
A syllogism (Greek: συλλογισμός – "conclusion," "inference") or logical appeal is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form.
In Aristotle's Prior Analytics, he defines syllogism as "a discourse in which, certain things having been supposed, something different from the things' supposed results of necessity because these things are so." (24b18–20)
Despite this very general definition, he limits himself first to categorical syllogisms^{[1]} (and later to modal syllogisms). The syllogism was at the core of traditional deductive reasoning, where facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations. Syllogism was superseded by firstorder predicate logic following the work of Frege, in particular 1879 Begriffsschrift (Concept Script) 1879.
A categorical syllogism consists of three parts: the major premise, the minor premise and the conclusion.
Each part thereof is a categorical proposition, and each categorical position containing two categorical terms.^{[2]} In Aristotle, each of the premises is in the form "Some/all A belong to B," or "Some/all A is/are [not]B," where "A" is one term and "B" is another, but 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, it is the minor term (the subject) of the conclusion. For example:
Each of the three distinct terms represents a category, in this example, "men," "mortal," and "Socrates." "Mortal" is the major term; "Socrates", the minor term. The premises also have one term in common with each other, which is known as the middle term in this example, "man." Here the major premise is universal and the minor particular, but this need not be so. For example:
Here, the major term is "die", the minor term is "men," and the middle term is "mortals". Both of the premises are universal.
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, if one argues that a given number of grains of sand does not make a heap and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a sorites argument.
Although there are infinitely many possible syllogisms, there are only a finite number of logically distinct types. We shall classify and enumerate them below. Note that the syllogism above has the abstract form:
The premises and conclusion of a syllogism can be any of four types, which are labelled by letters^{[3]} as follows. The meaning of the letters is given by the table:
code  quantifier  subject  copula  predicate  type  example  
a  All  S  are  P  universal affirmatives  All humans are mortal.  
e  No  S  are  P  universal negatives  No humans are perfect.  
i  Some  S  are  P  particular affirmatives  Some humans are healthy.  
o  Some  S  are not  P  particular negatives  Some humans are not clever. 
(See Square of opposition for a discussion of the logical relationships between these types of propositions.)
In Analytics, Aristotle mostly uses the letters A, B and C as term place holders, rather than giving concrete examples, an innovation at the time. 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 to enable the categorical statements to be written succinctly thus:
Form  Shorthand 

All A is B  AaB 
No A is B  AeB 
Some A is B  AiB 
Some A is not B  AoB 
Hence the form BARBARA can be written neatly as BaC,AaB > AaC
By definition, S is the subject of the conclusion, P is the predicate of the conclusion, 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 that it appears in. This gives rise to another classification of syllogisms known as the figure. Given that in each case the conclusion is SP, the four figures are:
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 
Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, although 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 syllogisms above are AAA1.
Of course, 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 of syllogism. 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.
Figure 1  Figure 2  Figure 3  Figure 4  
Barbara  Cesare  Darapti  Bramantip  
Celarent  Camestres  Disamis  Camenes  
Darii  Festino  Datisi  Dimaris  
Ferio  Baroco  Felapton  Fesapo  
Bocardo  Fresison  
Ferison 
The letters A, E, I, 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.
A sample syllogism of each type follows.
Barbara
Celarent
Darii
Ferio
Cesare
Camestres
Festino
Baroco
Darapti
Disamis
Datisi
Felapton
Bocardo
Ferison
Bramantip
Camenes
Dimaris
Fesapo
Fresison
Forms can be converted to other forms, following certain rules, and all forms can be converted into one of the firstfigure forms.
We may, with Aristotle, distinguish singular terms such as Socrates and general terms such as Greeks. Aristotle further distinguished (a) terms which could be the subject of predication, and (b) terms which could be predicated of others by the use of the copula (is are). (Such a predication is known as a distributive as opposed to nondistributive as in Greeks are numerous.) It is clear that Aristotle’s syllogism works only for distributive predication for 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 to enable a term to be interchangeable — that is 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 syllogism employing just 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, would not be a categorical syllogism. An example of such would be 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 it would be necessary to 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 noncategorical syllogism can be justified by use of the equivalence above and then citing BARBARA.
If a statement includes a term so that the statement is false if the term has no instances (is not instantiated) then the statement is said to entail existential import with respect to that term. In particular, a universal statement of the form All A is B has existential import with respect to A if All A is B is false if there are no As.
The following problems arise:
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 will entail 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:
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 Nobel 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 firstorder predicate calculus avoids the problems of such ambiguity by using formulae which carry no existential import with respect to universal statements; existential claims have to 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 exactly represented in first order predicate calculus in which any existential import with respect to terms A and/or B is made explicitly 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 that it can be established which construal if any would preserve the square of opposition and the validly of the traditionally valid syllogism. Strawson claims that such a construal is possible, but the results are such that, in his view, the answer to question (a) above is no.
Syllogism dominated Western philosophical thought until The Age of Enlightenment in the 17th Century. At that time, Sir Francis Bacon rejected the idea of syllogism and deductive reasoning by asserting that it was fallible and illogical^{[4]}. Bacon offered a more inductive approach to logic in which experiments were conducted and axioms were drawn from the observations discovered in them.
In the 19th Century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Kant famously claimed that logic was the one completed science, and that Aristotelian logic more or less included everything about logic there was to know. Though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kant's opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift (Concept Script). This introduced a calculus, a method of representing categorical statements — and statements which are not provided for in syllogism as well — by the use of quantifiers and variables. This led to the rapid development of sentential logic and firstorder 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, however, is the continued application of the intricate rules of Aristotelian logic, as taught by St. Thomas Aquinas, in the Vatican's Congregation for the Doctrine of the Faith, and the Apostolic Tribunal of the Roman Rota.
People often make mistakes when reasoning syllogistically.^{[5]}
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.^{[6]} However, this does not follow according to the rules of classical logic. For instance, while some cats (A) are black (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 first, the mood of the syllogism invoked is illicit (III), and second, the supposition of the middle term is variable between that of the middle term in the major premise, and that of the middle term in the minor premise (not all "some" cats are by necessity of logic the same "some black things").
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:
