridge, MA: [[Harvard University Press]].
* Patzig, Günter. 1968. ''Aristotle's theory of the syllogism: a logico-philological study of Book A of the Prior Analytics''. Dordrecht: Reidel.
* Rescher, Nicholas. 1966. ''Galen and the Syllogism''. University of Pittsburgh Press. {{ISBN|978-0822983958}}.
* [[Timothy Smiley|Smiley, Timothy]]. 1973. "What is a syllogism?" ''[[Journal of Philosophical Logic]]'' 2:136–54.
* Smith, Robin. 1986. "Immediate propositions and Aristotle's proof theory." ''Ancient Philosophy'' 6:47–68.
* Thom, Paul. 1981. "The Syllogism." ''[[Philosophia (journal)|Philosophia]]''. München. {{ISBN|3-88405-002-8}}.

==External links==
* {{cite SEP |url-id=aristotle-logic |title=Aristotle's Logic |last=Smith |first=Robin}}
* Koutsoukou-Argyraki, Angeliki. [https://www.isa-afp.org/entries/Aristotles_Assertoric_Syllogistic.html Aristotle's Assertoric Syllogistic (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)]
* {{cite SEP |url-id=medieval-syllogism |title=Medieval Theories of the Syllogism |last=Lagerlund |first=Henrik}}
* [https://www.historyoflogic.com/aristotle-syllogism-categorical.htm Aristotle's Prior Analytics: the Theory of Categorical Syllogism] an annotated bibliography on Aristotle's syllogistic
* [https://books.google.com/books?id=yfUysMS31PIC&dq=syllogisms+huseyin+cakir&pg=PA418 Fuzzy Syllogistic System]
* [https://gcris.iyte.edu.tr/handle/11147/6777 Development of Fuzzy Syllogistic Algorithms and Applications Distributed Reasoning Approaches]
* [https://www.rigpawiki.org/index.php?title=Logic Comparison between the Aristotelian Syllogism and the Indian/Tibetan Syllogism]
* [https://www.wisdomlib.org/buddhism/book/the-buddhist-philosophy-of-universal-flux/d/doc79648.html The Buddhist Philosophy of Universal Flux (Chapter XXIII – Members of a Syllogism (avayava))]
* [https://www.thefirstscience.org/syllogistic/ Online Syllogistic Machine] An interactive syllogistic machine for exploring all the fallacies, figures, terms, and modes of syllogisms.

{{Aristotelian logic}}
{{Mathematical logic}}
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[[Category:Syllogism| ]]
[[Category:Term logic]]
[[Category:Ancient Greek logic]]
[[Category:Arguments]]