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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." Aristotle, "Prior Analytics", 24b18–20 Despite this very general definition, in Aristotle's work Prior Analytics, he limits himself to categorical syllogisms that consist of three .[1] Stanford Encyclopedia of Philosophy: Ancient Logic Aristotle Non-Modal Syllogistic These include categorical modal syllogisms.[2] Stanford Encyclopedia of Philosophy: Ancient Logic Aristotle Modal Logic
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, where facts are determined by combining existing statements, in contrast to inductive reasoning where facts are determined by repeated observations.
Within academic contexts, the syllogism was superseded by first-order predicate logic following the work of Gottlob Frege, in particular his Begriffsschrift ( Concept Script) (1879), but syllogisms remain useful in some circumstances, and for general-audience introductions to logic.Hurley, Patrick J (2011). A Concise Introduction to Logic, Cengage Learning, ISBN 9780840034175Zegarelli, Mark (2010). Logic for Dummies, John Wiley & Sons, ISBN 9781118053072
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 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. "All A are B," and "No A are B" are termed universal propositions; "Some A are B" and "Some A are not B" are termed 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 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 the above example, humans, mortal, and Greeks. Mortal is the major term, 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.
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.
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 would make a heap is to construct a sorites argument.
(Note: M – Middle, S – subject, P – predicate. See below for more detailed explanation.)
The premises and conclusion of a syllogism can be any of four types, which are labeled by lettersAccording to Copi, p. 127: 'The letter names are presumed to come from the Latin words " Aff Irmo" and "n Eg O," which mean "I affirm" and "I deny," respectively; the first capitalized letter of each word is for universal, the second for particular' 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. |
In Analytics, Aristotle mostly uses the letters A, B and C (actually, the Greek letters alpha, beta and gamma) 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 so the categorical statements can be written succinctly:
| All A is B | AaB |
| No A is B | AeB |
| Some A is B | AiB |
| Some A is not B | AoB |
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:
| 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 John Buridan—reject the fourth figure as a figure distinct from the first. See entry on the Prior Analytics.)
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 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.
| Figure 1 | Figure 2 | Figure 3 | Figure 4 |
| B arb ar a | C es ar e | D at is i | C al em es |
| C el ar ent | C am estr es | D is am is | D im at is |
| D ar ii | F est in o | F er is on | Fr es is on |
| F er io | B ar oc o | B oc ard o | C al em os |
| B arb ar i | C es ar o | F el apt on | F es ap o |
| C el ar ont | C am estr os | D ar apt i | B am al ip |
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.
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 actually the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii) could also be written as "Some kittens are pets" (MiS is Datisi).
In the Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element.
| M:man S:greek P:mortal |
[[File:Modus Celarent.svg|thumb|right|150px|
| M:reptile S:snake P:fur |
| [[File:Modus Calemes.svg|thumb|right|150px | { style="text-align: center" |
| M:reptile S:fur P:snake |
[[File:Modus Darii.svg|thumb|right|150px|
| M:rabbit S:pet P:fur |
| [[File:Modus Dimatis.svg|thumb|right|150px | { style="text-align: center" |
| M:rabbit S:fur P:pet |
[[File:Modus Ferio.svg|thumb|right|150px|
| M:homework S:reading P:fun |
[[File:Modus Baroco.svg|thumb|right|150px|
| M:useful S:website P:informative |
[[File:Modus Bocardo.svg|thumb|right|150px|
| M:cat S:mammal P:tail |
[[File:Modus Barbari.svg|thumb|right|150px|
| M:man S:greek P:mortal |
| [[File:Modus Bamalip.svg|thumb|right|150px | { style="text-align: center" |
| M:man S:mortal P:greek |
[[File:Modus Celaront.svg|thumb|right|150px|
| M:reptile S:snake P:fur |
[[File:Modus Camestros.svg|thumb|right|150px|
| M:hooves S:human P:horse |
[[File:Modus Felapton.svg|thumb|right|150px|
| M:flower S:plant P:animal |
[[File:Modus Darapti.svg|thumb|right|150px|
| M:square S:rhomb P:rectangle |
| 1 | Barbara | Barbari | Darii | Ferio | Celaront | Celarent | ||||||||
| 2 | Festino | Cesaro | Cesare | Camestres | Camestros | Baroco | ||||||||
| 3 | Darapti | Datisi | Disamis | Felapton | Ferison | Bocardo | ||||||||
| 4 | Bamalip | Dimatis | Fesapo | Fresison | Calemes | Calemos |
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.
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 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:
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 validly 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.
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, in Logic (1800), that logic was the one completed science, and that Aristotelian logic more or less included everything about logic 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.) 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 that 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 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 arguments crafted by Advocates be presented in syllogistic format.
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—-by itself 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”.
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.See the meta-analysis by Khemlani, S. & Johnson-Laird, P.N. (2012). Theories of the syllogism: A meta-analysis. Psychological Bulletin, 138, 427-457.See the meta-analysis by Chater, N. & Oaksford, M. (1999). The Probability Heuristics Model of Syllogistic Reasoning. Cognitive Psychology, 38, 191–258. 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 first, the mood of the syllogism invoked (i.e. III-3) is illicit, 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:
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